Dependencies

Let \(A\) and \(\xi \) be as in Theorem 35.8.2.24, and fix \(0{\lt}\varepsilon \le 1/2\). There are \(a{\gt}0\), \(c\), \(C\) and \(N_0\) such that for every \(N\ge N_0\):

1. some state \(|\psi \rangle \) with \(\varepsilon (\psi ,\phi _N)\le \varepsilon \) is prepared in depth at most \(c\ln N\);

2. every state \(|\psi \rangle \) prepared in depth \(T\) with \(\varepsilon (\phi _N,\psi )\le \varepsilon \) has \(T\ge a\ln N-C\).

Any \(a{\lt}\xi /4\) is admissible. Thus the optimal depth is \(\Theta (\log N)\), as stated in the introduction of  [ 219 ] .

There is \(C\), depending only on \(d\) and \(D\), such that in the setting of Theorem 35.10.16, for any \(d\), if moreover every blocked tensor is injective in the sense of Definition 35.10.5, the vector \((\bigotimes _kV_k)|\Omega \rangle \) is a unit vector prepared in depth at most \(CL\) that satisfies (66) in place of \(|\psi \rangle \).

Let \(A\) and \(B\) be normal tensors with the same physical dimension \(d\). There is a constant \(c\) depending only on \(A\) and \(B\) such that, for every \(N\ge 2\) and \(0{\lt}\varepsilon \le 1\) for which \(|\phi _N(A)\rangle \) and \(|\phi _N(B)\rangle \) are nonzero, some local circuit \(U\) of depth at most \(c\log (N/\varepsilon )\) satisfies \(\varepsilon (U\hat\phi _N(A),\hat\phi _N(B))\le \varepsilon \), where \(\hat\phi _N(A)=|\phi _N(A)\rangle /\lVert \phi _N(A)\rVert \) and \(\hat\phi _N(B)=|\phi _N(B)\rangle /\lVert \phi _N(B)\rVert \) are the normalized states. If both bond dimensions are at least \(1\), there are \(c\) and \(N_0\) such that the same holds for every \(N\ge N_0\) and \(0{\lt}\varepsilon \le 1\) without the nonvanishing condition.

The paragraph “Discussion and outlook” of [ 219 ] states that its results imply that states in the same phase are transformed into each other by a circuit of logarithmic depth. Normal states lie in the trivial phase of the classification without symmetry; the corollary covers only this case. Non-normal states and phases defined with a symmetry are not treated [ 39 ] .

Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

A basis of normal tensors (BNT) for a total tensor \(A_{\mathrm{tot}}\) consists of \(g\) block tensors \((A_1, \ldots , A_g)\) with bond dimensions \((D_1, \ldots , D_g)\) such that each \(A_j\) is normal (Definition 2.5.5); at each system size \(N\geq 1\) the MPV of \(A_{\mathrm{tot}}\) lies in the span of the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\); and for sufficiently large \(N\) those block MPVs are linearly independent [ 17 , Definition IV.2 ] . In the canonical-form characterization of [ 13 , Section 2.3 ] , a BNT is a minimal family of representatives for \(\widetilde A_k^i=e^{i\phi _k}X_kA_j^iX_k^{-1}\) among the normal blocks.

For every \(\lambda _2\in \mathbb {C}\), define the associated correlation-length quantity by

For a nonzero subleading eigenvalue in the physical range \(0{\lt}|\lambda _2|{\lt}1\), this quantity is positive. If all subleading spectral values vanish and correlations therefore vanish after finitely many transfer steps, the total definition gives the limiting value \(\xi =0\).

In canonical form II of [ 13 , Appendix A ] , there are positive retained dimensions \(D_k\), normal tensors \(A_k\), weights \(\mu _k\), and, for \(S=\sum _kD_k\le D\), a matrix \(U\in \mathbb {C}^{S\times D}\) such that, for every physical index \(i\),

Thus omitted bond-space coordinates are literally zero. Every retained block additionally satisfies

where \(\Lambda _k\) is diagonal and positive definite. The two displayed equations are the only additional blockwise conditions; no ordering, injectivity, gauges, or basis-of-normal-tensors assumptions are included. In particular, a normal left-canonical tensor of positive bond dimension with a positive definite diagonal fixed matrix has a one-block presentation with weight one and no omitted bond coordinates. The line-246 convention (29) remains a separate condition on the underlying canonical-form data.

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

An MPS tensor \(A\) is irreducible if it has no nontrivial invariant projection.

For \(a+n\le N\), the interval tensor family is \(A[a,a+n)_j=A_{a+j}\), for \(0\le j{\lt}n\).

Let \(A\) be a tensor with a positive definite transfer fixed matrix \(\rho \) of trace one, and let \(\xi ,\eta {\gt}0\). For \(N\geq 2\), put

If \(q\leq N\), put \(M=\lfloor N/q\rfloor \) and take \(M-1\) blocks of length \(q\) and one block of length \(q'=q+(N\bmod q)\). Let \(V_k\) be the partial isometry in the polar decomposition of the tensor on the \(k\)th block, and let \(\omega _\rho \) be the fixed-point pair. Define

If \(q{\gt}N\), let \(\psi _{N,\eta }\) be the normalized periodic state itself. The block prescription is the one used in [ 219 , Supplemental Material, proof of Theorem 1 ] . The exact-state branch extends that prescription using the one-block preparation theorem.

With \(V\) from Definition 35.1.3.1 and \(N=qM\), the approximating state is \(|\widetilde\phi _N\rangle =|\phi _M(VP_\infty )\rangle /\lVert \phi _M(VP_\infty )\rVert \). By Lemma G.1.1.2 and Theorem 35.2.5, for \(M\ge 1\), \(|\phi _M(VP_\infty )\rangle =V^{\otimes M}\bigotimes _k|\omega \rangle _{R_kL_{k+1}}\). When \(\mathcal B_q\) is injective, \(V\) is an isometry and \(\lVert \phi _M(VP_\infty )\rVert =1\), so no normalization is needed. This is [ 219 , eqs. (9) and (10) ] .

Let \(n,k\ge 0\). A binary finite-range network with \(k+1\) isometry layers consists of a unitary \(u\) on \(\mathbb {C}^D\otimes \mathbb {C}^D\), isometries \(W_0,\dots ,W_{k-1}\colon \mathbb {C}^{D^2}\to \mathbb {C}^{D^2}\otimes \mathbb {C}^{D^2}\) and an isometry \(W\colon \mathbb {C}^{D^2}\to \mathbb {C}^n\otimes \mathbb {C}^n\). Its tree map is

where the Kronecker powers act on the regrouping of the \(2^{k+1}\) sites into neighbouring pairs, then pairs of pairs, and so on, as in (12). On \(M\) top sites \(\mathbb {C}^{D^2}=L_i\otimes R_i\), and for \(D\ge 1\), the state of the network on \(N=M2^{k+1}\) sites is

with \(L_{M+1}=L_1\). Every isometry maps one site to two neighbouring sites and every unitary acts on two neighbouring legs, so the network has range two and bond dimension \(D^2\) in every layer; the disentanglers below the first layer are the identity.

Cut the ring of \(N\) sites into \(M\ge 1\) consecutive blocks of lengths \(\ell _1,\dots ,\ell _M\) with \(\sum _k\ell _k=N\), block \(k\) consisting of the sites \(o_k+1,\dots ,o_k+\ell _k\) with \(o_k=\sum _{i{\lt}k}\ell _i\). Let \(\mathcal B_\ell =V_\ell P_\ell \) be the polar decomposition of the \(\ell \)-site blocked tensor, and let \(\omega \) be a vector on \(\mathbb {C}^D\otimes \mathbb {C}^D\). The state

has the coefficient \(\sum _c\prod _k\langle s_k|V_{\ell _k}|c_k\rangle \, \Omega (c)\) at the configuration \(s=(s_1,\dots ,s_M)\) read block by block, where \(c_k=(l_k,r_k)\) and \(\Omega \) is the pair product state (2) of \(\omega \). If the \(q\)-site blocked tensor is injective, \(\ell _k=q\) for every \(k\), and \(\omega \) is the pair of a fixed point \(\sigma \ge 0\) with \(\operatorname{Tr}\sigma =1\), it is the approximating state of Definition 35.3.8.

Fix a physical dimension \(d\), a virtual dimension \(\chi \) and a leaf bound \(R\). A depth-zero tree on \(n\) sites, with \(1\le n\le R\), is an isometry \(V\colon \mathbb {C}^\chi \to (\mathbb {C}^d)^{\otimes n}\). A tree of coarse depth \(h+1\) consists of two trees \(T_1,T_2\) of coarse depth \(h\), on \(n_1,n_2\) sites, and an isometry \(W\colon \mathbb {C}^\chi \to \mathbb {C}^\chi \otimes \mathbb {C}^\chi \). Its contraction is

The physical sites keep their order under concatenation. A tree of coarse depth \(h\) has \(2^h\) leaves and \(h+1\) isometry layers, counting the finest layer.

Let \(A^i=\bigoplus _j\operatorname {diag}(\mu _{j,1},\ldots ,\mu _{j,m_j}) \otimes A_j^i\) be in basis-of-normal-tensors canonical form, with each \(A_j\) left-canonical. Let \(\sigma _j\) be the unique density matrix satisfying \(\mathcal{E}_{A_j}(\sigma _j)=\sigma _j\). Irreducibility implies \(\sigma _j{\gt}0\). Choose one retained copy \(k_j\) of each block and place \(|\omega _j\rangle =\operatorname {vec}(\sqrt{\sigma _j})\) on both virtual coordinates of that copy. These are the block pairs used in [ 219 , eqs. (8) and (S7) ] .

Write \(E_{j,k_j}:\mathbb {C}^{D_j}\to \mathbb {C}^D\) for the corresponding coordinate inclusion.

\(=\)

Choose a central site of a chain with \(q=\ell +1+r\) sites. Write \(A\) for its tensor and list the left and right subchains from their respective outer boundaries inward. Thus the right product \(R(\sigma _R)\) is the transpose of the product in the original physical ordering. For a matrix \(G\colon \mathbb {C}^s\to \mathbb {C}^{D^2}\) from an arbitrary finite input to the virtual pairs, define

Let \(A_1,\dots ,A_N\) be tensors of physical dimension \(d\) and bond dimension \(D\), one on each site of a ring of \(N\) sites, and let \(|\phi _N\rangle =\sum _s\operatorname{Tr}(A_1^{s_1}\cdots A_N^{s_N})|s\rangle \) be their state. Cut the ring into \(M\ge 1\) consecutive blocks of lengths \(\ell _1,\dots ,\ell _M\) as in Definition 35.5.10. The blocked tensor \(B_k\) of block \(k\) has physical dimension \(d^{\ell _k}\) and the matrices \(B_k^{s_{o_k+1}\cdots s_{o_k+\ell _k}}=A_{o_k+1}^{s_{o_k+1}}\cdots A_{o_k+\ell _k}^{s_{o_k+\ell _k}}\). When all tensors are equal to \(A\), it is the \(\ell _k\)-site blocked tensor \(\mathcal B_{\ell _k}\) of \(A\).

Let \(B_k=V_kP_k\) be the polar decompositions of the blocked tensors of Definition 35.10.1, with \(P_k\) read as a tensor of physical dimension \(D^2\) and bond dimension \(D\) as in Theorem 35.1.3.3. The state of the positive parts on the \(M\) blocks is

the state of [ 219 , eq. (18) ] . For vectors \(\omega ^1,\dots ,\omega ^M\) on \(\mathbb {C}^D\otimes \mathbb {C}^D\) let \(|\Omega \rangle =\bigotimes _{k=1}^M|\omega ^k\rangle _{R_kL_{k+1}}\), the pair \(\omega ^k\) joining the right leg of block \(k\) to the left leg of block \(k+1\), cyclically; when all \(\omega ^k\) are equal to \(\omega \) it is the pair product state of Definition 35.2.3. The state \((\bigotimes _kV_k)|\Omega \rangle \) has the coefficient \(\sum _c\prod _k\langle s_k|V_k|c_k\rangle \, \Omega (c)\) at the configuration \(s\) read block by block; for equal tensors and equal pairs it is the state \(|\psi _{\ell ,\omega }\rangle \) of Definition 35.5.10.

Set \(\widehat\sigma _i=J_i\sigma _iJ_i^\dagger \). The padded reference for an interval \([a,b)\) is \(Q_{a:b}(X)=\operatorname {Tr}(P_bX)\widehat\sigma _a\). It vanishes on inputs outside the outgoing bond corner.

At \(N=qM\), put \(\beta '_j=\beta _j(c_j/\mu _{j,k_j}^q)^M\) and \(\alpha '_j=\beta '_j/(\sum _l|\beta '_l|^2)^{1/2}\), with \(c_j=(\sum _k|\mu _{j,k}|^{2q})^{1/2}\). If every \(\beta '_j\) vanishes, set \(\alpha '=0\). The corrected state is

The coefficient correction is necessary also for multiplicity one: then \(\beta '_j=|\mu _j|^{qM}\) [ 26 , 142 ] .

For maps \(\iota _j:\{ 1,\ldots ,D_j\} \to \{ 1,\ldots ,D\} \), let \(E_j|a\rangle =|\iota _j(a)\rangle \) and \(K_j=E_j\otimes E_j\). The weighted direct sum has letters \(B^i=\sum _j\mu _jE_jA_j^iE_j^\dagger \). Its sector-pair set is \(S=\{ (\iota _j(a),\iota _j(c)):j,a,c\} \).

For any tensor \(B\), read its blocked physical matrix on configurations: \(C_N(B)_{s,(a,c)}=(B^{s_1}\cdots B^{s_N})_{a,c}\). For the unweighted sum \(B=\bigoplus _jA_j\), write \(C_N=C_N(B)\), \(P_N=(C_N^\dagger C_N)^{1/2}\) and let \(S_N\) be the polar partial isometry of \(C_N\). Define

Both \(T\) and \(R_N\) map \(\mathbb {C}^b\) to the same virtual space \(\mathbb {C}^{D^2}\), whose dimension is independent of \(N\). Let \(Q_N\) be the polar partial isometry of \(R_N\).

With physical dimension four and bond dimension two, set \(B=(\mathbb {1},E_{01},E_{10},0)/\sqrt2\), where \(E_{ij}\) are matrix units. A single physical site already satisfies the capacity condition \(d^q=D^2=4\).

Fix a central site of a chain of \(q=\ell +1+r\) sites and a positive bond dimension \(D\). A mixed sequential factorization of \(V\colon \mathbb {C}^s\to (\mathbb {C}^d)^{\otimes q}\) consists of two families of local isometries \(Q^L_p\) and \(Q^R_p\), with outer bond dimensions \(b^L_0=b^R_0=1\) and every bond dimension at most \(D^2\), and an isometry \(C\colon \mathbb {C}^s\to \mathbb {C}^{b^L_\ell }\otimes \mathbb {C}^d\otimes \mathbb {C}^{b^R_r}\). Define \((E_L)_{\sigma _L,a}=(Q^L_1(\sigma _{L,1})\cdots Q^L_\ell (\sigma _{L,\ell }))_{1a}\) and define \(E_R\) in the same manner from the right boundary. The required contraction is

The input of dimension \(s\) is attached to the centre, rather than to either exterior boundary.

Let \(A^i=\bigoplus _{j=1}^b\operatorname {diag}(\mu _{j,1},\dots ,\mu _{j,m_j}) \otimes A_j^i\) with normal \(A_j\), each in the gauge (3), forming a basis of normal tensors, and put \(\beta _j=\sum _{k=1}^{m_j}\mu _{j,k}^N\). Consider lengths \(N=qM\) with \((\beta _1,\dots ,\beta _b)\neq 0\). Write \(\xi _j\) for the correlation length of the normal tensor \(A_j\), formed from the subleading eigenvalue of its own transfer map \(\mathcal{E}_{A_j}\). With \(|\omega _j\rangle \) the pair state of \(A_j\) and \(\alpha ^{(N)}_j=\beta _j/(\sum _l\lvert \beta _l\rvert ^2)^{1/2}\), the fixed-point state is

With \(V\) from Theorem 35.1.3.3 for the blocked map of \(A\), the approximating state is \(|\widetilde\phi _N\rangle =V^{\otimes M}|\Omega '\rangle / \lVert V^{\otimes M}|\Omega '\rangle \rVert \), whenever the denominator is nonzero. This is [ 219 , eqs. (19) and (S7) ] , with the canonical form of [ 14 ] .

When some block has multiplicity \(m_j\geq 2\), the source’s construction fails: no product of pairs carrying the weights \(\beta _j\) gives, after \(V^{\otimes M}\), an approximation of \(|\phi _N\rangle \). For \(A^0=\operatorname {diag}(1,1,0)\) and \(A^1=\operatorname {diag}(0,0,1)\) the overlap with \(|\phi _N\rangle \propto 2|0\cdots 0\rangle +|1\cdots 1\rangle \) tends to \(1/\sqrt5\) as \(M\to \infty \), for every \(q\) [ 128 ] . The failure lies in the coefficients \(\beta _j\). With the pairs of each block on one of its copies \(k_j\) and the coefficients \(\beta '_j=\beta _j(c_j/\mu _{j,k_j}^q)^M\) of Definition 35.17.4.3, with the copy norms \(c_j\) of Definition 35.17.1.1, in place of \(\beta _j\), the construction converges for all nonzero complex weights, also when the \(q\)-site states of distinct blocks overlap, at a rate that includes the correlation lengths of the mixed transfer maps (Lemma 35.17.4.4 and Theorem 35.17.4.5) [ 26 , 128 ] .

When a weight is not a nonnegative real number, the coefficients \(\beta _j=\mu _j^N\) count its phase twice, since \(V\) already carries it. For \(A^0=\operatorname {diag}(1,0)\) and \(A^1=\operatorname {diag}(0,-1)\) and odd \(N\) the approximating state is orthogonal to \(|\phi _N\rangle \); for multiplicity one the coefficients \(\beta _j=\lvert \mu _j\rvert ^N\) repair the construction (Section 35.17.4) [ 142 ] .

With the notation of Definition 35.17.1.1, choose a copy \(k_j\) of every block \(j\), and let \(N=qM\). The coefficient of block \(j\) is

If block \(j\) has the single copy \(k_j\), then \(\beta '_j=\lvert \mu _{j,k_j}\rvert ^N\), the coefficient of Theorem 35.17.4.2. Since the weights are nonzero, \(\beta '\neq 0\) whenever \(\beta \neq 0\).

For \(\delta \in \mathbb {R}\), a chain with bond dimensions at most \(D\) is approximated by pairs with error \(\delta \) on the given blocks if \(\phi _{\mathrm{pos}}\neq 0\) and there are unit vectors \(\omega ^k\) on \(\mathbb {C}^{b_k}\otimes \mathbb {C}^{b_k}\) with \(1-\lvert \langle \Omega ,\phi _{\mathrm{pos}}\rangle \rvert /\lVert \phi _{\mathrm{pos}}\rVert \le \delta \), the states being those of the padded chain and the padded pairs. The source defines a sequence of matrix product states with bond dimension at most \(D\) to have finite correlation length if, after blocking \(q=O(\log N)\) sites, this holds with an error tending to \(0\) as \(N\to \infty \)  [ 219 , paragraph “Inhomogeneous short-range correlated MPS” ] ; the definition here is the condition for one ring. The condition \(\phi _{\mathrm{pos}}\neq 0\) is presupposed by the normalized state in the source’s error, see  [ 66 ] .

Let \(\omega _1,\dots ,\omega _b\in \mathbb C^D\otimes \mathbb C^D\) be any vectors, \(\alpha \in \mathbb C^b\) any coefficients, and \(M\ge 0\). The state \(|\Omega '\rangle =\sum _j\alpha _j\bigotimes _{k=1}^M|\omega _j\rangle _{R_kL_{k+1}}\) is formed as in (88), and for a tensor \(A\) and a blocking length \(q\), with \(V\) from Theorem 35.1.3.3 for the \(q\)-site blocked map of \(A\), the approximating state is \(V^{\otimes M}|\Omega '\rangle /\lVert V^{\otimes M}|\Omega '\rangle \rVert \), and zero when the denominator vanishes. For multiplicity coefficients \(\mu _{j,k}\) put \(\beta _j=\sum _k\mu _{j,k}^N\) and \(\alpha ^{(N)}_j=\beta _j/(\sum _l\lvert \beta _l\rvert ^2)^{1/2}\). For any \(\beta \in \mathbb C^b\) with \(\beta \neq 0\), \(\sum _l\lvert \beta _l\rvert ^2{\gt}0\) and \(\sum _j\lvert \beta _j/(\sum _l\lvert \beta _l\rvert ^2)^{1/2}\rvert ^2=1\). Let \(W\colon \mathbb C^b\to \mathbb C^D\otimes \mathbb C^D\), \(|j\rangle \mapsto |\omega _j\rangle \), be the map whose \(j\)-th column is \(\omega _j\), let \(X^{\otimes M}\) denote the \(M\)-fold tensor power of a matrix \(X\), with entries \((X^{\otimes M})_{p,s}=\prod _{k=1}^M X_{p_k,s_k}\), and put \(|\chi _M\rangle =\sum _j\alpha _j|j\rangle ^{\otimes M}\in (\mathbb C^b)^{\otimes M}\). For \(\omega _j\) given by the embedded pair state of each normal block \(A_j\), \(\alpha =\alpha ^{(N)}\), and \(N=qM\), this specializes to Definition 35.17.1.

Natural-number cuts carry the actual cyclic bond dimensions. The rectangular channel of a partition block is \(F_{a,b}\), after equality-based identification of its final bond. Its physical indices are those of the original site tensors.

Let a positive ring have positive many blocks of lengths \(\ell _k\), and choose an injective encoding of the logical labels in registers of length \(r\), with \(2r\le \ell _k\). Define \(J:\mathbb {C}^b\to \mathcal H_N\) by \(Je_j=|\operatorname {reg}(j)\rangle \), repeating the same register word in every block and setting all unused sites to zero. For a tensor \(A\) and pair vectors \(\omega _j\), define

where \(V_k\) is the polar isometry of the \(\ell _k\)-site blocked tensor.

Let \(s\ge 1\). A binary finite-range network with sites of dimension \(d^s\) has its finest isometry \(W\colon \mathbb {C}^{D^2}\to \mathbb {C}^{d^s}\otimes \mathbb {C}^{d^s}\), which maps one site to two neighbouring registers of \(s\) sites of dimension \(d\). Its state on \(M2^{k+1}\) registers (59), read on the \(N=Ms2^{k+1}\) sites by splitting every register into its \(s\) sites, is its state on registers.

With the notation of Definition 35.17.1.1 and Theorem 35.17.1.2, fixed points \(\rho _j\) of the blocks, and \(|\Omega _j\rangle \) the product of the pairs of \(\rho _j\) over \(M\) sites in the bond space of block \(j\), the corrected approximating state on \(N=qM\) sites with coefficients \(\alpha _j\) is

When the weights \(\beta _j\) of Theorem 35.17.1.2 are not all zero, the source’s coefficients are \(\alpha _j=\beta _j/\bigl(\sum _l\lvert \beta _l\rvert ^2\bigr)^{1/2}\); if all \(\beta _j\) vanish these coefficients are undefined. It is obtained from the corrected block form of Theorem 35.17.1.2 by replacing each \(P_j\) by its fixed point, as [ 219 , eq. (S7) ] does with the printed block form. The weights are those of the source. The isometry \(L_j\) acts on the two legs of each site, so for a repeated block the state \(L_j^{\otimes M}|\Omega _j\rangle \) is not a product of nearest-neighbour pairs of the full bond space; for multiplicity one and \(\mu _j{\gt}0\) it is the state of Definition 35.17.1 [ 26 ] .

Let \(A_1,\dots ,A_b\) be tensors, let the isometries \(E_{j,k}\), \(1\leq k\leq m_j\), place the copy \(k\) of block \(j\) in the bond space, with \(E_{j,k}^\dagger E_{j',k'}=\delta _{jj'}\delta _{kk'}\mathbf1\), and let \(\mu _{j,k}\) be nonzero weights, with at least one copy of every block. The direct sum with multiplicities is \(A^i=\sum _{j,k}\mu _{j,k}E_{j,k}A_j^iE_{j,k}^\dagger \). For \(q\geq 1\) put \(K_{j,k}=E_{j,k}\otimes E_{j,k}\),

For a family \(A_j\), \(1\le j\le b\), of tensors with physical dimension \(d\), let \(\mathcal H_N=(\mathbb {C}^d)^{\otimes N}\) and define

The inverse square root is zero on the kernel, so \(F_{A,N}\) is the polar partial isometry also when the columns are dependent. The columns of \(V_{A,N}\) are the actual unnormalized periodic vectors, with their given phases. The endpoint \(F_{A,N}\) depends only on the family and \(N\), not on an accuracy or a block partition.

Let \(A\) be given by a sector decomposition with blocks \(A_1,\dots ,A_b\) of bond dimensions \(D_j\), multiplicities \(m_j\) and coefficients \(\mu _{j,k}\), whose flattened direct sum \(\bigoplus _{(j,k)}\mu _{j,k}A_j\) is the tensor of Definition 35.17.1 up to the order of the bond coordinates, so the bond space of \(A\) is \(\mathbb {C}^D\) with \(D=\sum _jm_jD_j\). The blocks carry no normality, basis-of-normal-tensors or gauge condition. For each \(j\) let \(\sigma _j\) be any matrix on \(\mathbb {C}^{D_j}\) and choose a copy \(k_j\in \{ 1,\dots ,m_j\} \). The pair of block \(j\) is \(\sum _{r,l}(\sqrt{\sigma _j})_{rl}|r\rangle |l\rangle \) placed, on both legs, on the bond coordinates of copy \(k_j\), which gives \(|\omega _j\rangle \in \mathbb {C}^D\otimes \mathbb {C}^D\). With \(\beta _j\) and \(\alpha ^{(N)}_j\) as in Definition 35.17.1, the fixed-point state \(|\Omega '\rangle \) is formed from these pairs by (88). Here each \(\sigma _j\) is an arbitrary matrix on \(\mathbb {C}^{D_j}\), and the copy \(k_j\) is specified explicitly.

A tree layout of a block of \(n\) sites with \(h+1\) coarse depths is a sequence of \(2^{h+1}\) even widths \(w_0,w_1,\dots \), each at least \(2s\), with \(o_{2^{h+1}}\le n\), where \(o_m=w_0+\cdots +w_{m-1}\). The leaf \(i\) covers the sites \(o_i,\dots ,o_{i+1}-1\), the last leaf also the sites \(o_{2^{h+1}},\dots , n-1\). For \(0\le j\le h+1\) and \(0\le p{\lt}2^j\), the node \(p\) of depth \(j\) covers the leaves \(p2^{h+1-j},\dots ,(p+1)2^{h+1-j}-1\); its two registers are the sites \(o_{p2^{h+1-j}},\dots ,o_{p2^{h+1-j}}+s-1\) and the \(s\) sites before \(o_{(p+1)2^{h+1-j}}\). The registers of a node are an even number of sites apart, and they are the first register of its first half and the last register of its second half. A register of a node of depth \(j'\) is a register of a node of every depth \(j\ge j'\), and no register of a node of depth \(j\) lies strictly between the two registers of a node of depth \(j\).

Let the two-site blocked tensor of \(A\) be injective and let \(u\) be a unitary on \(\mathbb {C}^D\otimes \mathbb {C}^D\). The tree network of \(A\) with \(k+1\) isometry layers is the binary finite-range network with isometries \(W=V^{(1)}\), \(W_i=V^{(i+2)}\) for \(0\le i{\lt}k\), and unitary \(u\).

Let \(D,M\ge 1\), let \(\ell _0+\cdots +\ell _{M-1}=N\), and let \(T_k\) be trees of common coarse depth \(h\), virtual dimension \(D^2\) and physical leaf bound \(R\), on their respective blocks of \(\ell _k\) sites. A unitary \(u\) on \(\mathbb {C}^D\otimes \mathbb {C}^D\) prepares a pair \(|\omega \rangle =u|0,0\rangle \). The network state is

with cyclic indexing of the pair legs. All lower disentanglers are identities. The network has \(h+2\) layers, including the finest isometries and the top unitary layer. Internal tree legs have dimension \(D^2\); every physical leaf has dimension at most \(d^R\).

Let \(D_1,\dots ,D_N\le D\) be the dimensions of the bonds of the ring, the bond \(k\) joining site \(k-1\) to site \(k\), cyclically, and let site \(k\) carry matrices \(A_k^i\) of size \(D_k\times D_{k+1}\). The state of the chain is \(|\phi _N\rangle =\sum _s\operatorname{Tr}(A_1^{s_1}\cdots A_N^{s_N})|s\rangle \), the setting of the paragraph “Inhomogeneous short-range correlated MPS” of  [ 219 ] . Padding every \(A_k^i\) with zeros to a \(D\times D\) matrix gives a chain of bond dimension \(D\). A chain with common bond dimension \(D\) is the special case \(D_k=D\) with unchanged matrices. For blocks as in Definition 35.10.1, let \(a_k=D_{o_k+1}\) and \(b_k=a_{k+1}\) be the dimensions of the bonds at the left and right end of block \(k\), cyclically, and let \(R_k\) be the rectangle of pairs \((\alpha ,\beta )\) with \(\alpha {\lt}a_k\) and \(\beta {\lt}b_k\). The blocked tensors are injective if, for every \(k\), the blocked tensor \(B_k\) of block \(k\) of the padded chain spans every matrix unit \(|\alpha \rangle \langle \beta |\) with \((\alpha ,\beta )\in R_k\). For blocks of at least one site, \(B_k\) vanishes outside \(R_k\) and its entries on \(R_k\) are those of the product of the rectangular matrices along the block, so the condition may be read as the source’s injectivity of the rectangular blocked tensor, whose matrices of size \(a_k\times b_k\) span all matrices of that size; that identification of the entries on \(R_k\) is not part of the formalization, see the section on zero padding in  [ 66 ] . For a block of length zero the two readings differ, the padded blocked tensor being the identity matrix of size \(D\). A pair \(\omega ^k\) on \(\mathbb {C}^{b_k}\otimes \mathbb {C}^{b_k}\) is padded with zeros to a vector on \(\mathbb {C}^D\otimes \mathbb {C}^D\). The states \(|\phi _{\mathrm{pos}}\rangle \), \(|\Omega \rangle \) and \((\bigotimes _kV_k)|\Omega \rangle \) of the chain are those of Definition 35.10.2 for the padded chain and the padded pairs.

Cut the ring of \(N\) sites into \(M\geq 1\) blocks of lengths \(\ell _k\geq 2r_1\). The register \(R_k\) consists of the last \(r_1\) sites of block \(k\), and its ancilla of the first \(r_1\) sites of block \(k+1\), cyclically; together they form the pair window \(k\). For a vector \(\alpha \) on \((\mathbb {C}^d)^{\otimes r_1}\), the GHZ-type state on the registers has the amplitude \(\alpha (u)\) at a configuration carrying the same configuration \(u\) on every register and \(0\) at every other site, and the amplitude \(0\) at every other configuration. If the labels \(j\) of \(b\) blocks are encoded injectively as configurations \(\delta (j)\) of \(r_1\) sites and \(\alpha (\delta (j))=\alpha _j\), \(\alpha (u)=0\) for every other \(u\), it is \(\sum _j\alpha _j|x_j\rangle \), where \(x_j\) carries \(\delta (j)\) on every register and \(0\) elsewhere: the state \(|\chi _M\rangle \) of Definition 35.17.3, with the label of each block stored in its register. The configuration \(x_j\) carries \((\delta (j),0,\dots ,0)\) on every pair window and \(0\) outside the windows.

Let \(S\) be a set of pairs of bond indices. A tensor \(B\) of physical dimension \(n\) and bond dimension \(D\) is injective on \(S\) if \(B^i_{\alpha \beta }=0\) for all \(i\) and \((\alpha ,\beta )\notin S\), and the matrices \(B^i\) span every matrix unit \(|\alpha \rangle \langle \beta |\) with \((\alpha ,\beta )\in S\). For the rectangle \(S=\{ (\alpha ,\beta ):\alpha {\lt}a,\ \beta {\lt}b\} \) the condition may be read as injectivity of a tensor with bond dimensions \(a\) and \(b\) whose matrices are padded with zeros; no such rectangular tensor enters the formal statement, and this reading is not proved.

Let \(M\in \mathbb {C}^{m\times k}\). The positive part is \(P=(M^\dagger M)^{1/2}\), the support projector is \(\Pi =\chi (M^\dagger M)\) with \(\chi (0)=0\) and \(\chi (x)=1\) for \(x\ne 0\), and the partial isometry is \(V=M\, g(M^\dagger M)\) with \(g(x)=x^{-1/2}\) for \(x{\gt}0\) and \(g(0)=0\); the functions \(\chi \) and \(g\) are applied to the positive semidefinite matrix \(M^\dagger M\) by the functional calculus. This is the decomposition \(B=VP\) of [ 219 , eq. (8) and Supplemental Material ] .

The physical matrix of a tensor \(B=\{ B^i\} _{i=0}^{n-1}\) with \(B^i\in M_{D}(\mathbb {C})\) is the matrix \(\widehat B\in \mathbb {C}^{n\times D^2}\) with entries \(\widehat B_{i,(\alpha ,\beta )}=B^i_{\alpha \beta }\), the matrix of \(B\) as a map from the virtual pair space \(\mathbb {C}^{D^2}\) to the physical space \(\mathbb {C}^n\). Conversely, every \(R\in \mathbb {C}^{n\times D^2}\) is the physical matrix of the tensor \(B^i_{\alpha \beta }=R_{i,(\alpha ,\beta )}\).

Identify \(\{ 0,\dots ,D^2-1\} \) with the pairs \((\alpha ,\beta )\) of virtual indices, and let \(V\), \(P\) and \(\Pi \) be the polar factors of \(\widehat B\), read as matrices indexed by \(\{ 0,\dots ,D^2-1\} \) on the virtual side. The positive-part tensor of \(B\) has physical dimension \(D^2\) and bond dimension \(D\), with \((P^k)_{\alpha \beta }=P_{k,(\alpha ,\beta )}\); its matrices are the rows of \(P\). The isometric factor is \(V\in \mathbb {C}^{n\times D^2}\). This is the tensor \(P\) of [ 219 , eqs. (9) and (18) ] .

An MPS tensor \(A=(A^i)_i\) is left-canonical when

Equivalently, its Kraus map is trace-preserving.

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

The transfer map \(\mathcal{E}_A\) of an MPS tensor \(A\) satisfies transfer map idempotence when \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\).

The matrix product vector (MPV) of a tensor \(A\) at system size \(N\) is the vector

Equivalently, for a configuration \(\sigma = (i_1, \ldots , i_N) \in \{ 0,\ldots ,d{-}1\} ^N\), we write

The coefficient function \(\sigma \mapsto V^{(N)}(A)_\sigma \) gives the components of the vector in (2); the displayed ket is the corresponding vector in \((\mathbb {C}^d)^{\otimes N}\). For a general word \(w\), we also write \(c_w(A) := \operatorname{tr}(A^w)\). The MPV family generated by \(A\) is the collection \(\mathcal{V}(A) = \bigl\{ |V^{(N)}(A)\rangle \bigr\} _{N \ge 1}\). The coefficient \(V^{(N)}(A)_\sigma \) is the periodic contraction

of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ 13 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

An MPS tensor \(A\) is a renormalization fixed point if there exists an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\) with coefficients \(V_{(i_1,i_2),j}\) satisfying \(V^\dagger V=\mathbb {1}\) such that for all physical indices \(i_1,i_2\in \{ 0,\ldots ,d-1\} \),

This expresses the physical two-site blocking relation \(AA=A\) of [ 13 , Definition 3.2 ] .

The bond regrouping is the bijection of configurations on a ring of \(N\) sites sending \(c=(l_k,r_k)_k\) to \((r_k,l_{k+1})_k\), with indices modulo \(N\). Its inverse sends \((p_k)_k=(p_k^1,p_k^2)_k\) to \((p_{k-1}^2,p_k^1)_k\).

Let \(\sigma \in M_{D}(\mathbb {C})\). The fixed-point tensor \(P_\infty \) of \(\sigma \) has physical space \(\mathbb {C}^D\otimes \mathbb {C}^D\) and bond dimension \(D\); its matrix at the physical pair \((l,r)\) is

This is the tensor \(P_\infty \) of [ 219 ] .

Let \(\sigma \in M_{D}(\mathbb {C})\). The entangled pair \(\omega \) on \(R\otimes L=\mathbb {C}^D\otimes \mathbb {C}^D\) has coefficient \(\omega _{(r,l)}=(\sqrt\sigma )_{rl}\). For a vector \(\omega \in \mathbb {C}^D\otimes \mathbb {C}^D\) and a ring of \(N\) sites, where site \(k\) carries the physical pair \(c_k=(l_k,r_k)\), the pair product state is

with indices of sites taken modulo \(N\). In terms of \(\rho =\sigma ^T\), \(\omega \) corresponds to \((\mathbb {1}\otimes \sqrt\rho )\sum _i|ii\rangle \), with pair product state \(\Omega =\bigotimes _k|\omega \rangle _{R_kL_{k+1}}\) [ 219 ] .

Let \(f\colon \mathbb N\to \mathbb R\) and let \(|\varphi _N\rangle \) be a vector on the ring of \(N\) sites for every \(N\). The sequence \((|\varphi _N\rangle )_N\) is asymptotically prepared with measurements in depth \(f\) if there are vectors \(|\psi _N\rangle \) such that for all large \(N\) the vector \(|\psi _N\rangle \) is a unit vector prepared with measurements and a circuit in depth at most \(f(N)\), and \(\lVert |\psi _N\rangle \langle \psi _N|-|\varphi _N\rangle \langle \varphi _N|\rVert _1\to 0\).

In the paragraph “Phases of matter” of [ 227 ] , a sequence \((|\psi _M\rangle )_M\) is mapped to a sequence \((|\varphi _M\rangle )_M\) if, for some \(k\), compositions of \(k\) channels, each a circuit of depth \(f(M)\) assisted by measurements and classical communication, map \(|\psi _M\rangle \) to states \(\sigma _M\) with \(\lVert \sigma _M-|\varphi _M\rangle \langle \varphi _M|\rVert _1\to 0\), where \(f\) is polylogarithmic. The definition is this relation from the sequence of product states, with \(k=2\) (measurements, then a circuit) and \(\sigma _N=|\psi _N\rangle \langle \psi _N|\).

Let \(b,M\geq 0\) and \(\alpha \in \mathbb C^b\). The GHZ-type state on \(M\) sites of dimension \(b\) is \(|\chi _M\rangle =\sum _{i=0}^{b-1}\alpha _i|i\rangle ^{\otimes M}\). For \(M\geq 1\) and the sites numbered \(0,\dots ,M-1\), its amplitude at a configuration \(s\) is \(\alpha _{s_0}\) if all \(s_n\) are equal and \(0\) otherwise.

A local circuit of depth \(T\) on a chain of \(N\) sites is a product of \(T\) layers, each a tensor product of unitaries acting on disjoint pairs of neighbouring sites. A state is prepared in depth \(T\) if it equals a local circuit of depth \(T\) applied to a product state.

More generally, a bond geometry on a finite set of sites \(I\) is a family \((B_b)_{b\in \beta }\) of subsets of \(I\), the bonds, and a layer is a product of unitaries acting on pairwise disjoint bonds. The model of [ 227 ] on a lattice takes for bonds the pairs of nearest-neighbour sites, that is the edges of a graph. Local circuits of depth \(T\) and states prepared in depth \(T\) are defined in the same way for every bond geometry.

The chain is a ring with sites \(1,\dots ,N\): the neighbouring pairs are \(\{ k,k+1\} \) with indices modulo \(N\); for \(N=1\) the only pair is \(\{ 1\} \), and for \(N=2\) both pairs are \(\{ 1,2\} \). The open chain has the pairs \(\{ k,k+1\} \) with \(k{\lt}N\). An operator acts on a set of sites \(X\) if it lies in \(M_{d}(\mathbb {C})^{\otimes X}\otimes \mathbb 1\), the span of the products \(\bigotimes _i m_i\) with \(m_i=\mathbb 1\) for \(i\notin X\).

Acting on a set of sites needs no geometry: for any finite set of sites \(I\) and \(X\subseteq I\), an operator on \((\mathbb {C}^d)^{\otimes I}\) acts on \(X\) if it lies in the span of the products \(\bigotimes _{i\in I}m_i\) with \(m_i=\mathbb 1\) for \(i\notin X\). The chain of \(N\) sites is the case \(I=\{ 1,\dots ,N\} \).

A vector \(|\psi \rangle \) is prepared with measurements and a circuit in depth \(T\) if \(|\psi \rangle =U|\varphi \rangle \) for a vector \(|\varphi \rangle \) prepared with measurements in depth \(T_1\) and a local circuit \(U\) of depth \(T_2\), with \(T_1+T_2\leq T\). The circuit \(U\) does not depend on the outcomes of the measurements.

The local labels are \(\{ 0,\dots ,d-1\} \). For a set \(S\) of sites of the ring of \(N\) sites and a string \(m\colon S\to \{ 0,\dots ,d-1\} \), let \(P_m\) be the projection onto the span of the basis vectors \(|x\rangle \) with \(x_i=m_i\) for every \(i\in S\); these are the outcome projections of a measurement of the sites of \(S\) in the computational basis, and \(P_m\) is the projection onto the configurations agreeing on \(S\) with any configuration extending \(m\). A protocol consists of a product vector \(|\pi \rangle \), a local circuit \(U\) of depth \(T_1\), a set \(S\) of measured sites, and for every \(m\) unitaries \(v_{m,i}\) on the sites \(i\), with product \(V_m=\bigotimes _iv_{m,i}\). It prepares a vector \(|\psi \rangle \) if \(|\pi \rangle \neq 0\) and for every \(m\) with \(P_mU|\pi \rangle \neq 0\) there is \(c\in \mathbb C\) with \(V_mP_mU|\pi \rangle =c|\psi \rangle \). A vector is prepared with measurements in depth \(T\) if a protocol with \(T_1\leq T\) prepares it.

The model is that of [ 227 ] , where the class of such preparations is called \(\mathrm{QCcc}\). The correction \(V_m\) may depend on the whole string \(m\): classical communication is global and free, and the single-site corrections are not counted. Ancillas are sites of the ring, and the local unitaries between the layers of the circuit, acting on a site and its ancillas and free in [ 227 ] , are counted here as gates. The correction \(V_m\) is a local unitary in the sense of [ 227 ] . So every preparation in this sense is one in the sense of [ 227 ] , of no larger depth.

A measurement round consists of a local circuit \(U\), a set \(S\) of sites measured in the computational basis, and for every outcome \(m\colon S\to \mathbb Z_d\) unitaries \(v_{m,i}\) on the sites \(i\), with product \(V_m=\bigotimes _{i}v_{m,i}\). Its operator for the outcome \(m\) is \(V_{m}P_{m}U\), with \(P_m\) the outcome projection of Definition 33.3.1, and its depth is the number of layers of \(U\). With a product vector it is a protocol in the sense of that definition. The outputs of a sequence of rounds \(R_1,\dots ,R_n\) from a vector \(|v\rangle \) are the vectors \(K_n\cdots K_1|v\rangle \), with \(K_j\) the operator of \(R_j\) for some outcome. A vector \(|\psi \rangle \) is prepared with measurement rounds in depth \(T\) if there are a product vector \(|\pi \rangle \neq 0\) and rounds of total depth at most \(T\) such that every nonzero output from \(|\pi \rangle \) is a multiple of \(|\psi \rangle \).

Every round is a protocol of \(\mathrm{QCcc}_\ell \) of [ 227 ] of no larger depth, in the sense of Definition 33.3.1; that paper performs a single round and notes that “one could also define a more general scheme with multiple rounds of LOCC”. A sequence of rounds is a composition of such transformations, with the depth counted as the total number of layers. The number of rounds is not bounded: the tree circuit of Chapter 35 with \(k\) levels uses \(2k\) rounds, whereas in the class \(\mathrm{QCcc}^{(k)}_\ell \) of the paragraph “Phases of matter” of [ 227 ] the number \(k\) of composed transformations does not depend on the system size. Several rounds are needed when a correction must precede gates that do not commute with it, as for a teleported register on which an isometry acts.

Let \(E\) be a set of vectors and \(W\) a matrix. A round implements \(W\) on \(E\) if for every outcome \(m\) there is \(c\in \mathbb C\) with \(V_{m}P_{m}U|v\rangle =cW|v\rangle \) for every \(|v\rangle \in E\). A sequence of rounds implements \(W\) on \(E\) if every output from every \(|v\rangle \in E\) is a multiple of \(W|v\rangle \).

Given a matrix \(M \in \mathbb {C}^{m\times d}\) and an MPS tensor \(A\) with physical dimension \(d\), the physical-index rotation is the tensor with physical dimension \(m\) given by \((A_M)^i := \sum _j M_{ij}\, A^j\) for \(i=0,\ldots ,m-1\). When the matrix is a unitary \(u\), the rotation is written \(A_u\).

Let \(P\) be a sector decomposition. In addition to spanning and eventual linear independence, the canonical form specifies irreducible left-canonical representatives, self-overlap normalization, pairwise gauge-phase separation, and weight normalization. Explicitly, each basis block has positive bond dimension, is irreducible, and is left-canonical; the normalized self-overlap of every basis block converges to \(1\); the basis MPV family is eventually linearly independent; distinct same-dimension basis blocks are not gauge-phase equivalent after identifying equal bond dimensions; and the [ 13 , Section 2.3 ] canonical-form normalization holds:

The MPV expansion takes the two-layer form

matching [ 13 , Section 2.3 ] and [ 17 , Definition IV.2 ] . The canonical-form normalization accommodates equal-modulus and repeated-copy sectors, such as \(C \oplus (-C)\) and \(C \oplus (1/2) C\).

A sector-weight tuple over \(g\) basis blocks consists of multiplicities \(r_j {\gt} 0\), nonzero weights \(\mu _{j,q}\) for \(q \in \{ 1,\ldots ,r_j\} \), and the sector coefficients

A sector decomposition consists of the basis blocks \((A_j)_{j=1}^g\) together with these multiplicities and weights. If \(A_j\) has bond dimension \(D_j\), then the flattened copy family has total bond dimension

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).

A tensor \(A\) is \(L\)-block injective if and only if its \(L\)-blocked tensor \(A^{[L]}\) is injective.

Let \(\sigma \ge 0\) with \(\operatorname{Tr}\sigma =1\), let the \(q\)-site blocked tensor \(\mathcal B_q\) be injective, and let \(M\ge 1\). Then \(\lVert \phi _M(VP_\infty )\rVert =1\), so the approximating state \(|\widetilde\phi _N\rangle \) of Definition 35.3.8 equals \(|\phi _M(VP_\infty )\rangle \) and is a unit vector.

Let \(B\) be a normal tensor in the gauge (3), with a positive definite fixed point \(\sigma \) of trace one, and let \(A\) be a tensor with \(|\phi _N(B)\rangle =\zeta ^N|\phi _N(A)\rangle \) for all \(N\) and some \(\zeta \neq 0\). Let \(0{\lt}t{\lt}1\) bound the moduli of the eigenvalues of \(\mathcal{E}_B\) other than \(1\), and let \(\xi =-1/\log t\). For every \(a{\gt}\xi /2\) there is \(b_0\ge 0\) such that for \(\varepsilon {\gt}0\), every \(q\ge 1\) and every \(M\ge 1\) with \(q\ge a\log (Mq/\varepsilon )+b_0\), the approximating state of \(B\) with blocks of \(q\) sites on \(N=Mq\) sites has error at most \(\varepsilon \) against \(|\phi _N(A)\rangle \).

Let \(L\ge 2^{h+2}s\). The widths \(w_i=2\lfloor \lfloor L/2\rfloor /2^{h+1}\rfloor \), increased by \(2\) for the first \(\lfloor L/2\rfloor \bmod 2^{h+1}\) leaves, are a tree layout of a block of \(L\) sites with \(h+1\) coarse depths, whose leaves cover all the sites but at most one. If \(L\le 2^{h+1}c\), every leaf has at most \(c+3\) sites.

The tree map (58) of a binary finite-range network is an isometry, and the state (59) is a unit vector.

Suppose every blocked tensor \(\mathcal B_{\ell _k}\) is injective. Then \(\langle \psi _{\ell ,\omega } | \psi _{\ell ,\omega } \rangle =\langle \omega | \omega \rangle ^M\), and for the pair \(\omega \) of the fixed point

where \(\tau _\ell \) is the transfer matrix of the mixed transfer map \(X\mapsto \sum _iP_\ell ^iX(P_\infty ^i)^\dagger \).

Cut the ring of \(N\) sites into \(M\) blocks of lengths \(\ell _k\). If operators \(U_k\) on the \(\ell _k\) sites of the blocks act by maps \(f_k\) of the configurations of the blocks, \(U_k(v)=v\circ f_k\), then the layer of the \(U_k\) on the blocks acts by the map changing the configuration of every block \(k\) by \(f_k\); the blocks cover the ring. If \(\ell _k\geq 2r_1\) and operators \(W_k\) on \(2r_1\) sites act by maps \(f_k\), the layer of the \(W_k\) on the pair windows of Definition 35.21.1.1 acts by the map changing the configuration of every pair window \(k\) by \(f_k\) and keeping the other sites.

Let \(K,a,\varepsilon {\gt}0\), \(M,q\geq 1\), \(b\geq a\max (\log K,0)\), and \(a\log (Mq/\varepsilon )+b\leq q\). Then \(KMe^{-q/a}\leq \varepsilon \).

Let \(r_1\geq 1\) and values be taken in \(\mathbb Z_d\). There is \(C\) such that for every \(q\geq 3r_1\) some unitary on \(q\) sites, a product of at most \(Cq\) gates on neighbouring sites, acts by the map of configurations that adds the last \(r_1\) sites to the first \(r_1\), site by site, and leaves the other sites unchanged.

Let \(D\ge 1\), encode the bond indices injectively in \(r_1\ge 1\) sites, let \(\pi \) enumerate the bond pairs, and let \(Q_1,\dots ,Q_q\) be an isometric chain with \(b_0=1\), as in Theorem 35.21.1.7. There is \(C\), independent of \(q\ge 3r_1\), of the chain and of \(G\), such that for every unitary \(G\) on \(\mathbb {C}^{D^2}\) some unitary \(U\) on \(q\) sites, a product of at most \(Cq\) gates on neighbouring sites, and some amplitudes \(Z(\sigma ,y)\) with \(Z(\sigma ,y)=(Q_1^{\sigma _1}\cdots Q_q^{\sigma _q})_{1y}\) for \(y{\lt}b_q\) satisfy \(\langle \sigma |U|l,0,r\rangle =\sum _yZ(\sigma ,y)G_{yx}\) whenever \(\pi (x)=(l,r)\). A unitary on the levels of \(\mathbb {C}^m\) encoded injectively in the configurations of a window extends to a unitary on the window.

For finitely many normal left-canonical tensors \(A_1,\dots ,A_b\), \(b\geq 1\), of positive bond dimensions, there is \(0{\lt}t{\lt}1\) bounding the moduli of the eigenvalues other than \(1\) of every transfer map \(\mathcal{E}_{A_j}\).

For finitely many normal left-canonical tensors \(A_1,\dots ,A_b\), \(b\geq 1\), of positive bond dimensions, whose mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), have all eigenvalues of modulus below one, there is \(0{\lt}t{\lt}1\) bounding the moduli of the eigenvalues other than one of every \(\mathcal{E}_{A_j}\) and of all eigenvalues of every \(\mathcal{E}_{jj'}\).

Let \(C,q\) be natural numbers, \(N\geq 2\), \(0{\lt}\varepsilon \leq 1\), \(b\geq 1\), and \(q\leq 2(a\log (N/\varepsilon )+b)\). Then \(Cq\leq C(2a+2b/\log 2)\log (N/\varepsilon )\).

For unit vectors \(a\), \(b\) in a complex inner product space and every scalar \(c\) with \(\lvert c\rvert =1\), \(\varepsilon (a,b)\le \lVert a-cb\rVert ^2/2\), with equality for some such \(c\). For unit vectors \(x\), \(y\), \(z\), \(\varepsilon (x,z)\le 2\bigl(\varepsilon (x,y)+\varepsilon (y,z)\bigr)\). A unitary matrix \(U\) satisfies \(\overline{(Ux)}\cdot (Uy)=\overline{x}\cdot y\) for coordinate vectors \(x\), \(y\), and so preserves inner products and norms.

For a Hermitian operator \(Z\) and unit vectors \(\phi ,\psi \), with \(\sigma _\chi (Z)=\lVert (Z-\langle Z\rangle _\chi )\chi \rVert \), the inequality (50) holds: \(\lvert \langle \phi | \psi \rangle \rvert \, \lvert \langle Z\rangle _\phi -\langle Z\rangle _\psi \rvert \le \sigma _\phi (Z)+\sigma _\psi (Z)\).

Let \(A\) be in the gauge (3), with the products of \(L\ge 1\) matrices spanning \(M_{D}(\mathbb {C})\), let \(P(Z)=\operatorname{Tr}(Z)\, \rho \), and let \(\lambda _2\) be an eigenvalue of \(\mathcal{E}_A\) of largest modulus among those different from \(1\).

1. For every \(r{\gt}\lvert \lambda _2\rvert \) there is \(C_r{\gt}0\) with \(\lVert (\mathcal{E}_A-P)^n\rVert \le C_rr^n\) for all \(n\ge 0\).

2. For \(t,n\ge 1\) and \(N=2L+t+n\), the connected correlator \(G_N(X,Y;L+t+1)\) of (46) equals \((\alpha \beta +G_\infty +\delta _{XY})/(1+\delta ) -(\alpha +\delta _X)(\beta +\delta _Y)/(1+\delta )^2\), where \(\alpha =\operatorname{Tr}\mathcal{E}_X(\rho )\), \(\beta =\operatorname{Tr}\mathcal{E}_Y(\rho )\), \(G_\infty \) is the limit correlator (48) at separation \(t\), and \(\delta ,\delta _X,\delta _Y,\delta _{XY}\) are traces of linear maps containing \((\mathcal{E}_A-P)^n\), \((\mathcal{E}_A-P)^N\) or a power of \(\mathcal{E}_A-P\) of exponent at least \(n\).

3. If \(\lvert z-1\rvert \), \(\lvert n_{XY}-(ab+g)\rvert \), \(\lvert n_X-a\rvert \) and \(\lvert n_Y-b\rvert \) are at most \(\varepsilon \le \tfrac 12\) and \(\lvert a\rvert ,\lvert b\rvert , \lvert g\rvert \le M\), then \(\lvert n_{XY}/z-(n_X/z)(n_Y/z)-g\rvert \le 4(M+2)^2\varepsilon \).

4. The trace of a linear map on \(M_{D}(\mathbb {C})\) is bounded by a constant depending only on \(D\) times its operator norm, and \(e^{-(s-1)/\xi }\le \lvert \lambda _2\rvert ^t\) whenever \(t+1\le s\) and \(0{\lt}\lvert \lambda _2\rvert {\lt}1\).

Suppose the products of \(L\) matrices of \(A\) span \(M_{D}(\mathbb {C})\) and \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), let \(\rho \) be Hermitian with \(\operatorname{Tr}\rho =1\), put \(P(Z)=\operatorname{Tr}(Z)\, \rho \), and let \(\lambda \neq 0,1\) be an eigenvalue of \(\mathcal{E}_A\). Then there are Hermitian operators \(\mathcal O,\mathcal O'\) on \((\mathbb {C}^d)^{\otimes L}\) of norm one, an integer \(K\in \{ 1,2\} \) with \(K=1\) when \(\lambda \) is real, distinct complex numbers \(\mu _1,\dots ,\mu _K\) of modulus one and a nonzero \(a\in \mathbb {C}^K\) such that for every \(t\ge 0\)

The inner insertion is traceless, so the displayed expression equals the centered connected contraction also at \(t=0\). This replaces the choice of observables in the proof of [ 219 , Lemma 2 ] by one with Hermitian observables: for non-real \(\lambda \) the eigenvector is not Hermitian, and a Hermitian observable pairs it with its adjoint, an eigenvector for \(\bar\lambda \).

Suppose the products of \(L\) matrices of \(A\) span \(M_{D}(\mathbb {C})\) and \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), let \(\operatorname{Tr}\rho =1\), put \(P(Z)=\operatorname{Tr}(Z)\, \rho \), and let \(R\neq 0\) satisfy \(\mathcal{E}_A(R)=\lambda R\) with \(\lambda \neq 1\). Then there are operators \(X,Y\) on \((\mathbb {C}^d)^{\otimes L}\) such that for every \(t\ge 0\)

When \(\rho \) is the fixed point of the gauge (3), the left side is the limit correlator (48) at \(t=s-L-1\); it is bilinear in \((X,Y)\). The inner insertion has zero trace, so centering it leaves the formula unchanged even at \(t=0\). This is the choice of observables in the proof of [ 219 , Lemma 2 ] , without the source’s requirement that the one-point functions vanish.

Let \(K,M,\varepsilon {\gt}0\) and \(\log K+\log M-\log \varepsilon \leq x\). Then \(KMe^{-x}\leq \varepsilon \).

Let \(\xi ,C,M,\varepsilon {\gt}0\) and \(s\ge 1\). There is \(k\ge 0\) with \(x_k\le s2^{k+1}\) and \(k\le \log _2\max (1,x_k)\), where \(x_k=\xi \log (CMs2^{k+1}/\varepsilon )\).

Let \(K,\xi ,\varepsilon {\gt}0\), \(M,q\ge 1\), and \(\xi \log (KMq/\varepsilon )\le q\). Then \(KMe^{-q/\xi }\le \varepsilon \).

Let \(X\) and \(Y\) be tensors with \(d\) physical indices and bond dimensions \(D_1,D_2\geq 1\), with mixed transfer map \(\mathcal{E}_{XY}(Z)=\sum _iX^iZ(Y^i)^\dagger \) on \(D_1\times D_2\) matrices, and let \(X_q\), \(Y_q\) be their blocked tensors on \(q\) sites. Then \(\mathcal{E}_{X_qY_q}=\mathcal{E}_{XY}^{\, q}\), and for tensors with a common number of physical indices the mixed Gram matrix of their physical matrices, read as maps from bond pairs to physical indices, is a rearrangement of the mixed transfer map, \((\mathcal B_X^\dagger \mathcal B_Y)_{(a,b),(a',b')} =\mathcal{E}_{YX}(|b'\rangle \langle b|)_{a'a}\). For \(\lambda _2\in \mathbb {C}\) with \(\lvert \lambda _2\rvert \leq 1\), correlation length \(\xi \), and \(\gamma \geq 0\), \(e^{-\gamma /\xi }\leq 1\). Now let \(\lvert \lambda _2\rvert {\lt}1\) and \(0{\lt}\gamma {\lt}1\), and write \(r=e^{-\gamma /\xi }\).

1. If every eigenvalue of \(\mathcal{E}_{XY}\) has modulus at most \(\lvert \lambda _2\rvert \), there are \(C,K\) with \(\lVert \mathcal{E}_{XY}^{\, n}(Z)\rVert \leq Cr^n\lVert Z\rVert \) for all \(n\) and \(Z\), and \(\bigl\lvert \sum _s\phi _N(X)_s\, \overline{\phi _N(Y)_s}\bigr\rvert \leq Kr^N\) for all \(N\).

2. If every eigenvalue of \(\mathcal{E}_{YX}\) has modulus at most \(\lvert \lambda _2\rvert \), there is \(K'\) with \(\lVert \mathcal B_X^\dagger \mathcal B_Y\rVert \leq K'r^q\) for all \(q\).

For distinct blocks \(A_j\), \(A_{j'}\) of a basis of normal tensors the spectral radius \(\tau _{jj'}\) of \(\mathcal{E}_{jj'}=\mathcal{E}_{A_jA_{j'}}\) is below one, and the source writes \(\xi _{jj'}=-1/\ln \tau _{\max }\) with \(\tau _{\max }=\tau _{jj'}\) [ 219 , eqs. (S14)–(S15) ] .

For a normal left-canonical tensor \(A\) with \(D\geq 1\) there is \(0{\lt}t{\lt}1\) bounding the moduli of the eigenvalues other than \(1\) of \(\mathcal{E}_A\).

Let the products of some fixed number of matrices of \(A\) span \(M_{D}(\mathbb {C})\), and let \(\lambda _1\) be an eigenvalue of \(\mathcal{E}_A\) of largest modulus. Then \(\lambda _1\ne 0\), and there are \(\zeta \ne 0\), an invertible \(X\) and \(L\ge 1\) such that \(B^i=X\zeta A^iX^{-1}\) satisfies the gauge (3) for some \(\rho \), the products of \(L\) matrices of \(B\) span \(M_{D}(\mathbb {C})\), the eigenvalues of \(\mathcal{E}_B\) are those of \(\mathcal{E}_A\) divided by \(\lambda _1\), and \(|\phi _N(B)\rangle =\zeta ^N|\phi _N(A)\rangle \) for every \(N\). In particular, whenever \(|\phi _N(A)\rangle \neq 0\), \(\lvert \langle \phi _N(B) | \psi \rangle \rvert /\lVert \phi _N(B)\rVert =\lvert \langle \phi _N(A) | \psi \rangle \rvert /\lVert \phi _N(A)\rVert \) for every vector \(|\psi \rangle \); when \(|\phi _N(A)\rangle =0\), also \(|\phi _N(B)\rangle =0\). If instead \(A\) is irreducible and \(\mathcal{E}_A\) has the unique eigenvalue \(1\) of largest modulus, the same holds with \(\zeta =1\).

Let \(A\) and \(B\) be tensors with \(|\phi _N(B)\rangle =\zeta ^N|\phi _N(A)\rangle \) for some \(\zeta \neq 0\), and write \(|\hat\phi _N(A)\rangle \) for the normalization of \(|\phi _N(A)\rangle \) (zero when \(|\phi _N(A)\rangle =0\)). Then every vector \(|\psi \rangle \) on \(N\) sites satisfies \(\lvert \langle \psi | \hat\phi _N(B) \rangle \rvert =\lvert \langle \psi | \hat\phi _N(A) \rangle \rvert \), so the error of \(|\psi \rangle \) is the same against both normalized states.

For operators \(X,X'\) on \((\mathbb {C}^d)^{\otimes L_1}\) and \(Y,Y'\) on \((\mathbb {C}^d)^{\otimes L_2}\), with \(X\otimes Y\) acting on \(L_1+L_2\) consecutive sites,

For an operator \(O\) on \((\mathbb {C}^d)^{\otimes N}\) and the unnormalized vector \(|\phi _N(A)\rangle \) of (1), \(\langle \phi _N(A)|O|\phi _N(A)\rangle =\operatorname {tr}\mathcal{E}_O\), the trace of a linear map on \(M_{D}(\mathbb {C})\).

Suppose the products of \(L\) matrices of \(A\) span \(M_{D}(\mathbb {C})\). Then every linear map \(\Phi \) on \(M_{D}(\mathbb {C})\) equals \(\mathcal{E}_X\) for some operator \(X\) on \((\mathbb {C}^d)^{\otimes L}\). For \(L=1\) and injective \(A\), this corresponds to the choice of transfer matrices in the proof of [ 219 , Lemma 2 ] . If moreover \(\Phi (Z^\dagger )=\Phi (Z)^\dagger \) for every \(Z\), then \(X\) can be chosen Hermitian.

Let \(A\) be the direct sum with multiplicities of Definition 35.17.1.1, let \(q\geq 1\), and let \(B=VP\) be the polar decomposition of its blocked map. The states of distinct blocks need not be orthogonal.

1. If \(MY=0\) for a matrix \(M\) with \(M=WQ\) the polar decomposition of Theorem 35.1.1.4, so that \(W^\dagger W\) is the support projector of \(Q\), then \(WY=0\).

2. \(L_j^\dagger K_{j,k}=(\mu _{j,k}^q/c_j)\mathbf1\), and \(L_{j'}^\dagger K_{j,k}=0\) for \(j'\neq j\).

3. \(VK_{j,k}=(\mu _{j,k}^q/c_j)\, VL_j\).

4. For any matrices \(\sigma _j\) and coefficients \(\alpha _j\), with \(|\Omega _j\rangle \) the product of the pairs of \(\sigma _j\) over \(M\) sites,

   \begin{align*} V^{\otimes M}\sum _j\alpha _jK_{j,k_j}^{\otimes M}|\Omega _j\rangle =V^{\otimes M}\sum _j\alpha _j\Bigl(\frac{\mu _{j,k_j}^q}{c_j}\Bigr)^M L_j^{\otimes M}|\Omega _j\rangle . \end{align*}

5. Multiplying all coefficients of the corrected state by one \(t{\gt}0\) does not change its overlap with \(|\phi _N\rangle \).

6. The approximating state of Definition 35.17.1 with the pairs of \(\sigma _j\) on the copies \(k_j\) and the coefficients \(\beta '_j\) in place of \(\beta _j\) has the same overlap with \(|\phi _N\rangle \) as the corrected state of Definition 35.17.1.3 with the coefficients \(\beta _j\).

Let \(B=\sum _jc_jB_jK_j^\dagger \) with \(c_j{\gt}0\), isometries \(K_j\) with orthogonal ranges, so that \(K_j^\dagger K_{j'}=\delta _{jj'}\), and \(B_j^\dagger B_{j'}=0\) for \(j\neq j'\). With \(B_j=V_jP_j\) the polar decompositions of the summands, the polar decomposition of \(B\) has

This is the block form of the positive part asserted in [ 219 , eq. (S5) ] (“Since \(V\) is an isometry, \(P\) inherits the block structure of \(A\)”), for blocks of multiplicity one under the orthogonality of their states, which the source does not state and without which the block form fails [ 142 ] .

Let \(S\) be the set of bond pairs \((E_ja,E_jc)\) whose two indices lie in the same block.

1. If the matrices of a tensor \(B\) vanish outside a set \(S\) of bond pairs and \(\mathcal Bx=0\) has no solution \(x\neq 0\) supported on \(S\), then \(B\) is injective on \(S\).

2. With \(|\lambda _2|{\lt}1\) there is \(L_0\) such that for every \(\ell \geq L_0\) the \(\ell \)-site blocked tensor of \(\bigoplus _jA_j\) is injective on \(S\).

1. For a finite index set there is \(K\geq 0\) such that for every matrix \(E\) whose powers have norm at most \(c\) and all matrices \(X_0,\dots ,X_{M-1}\) within \(\delta \) of \(E\), \(\lvert \operatorname{Tr}(X_0\cdots X_{M-1})-\operatorname{Tr}E^M\rvert \leq Kc^2(M\delta ) e^{cM\delta }\).

2. The mixed transfer map of \(P_\infty K_{j'}K_{j'}^\dagger \) against the fixed-point tensor of \(E_j\rho _jE_j^\dagger \) is \(X\mapsto \operatorname{Tr}(E_jE_j^\dagger X)\, E_j\rho _jE_j^\dagger \) for \(j=j'\) and zero for \(j\neq j'\).

3. With \(|\lambda _2|{\lt}1\) and \(0{\lt}\gamma {\lt}1\), for all \(j,j'\) there is \(C{\gt}0\) such that for every cut into \(M\geq 1\) blocks of lengths \(\ell _k\geq q\geq 1\), with \(y=Me^{-\gamma q/\xi }\), the trace \(z_{jj'}\) of Lemma 35.21.2.2(2) satisfies \(\lvert z_{jj'}-\delta _{jj'}\rvert \leq Cy\, e^{Cy}\).

Cut a ring of \(N\geq 1\) sites into \(M\geq 1\) blocks of lengths \(\ell _k\geq 1\), and let \(\omega _j\) be the pair of \(\rho _j\) placed along \(E_j\).

1. The \(\ell \)-site blocked map of \(A_j\), placed along \(E_j\), is \(V^{(\ell )}P^{(\ell )}K_jK_j^\dagger \), so \(|\phi _N(A_j)\rangle =\bigl(\bigotimes _kV^{(\ell _k)}\bigr) |\phi _M(P^{(\ell _k)}K_jK_j^\dagger )\rangle \), where the second factor is the periodic state of the family of tensors \(P^{(\ell _k)}K_jK_j^\dagger \) on the blocks.

2. If every blocked map \(\mathcal B^{(\ell _k)}\) is injective on the bond pairs of the blocks, then the states \(|\psi _j\rangle =\bigl(\bigotimes _kV^{(\ell _k)}\bigr) \bigotimes _k|\omega _j\rangle \) are orthonormal, so \(\sum _j\alpha _j|\psi _j\rangle \) is a unit vector for every unit vector \(\alpha \), and \(z_{jj'}=\langle \psi _j | \phi _N(A_{j'}) \rangle \) is the trace of the ordered product over the blocks of the mixed transfer matrices of \(P^{(\ell _k)}K_{j'}K_{j'}^\dagger \) against the fixed-point tensor of \(E_j\rho _jE_j^\dagger \).

Let the \(N\)-site blocked map of \(\bigoplus _jA_j\) be injective on the bond pairs of the blocks, and let \(|\phi \rangle =\sum _j\beta _j|\phi _N(A_j)\rangle \neq 0\). Then \(|\phi \rangle /\lVert \phi \rVert =V^{(N)}|\omega \rangle \) for a unit pair \(\omega \) supported on the bond pairs of the blocks.

In the setting of Lemma 35.17.2.3 with weights one, let \(\phi _N\) be the periodic state of \(A\).

1. There is \(K\) with \(\bigl\lvert \lVert \phi _N\rVert ^2-b\bigr\rvert \le Ke^{-\gamma N/\xi }\) for every \(N\ge 1\).

2. The states \(\phi _N(P_{\infty ,\tilde\rho _j})\) are orthonormal for every \(N\ge 1\).

3. For a vector \(W\) with \(\lVert W\rVert ^2=c{\gt}0\) and every \(u\), \(\lVert u\rVert ^2-\lvert \langle W | u \rangle \rvert ^2/c\le \lVert u-W\rVert ^2\).

4. If \(\lambda _2\neq 0\), there is \(K'\) such that for every \(q\ge 1\)

   \begin{align*} \lVert \phi _q\rVert ^2-\frac1b\Bigl\lvert \sum _j \langle \phi _1(P_{\infty ,\tilde\rho _j}) | \phi _1(P) \rangle \Bigr\rvert ^2 \le K’e^{-2\gamma q/\xi }. \end{align*}

1. For an isometry \(E\) placing the bond coordinates of a block and a positive semidefinite \(\sigma \), the pair of \(\sigma \) placed along \(E\) on both legs is the pair of \(\tilde\sigma =E\sigma E^\dagger \).

2. For a tensor \(A\), matrices \(\sigma '_1,\dots ,\sigma '_b\) on the bond space, coefficients \(\alpha _j\), and \(M\geq 1\), the state \(V^{\otimes M}\sum _j\alpha _j\bigotimes _k|\omega (\sigma '_j)\rangle _{R_kL_{k+1}}\) has inner product \(\sum _j\overline{\alpha _j}\, \langle \phi _M(P_{\infty ,\sigma '_j}) | \phi _M(P) \rangle \) with \(|\phi _N(A)\rangle \), \(N=qM\), where \(P\) is the positive part of the blocked map and \(P_{\infty ,\sigma '_j}\) the fixed-point tensor of \(\sigma '_j\).

3. For blocks placed as in Lemma 35.17.2.3 and positive semidefinite \(\rho _j\), with \(\tilde\rho _j=E_j\rho _jE_j^\dagger \) and \(\Pi _j=E_jE_j^\dagger \), the mixed transfer map of \(P_\infty \) against the fixed-point tensor of \(\tilde\rho _j\) is \(X\mapsto \operatorname{Tr}(\Pi _jX)\, \tilde\rho _j\).

4. In the setting of Lemma 35.17.2.3, for every block \(j\) there is \(C_j{\gt}0\) such that for every \(q\geq 1\) and every \(M\geq 1\), with \(y=Me^{-\gamma q/\xi }\),

   \begin{align*} \bigl\lvert \langle \phi _M(P_{\infty ,\tilde\rho _j}) | \phi _M(P) \rangle -1\bigr\rvert \leq C_jy\, e^{C_jy}. \end{align*}

5. In the setting of Lemma 35.17.2.3 with \(0{\lt}\gamma {\lt}1\), there is \(K\) with \(\bigl\lvert \lVert \phi _N(A)\rVert ^2-b\bigr\rvert \leq Ke^{-\gamma N/\xi }\) for every \(N\geq 1\).

1. For a \(D^2\times D^2\) matrix \(H\), read as a tensor \(X\) with physical index the output pair, and a \(D\times D\) matrix \(\sigma \), the mixed transfer map of \(X\) against the fixed-point tensor \(P_{\infty ,\sigma }\) satisfies \(\operatorname{Tr}\sum _iX^i\sigma (P_{\infty ,\sigma }^i)^\dagger =\operatorname{Tr}\bigl(H(\sqrt\sigma ^{\, \mathsf T}\otimes \sigma )\bigr)\), which is real when \(H\) and \(\sigma \) are Hermitian.

2. For an isometry \(E\) placing a block, \(\sqrt{E\sigma E^\dagger }=E\sqrt\sigma E^\dagger \) for \(\sigma \ge 0\), \(\operatorname{Tr}(E\sigma E^\dagger )=\operatorname{Tr}(\Pi E\sigma E^\dagger )=\operatorname{Tr}\sigma \), and \(EXE^\dagger \, \Pi =EXE^\dagger \) for \(\Pi =EE^\dagger \).

3. For blocks placed as in Lemma 35.17.2.3 with weights one, \(A^iE_j=E_jA_j^i\), and the transfer map of \(A\) satisfies \(\mathcal{E}_A(E_jXE_j^\dagger )=E_j\mathcal{E}_{A_j}(X)E_j^\dagger \); in particular \(\tilde\rho _j=E_j\rho _jE_j^\dagger \) is a fixed point of \(\mathcal{E}_A\).

4. A linear map \(R:X\mapsto \operatorname{Tr}(QX)\sigma \) has trace \(\operatorname{Tr}(Q\sigma )\) and is idempotent when \(\operatorname{Tr}(Q\sigma )=1\), and \(RTR=\operatorname{Tr}(QT(\sigma ))R\) for every linear map \(T\). If \(\sqrt\sigma Q=\sqrt\sigma \), then \(\operatorname{Tr}(QY)=\operatorname{Tr}Y\) for every value \(Y\) of a mixed transfer map against \(P_{\infty ,\sigma }\).

5. The mixed transfer matrix \(R_j\) of \(P_\infty \) against \(P_{\infty ,\tilde\rho _j}\) is idempotent with trace one, and for every \(D^2\times D^2\) matrix \(H\), read as a tensor \(X\), the mixed transfer matrix \(\tau \) of \(X\) against \(P_{\infty ,\tilde\rho _j}\) satisfies \(R_j\tau R_j=\alpha R_j\) with \(\alpha =\operatorname{Tr}\sum _iX^i\tilde\rho _j(P_{\infty ,\tilde\rho _j}^i)^\dagger \).

6. For a state \(\sigma \), the vector \((P_{\infty ,\sigma }^i\sqrt\sigma )_i\) has norm one, and so has \((X^i\sqrt\sigma )_i\) whenever \(\operatorname{Tr}\mathcal{E}_X(\sigma )=1\). If \(\lvert \lVert u_q\rVert -c\rvert \le K'x^q\) and \(\lVert u_q-v\rVert \le Kx^q\) for every \(q\ge 1\), with \(0\le x{\lt}1\), then \(\lVert v\rVert =c\).

7. In the setting of Lemma 35.17.2.3 with weights one, if \(\lambda _2\neq 0\), then for every block \(j\) there is \(C_j{\gt}0\) such that for every \(q\ge 1\) and every \(M\ge 2\) with \(C_jy{\lt}1\), where \(y=Me^{-2\gamma q/\xi }\),

   \begin{align*} \bigl\lvert \langle \phi _M(P_{\infty ,\tilde\rho _j}) | \phi _M(P) \rangle -1\bigr\rvert \le C_jy\, e^{C_jy}. \end{align*}

1. For blocks placed as in Lemma 35.17.2.3, positive semidefinite \(\rho _k\) and reals \(t_k\geq 0\), the mixed transfer map of \(P_\infty ^{t}=\sum _kK_k\bigl(\sqrt{t_k^2\rho _k}^{\, \mathsf T}\otimes \mathbb {1}\bigr) K_k^\dagger \) against the fixed-point tensor of \(\tilde\rho _j\) is \(X\mapsto t_j\operatorname{Tr}(\Pi _jX)\, \tilde\rho _j\).

2. For blocks placed as there, positive semidefinite \(\rho _k\) of trace one, and a block \(j\), there is \(C{\gt}0\) such that for all reals \(t_k\geq 0\) with \(t_j\leq 1\), all \(\delta \), all \(M\geq 1\), and every matrix \(P\) on the bond pairs with \(\lVert P-P_\infty ^{t}\rVert \leq \delta \),

   \begin{align*} \bigl\lvert \langle \phi _M(P_{\infty ,\tilde\rho _j}) | \phi _M(P) \rangle -t_j^M\bigr\rvert \leq C(M\delta )\, e^{CM\delta }. \end{align*}

3. In the setting of Lemma 35.17.2.3 with \(0{\lt}\gamma {\lt}1\), there is \(K\) such that for all weights with \(\lvert \mu _j\rvert \leq 1\) and every \(N\geq 1\), \(\bigl\lvert \lVert \phi _N(A)\rVert ^2-\sum _j\lvert \mu _j^N\rvert ^2\bigr\rvert \leq Ke^{-\gamma N/\xi }\) for \(A=\bigoplus _j\mu _jA_j\).

Let \(A^i=\bigoplus _{j=1}^b\mu _jA_j^i\) be the direct sum with weights \(\mu _j\in \mathbb {C}\) of blocks \(A_j\) of bond dimensions \(D_j\), each placed on its own bond coordinates by an isometry \(E_j\), and put \(K_j=E_j\otimes E_j\). Let every \(A_j\) be normal in the gauge (3) with positive definite fixed point \(\rho _j\) of trace one. Let \(\lambda _2\in \mathbb {C}\) with \(\lvert \lambda _2\rvert {\lt}1\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\) and the moduli of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), let \(\xi \) be its correlation length, and let \(0{\lt}\gamma {\lt}1\). For \(q\geq 1\) the blocked map of \(A\) is \(\mathcal B=\sum _j\mu _j^q\mathcal B_jK_j^\dagger \), with \(\mathcal B_j\) the blocked map of \(A_j\), so \(\mathcal B^\dagger \mathcal B=\sum _{j,j'}\overline{\mu _j^q}\mu _{j'}^q K_j\mathcal B_j^\dagger \mathcal B_{j'}K_{j'}^\dagger \). There is \(K\) such that for all weights with \(\lvert \mu _j\rvert \leq 1\) and every \(q\geq 1\)

For weights one, \(\mathcal B=\sum _j\mathcal B_jK_j^\dagger \), and with

there are \(K,K'\) such that for every \(q\geq 1\)

where \(P\) is the positive part of \(\mathcal B\). This replaces the block form asserted in [ 219 , eq. (S5) ] , which holds only when the states of distinct blocks on \(q\) sites are orthogonal [ 142 ] .

Let \(A\) be normal with \(D\ge 1\). There is \(N_0\) such that \(|\phi _N(A)\rangle \neq 0\) for every \(N\ge N_0\).

Let \(A\) be a tensor, \(n_1,n_2\ge 0\), and let \(W:\mathbb {C}^{D^2}\to \mathbb {C}^{D^2}\otimes \mathbb {C}^{D^2}\) be the partial isometry of the polar decomposition of the tensor \((a,b)\mapsto P_{n_1}^aP_{n_2}^b\) of two neighbouring sites. Then

and \(P_{n_1+n_2}\) is the positive part of the tensor \((a,b)\mapsto P_{n_1}^aP_{n_2}^b\). No injectivity is assumed, as in the footnote of  [ 219 , paragraph “The sequential-RG circuit” ] . For \(n_1=n_2=n\) the tensor \((a,b)\mapsto P_n^aP_n^b\) is the two-site blocked tensor of \(P_n\), and \(W\) is the partial isometry of its polar decomposition. For \(n=2^j\), \(j\ge 1\), the positive part \(P_n\) is the tensor \(T_j\) of the tree of [ 219 , eq. (16) ] , so \(W\) is its layer \(V^{(j+1)}\). If the blocked tensor over \(n_1+n_2\) sites is injective, so is the tensor \((a,b)\mapsto P_{n_1}^aP_{n_2}^b\), and \(W\) is an isometry.

Let \(A\) be normal in the gauge (3) with correlation length \(\xi \), and let \(0{\lt}\gamma {\lt}1\). Then, for \(N=qM\),

for \(q\ge 1\) [ 227 , Supplemental Material, eq. (S29) ] , quoted as [ 219 , eq. (S9) ] . The sources state this for \(0{\lt}\gamma {\lt}1/2\); the range \(0{\lt}\gamma {\lt}1\) follows from the Lipschitz bound of Theorem 35.3.7. Explicitly, there is a constant \(C{\gt}0\) depending only on \(A\), \(\rho \), and \(\gamma \) such that \(\bigl\lvert 1-\lvert \langle \phi _M(P_\infty )|\phi _M(P_q)\rangle \rvert \bigr\rvert \le Cy\, e^{Cy}\) for all \(q\ge 0\) and \(M\ge 1\), where \(y=M\, e^{-\gamma q/\xi }\). The same bound holds for the overlap itself: \(\bigl\lvert \langle \phi _M(P_\infty )|\phi _M(P_q)\rangle -1\bigr\rvert \le Cy\, e^{Cy}\).

For a Hermitian idempotent \(E\) on \(\mathbb {C}^m\) there are \(r\le m\) and a unitary \(T\) with \(ET=T\Lambda \), where \(\Lambda \) is the diagonal matrix with \(\Lambda _{yy}=1\) for \(y{\lt}r\) and \(\Lambda _{yy}=0\) otherwise. For the polar decomposition \(B=VP\) of a tensor, with \(V^\dagger V=\Pi \) as in Theorem 35.1.3.3, \(V\Pi =V\).

The state on registers of a binary finite-range network with sites of dimension \(d^s\) is a unit vector.

With the notation of Lemma 35.17.3.3, let \(q\geq 1\), write \(H_{kj}=L_k^\dagger PC^{-1}L_j\) for the relative blocks, and, for a positive semidefinite \(\sigma \), write \(\lVert X\rVert _\sigma ^2=\operatorname{Tr}\sum _iX^i\sigma (X^i)^\dagger \) for a tensor \(X\) read from a matrix. Let \(\sigma _j\) be a positive semidefinite fixed point of \(\mathcal{E}_{A_j}\) of trace one, \(F_j\) its fixed-point tensor, and \(\alpha _j=\operatorname{Tr}\mathcal{E}_{H_{jj}F_j}(\sigma _j)\).

1. \(\sum _kH_{kj}^\dagger H_{kj}=\mathcal B_j^\dagger \mathcal B_j\), where \(\mathcal B_j\) is the physical matrix of the \(q\)-blocked tensor \(A_j\), and \(\sum _k\lVert H_{kj}\rVert _{\sigma _j}^2=1\).

2. \(\alpha _j\) is real, \(0\leq \alpha _j\leq 1\), and \(1-\alpha _j=\tfrac 12\bigl(\lVert H_{jj}-F_j\rVert _{\sigma _j}^2 +\sum _{k\neq j}\lVert H_{kj}\rVert _{\sigma _j}^2\bigr)\).

3. There is \(C{\gt}0\) such that for all nonzero weights, all \(q\geq 1\) and all \(\delta \), if every \(H_{kj}\) is within \(\delta \) in operator norm of its limit from Lemma 35.17.3.3(2), then \(\lvert \alpha _j\rvert \leq 1\) and \(\lvert 1-\alpha _j\rvert \leq C\delta ^2\).

4. Under the hypotheses of Lemma 35.17.3.3(3), for every \(0{\lt}\gamma {\lt}1\) there are \(C{\gt}0\) and \(q_0\) such that for all nonzero weights and all \(q\geq \max (q_0,1)\), \(\lvert 1-\alpha _j\rvert \leq Ce^{-2\gamma q/\xi }\).

With the notation of Lemma 35.17.3.3, let \(q\geq 1\) and \(M\geq 1\), and put \(N=qM\).

1. The letters of the tensor read from \(YK^\dagger \), for a coordinate isometry \(K\) placing a bond pair, are those of the tensor read from \(Y\), placed on the bond coordinates of \(K\).

2. \(L_j^\dagger P=\sum _{j'}\sum _k\mu _{j',k}^q\, (L_j^\dagger PC^{-1}L_{j'}) K_{j',k}^\dagger \), so the tensor read from \(L_j^\dagger P\) is the direct sum over the copies \((j',k)\), with the weights \(\mu _{j',k}^q\), of the tensors read from the blocks \(L_j^\dagger PC^{-1}L_{j'}\).

3. For every tensor \(F\) on the bond pairs of block \(j\), \(\langle \phi _M(F) | \phi _M(L_j^\dagger P) \rangle =\sum _{j'}\beta _{j'}\langle \phi _M(F) | \phi _M(L_j^\dagger PC^{-1}L_{j'}) \rangle \) with \(\beta _{j'}=\sum _k\mu _{j',k}^N\).

4. For positive semidefinite \(\rho _j\) of trace one and blocks \(j,j'\) there is \(C{\gt}0\) such that for all nonzero weights, all \(q\), \(\delta \), \(M\geq 1\), and every \(G\) with \(\lVert G-L_j^\dagger P_\infty C^{-1}L_{j'}\rVert \leq \delta \), the overlap of the tensor read from \(G\) with the fixed-point tensor of \(\rho _j\) on \(M\) sites is \(\delta _{jj'}\) up to \(C(M\delta )\, e^{CM\delta }\).

With the notation of Lemma 35.17.3.5, let \(\sigma _k\) be a positive semidefinite fixed point of \(\mathcal{E}_{A_k}\) of trace one for every block \(k\), and let \(j,j'\) be blocks. There is \(C{\gt}0\) such that for all nonzero weights, all \(q\geq 1\), all \(\delta \) and all \(M\geq 2\): if every relative block \(H_{kl}\) is within \(\delta \) of its limit and \(CM\delta ^2{\lt}1\), then the overlap of the tensor read from \(H_{jj'}\) with \(F_j\) on \(M\) sites is \(\delta _{jj'}\) up to \(CM\delta ^2e^{CM\delta ^2}\). No quadratic bound is asserted for \(M=1\).

1. Let \(L_j\) be isometries with orthogonal ranges in dimension \(n\), \(c_j{\gt}0\), \(C=\sum _jc_jL_jL_j^\dagger \) and \(C^{-1}=\sum _jc_j^{-1}L_jL_j^\dagger \), let \(h_j\geq 0\) with \(h_j^2=g_j\geq \lambda \mathbb {1}\) for some \(\lambda {\gt}0\), and let \(\mathcal B=\widetilde{\mathcal B}C\). If \(\Delta =\widetilde{\mathcal B}^\dagger \widetilde{\mathcal B} -\sum _jL_jg_jL_j^\dagger \) satisfies \(\lVert \Delta \rVert \leq \lambda /2\), the positive part \(P\) of \(\mathcal B\) satisfies \(\lVert (P-S)C^{-1}\rVert \leq n^2(\lambda /2)^{-1/2}\lVert \Delta \rVert \) with \(S=\sum _jc_jL_jh_jL_j^\dagger \).

2. With the notation of Theorem 35.17.1.2 and Lemma 35.17.3.1, \(C^{-1}L_j=c_j^{-1}L_j\), the block \(L_j^\dagger PC^{-1}L_{j'}\) equals \(c_{j'}^{-1}L_j^\dagger PL_{j'}\), and the block \(L_j^\dagger P_\infty C^{-1}L_{j'}\) is \(\sqrt{\rho _j}^{\, \mathsf T}\otimes \mathbb {1}\) for \(j=j'\) and zero otherwise.

3. Let the blocks be normal in the gauge (3) with positive definite fixed points \(\rho _j\) of trace one, let \(\lambda _2\) with \(\lvert \lambda _2\rvert {\lt}1\) bound the moduli of the eigenvalues other than one of every \(\mathcal{E}_{A_j}\) and of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), and let \(0{\lt}\gamma {\lt}1\) with \(e^{-\gamma /\xi }{\lt}1\). There are \(K\) and \(q_0\) such that for all nonzero weights and all \(q\geq \max (q_0,1)\), \(\lVert (P-P_\infty )C^{-1}\rVert \leq Ke^{-\gamma q/\xi }\), and every block \(L_j^\dagger PC^{-1}L_{j'}\) is \(L_j^\dagger P_\infty C^{-1}L_{j'}\) up to \(Ke^{-\gamma q/\xi }\).

If \(0{\lt}q\leq N\), the block lengths in Definition 35.8.3.1 sum to \(N\).

With the notation of Theorem 35.17.1.2, let the blocks be as in Lemma 35.17.2.3, with \(c_j\) and \(L_j\) the norms and isometries of the copies on \(q\) sites, and put

1. \(P_\infty \) is positive semidefinite, its square is \(\sum _jc_j^2L_j(\rho _j^{\mathsf T}\otimes \mathbb {1})L_j^\dagger \), and for \(0{\lt}\gamma {\lt}1/2\) there is \(K\) such that for all weights with \(\lvert \mu _{j,k}\rvert \leq 1\) and every \(q\geq 1\) the positive part \(P\) of the blocked map satisfies \(\lVert P-P_\infty \rVert \leq Ke^{-\gamma q/\xi }\).

2. The mixed transfer map of \(L_j^\dagger P_\infty \), read as a tensor whose physical leg is a bond pair of block \(j\), against the fixed-point tensor of \(\rho _j\) is \(X\mapsto \sum _k\mu _{j,k}^q\operatorname{Tr}(E_{j,k}^\dagger X)\, E_{j,k}\rho _j\).

3. For a block \(j\) with at least one copy there is \(C_j{\gt}0\) such that for all weights with \(\lvert \mu _{j,k}\rvert \leq 1\), all \(q\), \(\delta \), \(M\geq 1\) and every \(G\) with \(\lVert G-L_j^\dagger P_\infty \rVert \leq \delta \), the overlap of \(G\) with the fixed-point tensor of \(\rho _j\) on \(M\) sites is \(\beta _j=\sum _k\mu _{j,k}^{qM}\) up to \(C_j(M\delta )\, e^{C_jM\delta }\).

4. The unnormalized corrected state has inner product \(\sum _j\overline{\alpha _j}\langle \phi _M(P_{\infty ,\rho _j}) | \phi _M(L_j^\dagger P) \rangle \) with \(|\phi _N(A)\rangle \), and for isometries \(L_j\) with orthogonal ranges, \(\beta \neq 0\) and \(\alpha _j=\beta _j/(\sum _l\lvert \beta _l\rvert ^2)^{1/2}\) it has norm at most one.

In Theorem 35.4.1, let the matrix elements of \(V\) have the stated form only for the first \(r\) inputs \(x\), and let these \(r\leq D^2\) columns of \(V\) be orthonormal. Then the same conclusion holds with \(b_q=r\), for these \(r\) inputs. For an injective blocked tensor \(B=VP\) of tensors \(A_1,\dots ,A_q\), one on each site, \(V=BP^{-1}\) has the matrix elements \(\langle i_1\cdots i_q|V|x\rangle =\sum _{\alpha ,\beta } (A_1^{i_1}\cdots A_q^{i_q})_{\alpha \beta }(P^{-1})_{(\alpha ,\beta ),x}\); for equal tensors \(A_p=A\) this reads \(\sum _{\alpha ,\beta }(A^{i_1}\cdots A^{i_q})_{\alpha \beta } (P^{-1})_{(\alpha ,\beta ),x}\) with \(B=\mathcal B_q\).

Let \(A_1,\dots ,A_b\) be tensors with \(d\) physical indices and bond dimensions \(D_j\), normal in the gauge (3) with positive definite fixed points \(\rho _j\) of trace one, with \(\mathcal B_j\) the blocked map of \(A_j\) on \(q\) sites. Let \(\lambda _2\in \mathbb {C}\) with \(\lvert \lambda _2\rvert {\lt}1\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\) and the moduli of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), and let \(\xi \) be its correlation length.

1. A matrix \(L\) with \(L^\dagger L=\mathbb {1}\) has operator norm at most one, and for isometries \(L_j\) with \(L_j^\dagger L_{j'}=0\) for \(j\neq j'\), \(L_j^\dagger \sum _kL_kY_k=Y_j\) and \(\bigl(\sum _jL_jY_jL_j^\dagger \bigr)\bigl(\sum _jL_jZ_jL_j^\dagger \bigr) =\sum _jL_jY_jZ_jL_j^\dagger \).

2. For matrices \(W,W'\) with the same rows and vectors \(\psi ,\varphi \) on \(M\) sites, \((W^{\otimes M})^\dagger =(W^\dagger )^{\otimes M}\), \(\langle W^{\otimes M}\psi | W'^{\otimes M}\varphi \rangle =\langle \psi | (W^\dagger W')^{\otimes M}\varphi \rangle \); if \(W^\dagger W\) is idempotent, then \(\lVert W^{\otimes M}\psi \rVert \leq \lVert \psi \rVert \); and on \(M\geq 1\) sites \(\mathbb {1}^{\otimes M}=\mathbb {1}\) and \(0^{\otimes M}=0\).

3. For a coordinate isometry \(E\), \(E^\dagger =E^{\mathsf T}\); for rectangular \(X\) and \(Y\), the entry of \(X^{\mathsf T}\otimes Y\) at \(((l,r),(a,c))\) is that of \(X|l\rangle \langle r|Y\) at \((a,c)\).

4. For \(0{\lt}\gamma {\lt}1\) there is \(K\) such that for all \(R\), all coefficients \(\lvert c_j\rvert \leq R\), all matrices \(L_j\) of norm at most one and every \(q\), the matrix \(\mathcal B=\sum _jc_j\mathcal B_jL_j^\dagger \) satisfies \(\bigl\lVert \mathcal B^\dagger \mathcal B-\sum _j\lvert c_j\rvert ^2 L_j(\rho _j^{\mathsf T}\otimes \mathbb {1})L_j^\dagger \bigr\rVert \leq R^2Ke^{-\gamma q/\xi }\).

5. For every matrix \(\mathcal B\) and positive semidefinite \(P_\infty \), the positive part \(P\) of \(\mathcal B\) satisfies \(\lVert P-P_\infty \rVert \leq \lVert \mathcal B^\dagger \mathcal B-P_\infty ^2 \rVert ^{1/2}\).

6. For \(0{\lt}\gamma {\lt}1\) there is \(K\) such that for all \(R\), all coefficients \(\lvert \beta _j\rvert \leq R\) and every \(N\), \(\bigl\lvert \lVert \sum _j\beta _j\phi _N(A_j)\rVert ^2-\sum _j\lvert \beta _j\rvert ^2 \bigr\rvert \leq R^2Ke^{-\gamma N/\xi }\).

7. For a tensor \(A\) with blocked map \(\mathcal B=VP\) on \(q\) sites and any vector \(F\) on the bond pairs of \(M\) sites, \(\langle V^{\otimes M}F | \phi _N(A) \rangle =\langle F | \phi _M(P) \rangle \) with \(N=qM\). If \(\lVert v\rVert \leq 1\) and \(t\geq 0\), then \(t^{-1}\lvert \langle v | w \rangle \rvert \leq \bigl\lvert \langle v | w \rangle \bigr\rvert / (\lVert v\rVert t)\), the right side read as zero when \(v=0\) or \(t=0\).

8. In a normed ring, if \(\lVert E^k\rVert \leq c\) for every \(k\geq 0\) and \(\lVert X_k-E\rVert \leq \delta \) for every \(k\), then \(\lVert X_1\cdots X_M-E^M\rVert \leq c((1+c\delta )^M-1)\). Consequently, for a tensor \(F\) of bond dimension \(D_2\) and \(c\geq 0\) there is \(C{\gt}0\) such that for all \(G,G'\) from bond pairs of dimension \(D_1\) to bond pairs of dimension \(D_2\) with \(\lVert G-G'\rVert \leq \delta \), if the powers of the mixed transfer map \(T'\) of \(G'\) against \(F\) satisfy \(\lVert T'^kX\rVert \leq c\lVert X\rVert \), then the overlap of \(G\) with \(F\) on \(M\) sites is \(\operatorname{Tr}T'^M\) up to \(C(M\delta )\, e^{CM\delta }\).

Let \(a,b\) be positive elements of a C\(^*\)-algebra, and let \(c,s\ge 0\) be real numbers with \(b\ge c\, \mathbb {1}\). Write \(\varepsilon =\lVert a-b\rVert \).

1. If \(\varepsilon \le 2s\sqrt c+s^2\), then \(\sqrt a-\sqrt b\le s\).

2. If \(s\le \sqrt c\) and \(\varepsilon +s^2\le 2s\sqrt c\), then \(\sqrt b-\sqrt a\le s\).

Let \(P\) and \(S\) be positive semidefinite \(n\times n\) matrices, let \(\lambda {\gt}0\) and \(R\) satisfy \(\lambda RR^\dagger \leq P^2\), and let \(X=PW_1+W_2S\) satisfy \(PX+XS=RZ\). Then \(\lVert X\rVert \leq n^2\lambda ^{-1/2}\lVert Z\rVert \).

The tree map (58) with finest layer \(W\) and coarser layers \(V_0,V_1,\dots \), at \(2^{j+1}\) leaves grouped into neighbouring pairs \(e_0,\dots ,e_{2^j-1}\), is \(\sum _{e'}\prod _pW(e_p,e'_p)\) times the tree map of the coarser layers at the outputs \(e'\) grouped into pairs in their turn, the sum running over these outputs.

Let the two-site blocked tensor of \(A\) be injective, and let \(V^{(j)}\) and \(T_j\) be as in Theorem 35.4.15. Then for every \(j\ge 1\) the two-site blocked tensor of \(T_{j-1}\) is injective, so each \(V^{(j)}\) is an isometry, and the tree map (58) with \(W=V^{(1)}\) and \(W_i=V^{(i+2)}\) is the product of the layers in (12) for \(q=2^{k+1}\).

Let \(D\ge 1\) and let \(\rho \ge 0\) with \(\operatorname{tr}\rho =1\). Then there is a unitary \(u\) on \(\mathbb {C}^D\otimes \mathbb {C}^D\) with \(u|0\rangle |0\rangle =|\omega \rangle \), the pair of the fixed-point state.

Let \(X,Y\) be operators on \((\mathbb {C}^d)^{\otimes L}\) with \(L\ge 1\), let \(m,n\ge 0\) and \(N=2L+m+n\), and let \(X_1\) act on the sites \(1,\dots ,L\) and \(Y_{L+m+1}\) on the sites \(L+m+1,\dots ,2L+m\) of a ring of \(N\) sites. Then

The expectation in the normalized vector \(|\phi _N\rangle \) is \(\langle \phi _N(A)|O|\phi _N(A)\rangle /\langle \phi _N(A) | \phi _N(A) \rangle \).

Let \(\mu _1,\dots ,\mu _K\) be distinct complex numbers of modulus one and \(a\in \mathbb {C}^K\) nonzero. There is \(c_0{\gt}0\) such that for every integer \(t\ge 0\) some \(u\in \{ 0,\dots ,K-1\} \) satisfies \(\bigl\lvert \sum _{j=1}^Ka_j\mu _j^{\, t+u}\bigr\rvert \ge c_0\). This gives the lower bound of (49) for some positive constant \(c_0\), not necessarily the explicit constant displayed there.

Let \(A\) be in the gauge (3), with the products of \(L\) matrices spanning \(M_{D}(\mathbb {C})\), and let \(\lvert \lambda _2\rvert {\lt}r{\lt}1\) and \(M\ge 0\). For an operator \(X\) on \((\mathbb {C}^d)^{\otimes w}\) let \(X^{(k)}\) act on the sites \(k\Delta +1,\dots ,k\Delta +w\) of the ring of \(N\) sites, for \(0\le k{\lt}n\) with \(n\Delta \le N\), and let \(Z=\sum _kc_kX^{(k)}\) with real \(\lvert c_k\rvert \le \gamma \).

1. Expectations in \(|\phi _N\rangle \) of operators on one window, or on two windows of the same length at a fixed distance, do not depend on the position of the windows, as long as the windows lie in the sites \(1,\dots ,N\) without wrapping around the ring.

2. In the gauge (3), \(\lVert \mathcal{E}_X\rVert \le D^3\lVert X\rVert \) whatever the number of sites \(X\) acts on.

3. There is \(C\) such that for \(\lVert X\rVert ,\lVert Y\rVert \le M\) on windows of \(w\) sites separated by \(g\ge 1\) and \(g'\ge 1\) sites around the ring, the connected correlation in \(|\phi _N\rangle \) is at most \(C(r^g+r^{g'})\).

4. There is \(C'\) such that for Hermitian \(X\) with \(\lVert X\rVert \le M\) and \(w{\lt}\Delta \), and for every \(N\) with \(|\phi _N(A)\rangle \neq 0\), \(\lVert (Z-\langle Z\rangle _{\phi _N})\phi _N\rVert ^2\le 3n\gamma ^2C'\).

5. If \(|\psi \rangle \) is a unit vector prepared in depth \(T\) and \(w+2T\le \Delta \), then \(\lVert (Z-\langle Z\rangle _\psi )\psi \rVert ^2\le 6n\gamma ^2\lVert X\rVert ^2\).

A product vector whose site vectors at the sites of a set \(Z\) are multiples of \(|0\rangle \) has \(|0\rangle \) at the sites of \(Z\), in the sense of Lemma 33.5.3.

The two constructions of Definition 35.1.2.1 are mutually inverse; in particular a tensor is determined by its physical matrix.

For \(W\in \mathbb {C}^{m\times n}\), the physical matrix of \(B_W\) is \(W\widehat B\).

The physical matrix of the positive-part tensor is \(P\), with its rows indexed by \(\{ 0,\dots ,D^2-1\} \).

For \(M\in \mathbb {C}^{m\times d}\) and a length-\(N\) physical configuration \(s=(s_0,\ldots ,s_{N-1})\in \{ 0,\ldots ,m-1\} ^N\), physical rotation satisfies

For \(\Lambda ,\Lambda '\in M_{d}(\mathbb {C})\),

For \(\Lambda \in M_{d}(\mathbb {C})\), \(|V^{(N)}(\Lambda A)\rangle =\Lambda ^{\otimes N}|V^{(N)}(A)\rangle \) and \(\Gamma _L^{\Lambda A}(X)=\Lambda ^{\otimes L}\, \Gamma _L^{A}(X)\) for every boundary matrix \(X\in M_{D}(\mathbb {C})\).

Let \(\varepsilon _N\to 0\), let \(|\varphi _N\rangle \) be unit vectors for all large \(N\), and suppose that for all large \(N\) some unit vector \(|\psi \rangle \) with \(1-\lvert \langle \psi |\varphi _N | \rvert \rangle \leq \varepsilon _N\) is prepared with measurements and a circuit in depth at most \(f(N)\). Then \((|\varphi _N\rangle )_N\) is asymptotically prepared with measurements in depth \(f\).

For a layer \(W\) with gates \(g_k\), the layer with the gates \(g_k^\dagger \) on the same bonds implements \(W^\dagger \), and the same holds for the product of the gates of \(W\) over any subset of its bonds. If a circuit has the layers \(W_1,\dots ,W_T\), applied in this order, and implements \(U\), then the circuit with the layers \(W_T^\dagger ,\dots ,W_1^\dagger \) implements \(U^\dagger \); the layers of a circuit implementing \(U\) followed by those of a circuit implementing \(U'\) implement \(U'U\). Consequently, a local circuit of depth \(T\) is unitary, and its adjoint is a local circuit of depth \(T\). If \(U\) and \(U'\) are local circuits of depths \(T\) and \(T'\), then \(U'U\) is a local circuit of depth \(T+T'\).

Let \(X_k\) be operators on pairwise disjoint sets \(S_k\) of sites, each acting by a map \(f_k\) of the configurations of \(S_k\), that is \(X_kv=v\circ f_k\) for every vector \(v\); then \(X_k\) has the entry \(1\) at \((u,f_k(u))\) and \(0\) elsewhere. The product of the \(X_k\) acts by the map \(\rho \) with \(\rho (x)|_{S_k}=f_k(x|_{S_k})\) for every \(k\) and \(\rho (x)_i=x_i\) at every site \(i\) outside all the \(S_k\). A permutation \(\tau \) of the sites acts as \(v\mapsto v\circ (x\mapsto x\circ \tau )\). If \(X\) acts on the sites of \(S\) and the product vector \(|\pi \rangle \) is \(|z\rangle \) on every site of \(S\), then \((X|\pi \rangle )(y)=X_{y|_S,(z,\dots ,z)}\pi '(y)\), where \(|\pi '\rangle \) is the product vector with the all-ones vector at the sites of \(S\) and the vectors of \(|\pi \rangle \) elsewhere.

Let \(|\psi \rangle \) be a unit vector prepared in depth \(T\) on a bond geometry, and let \(Z\), \(Z'\) act on sets of sites whose light cones of radius \(T\) are disjoint, for instance on the ring sets at distance larger than \(2T\). Then \(\langle \psi |ZZ'|\psi \rangle =\langle \psi |Z|\psi \rangle \langle \psi |Z'|\psi \rangle \). Without the normalization, every vector \(|\psi \rangle \) prepared in depth \(T\) satisfies \(\langle \psi |ZZ'|\psi \rangle \langle \psi | \psi \rangle =\langle \psi |Z|\psi \rangle \langle \psi |Z'|\psi \rangle \). This is the statement in the proof of [ 219 , Theorem 1 ] that every connected correlation of such operators vanishes. Its two ingredients on product vectors hold for any finite set of sites: the expectation of \(\bigotimes _im_i\) in \(|\pi \rangle =\bigotimes _i|v_i\rangle \) is \(\prod _i\langle v_i|m_i|v_i\rangle \), and \(\langle \pi |AB|\pi \rangle \langle \pi | \pi \rangle =\langle \pi |A|\pi \rangle \langle \pi |B|\pi \rangle \) for \(A\), \(B\) acting on disjoint sets of sites.

The output of the protocol of a round and a product vector \(|\pi \rangle \) for the outcome \(m\) is \(V_{m}P_{m}U|\pi \rangle \). For a nonzero vector some outcome of a round gives a nonzero vector, so a nonzero vector has a nonzero output under every sequence of rounds, and a vector prepared with measurement rounds is nonzero. The outputs of a multiple \(c|v\rangle \) are the multiples by \(c\) of the outputs of \(|v\rangle \), and the outputs of a concatenation of two sequences are the outputs of the second sequence from the outputs of the first. A vector prepared with measurements in depth \(T\) is prepared with one measurement round in depth \(T\).

A nonzero vector prepared in depth \(T\) is prepared with measurements in depth \(T\). Conversely, if a protocol measuring no site, with \(T_1\leq T\), prepares \(|\psi \rangle \), then there are unitaries \(u_i\) on the sites \(i\) such that \(\bigotimes _iu_i|\psi \rangle \) is prepared in depth \(T\).

A layer acts inside a set of sites \(R\) if each of its bonds is contained in \(R\), and \(U\) is a local circuit of depth \(T\) inside \(R\) if it is implemented by \(T\) layers acting inside \(R\). If \(U\) and \(U'\) are local circuits inside \(R\) of depths \(T\) and \(T'\), then \(U'U\) is a local circuit inside \(R\) of depth \(T+T'\). For a bond geometry whose bonds are nonempty, such as the ring, if \(U_i\), \(i\in I\) for a finite set \(I\), are pairwise commuting local circuits of depth \(T\) inside pairwise disjoint sets \(R_i\), then their product is a local circuit of depth \(T\) inside \(\bigcup _{i\in I}R_i\). Let \(e\) be an injective map from the sites of an open chain of \(m\) sites into the ring with \(e(j+1)=e(j)+1\). If \(X\) is a product of at most \(K\) unitaries, each acting on two neighbouring sites of the open chain, then \(X\) placed by \(e\) is a local circuit of depth \(K\) inside the range of \(e\). For a list of such maps \(e_i\) with pairwise disjoint ranges and such products \(X_i\), the product of the \(X_i\) placed by \(e_i\) is a local circuit of depth \(K\) inside the union of the ranges.

Let \(d\ge 1\), and let \(W_1,\dots ,W_n\) be windows of \(m\) sites of a chain, that is maps from \(\{ 1,\dots ,m\} \) into the sites such that \((p,t)\mapsto W_p(t)\) is injective; then every \(W_p\) is injective and the ranges of distinct windows are disjoint. For configurations \(c_p\) of \(m\) sites, the configuration with \(c_p\) on the windows carries \(c_p(t)\) at \(W_p(t)\) and \(0\) at every other site. It depends injectively on \((c_1,\dots ,c_n)\), and every configuration with \(0\) outside the windows is the configuration with its own restrictions on the windows.

1. Let \(Y_1,\dots ,Y_n\) be matrices on \(m\) sites with \(Y_p|\iota _p(c)\rangle =\sum _eV_p(e,c)|o_p(e)\rangle \) for injective maps \(o_p\). Then the product of the matrices \(Y_p\) placed by the windows maps the configuration with \(\iota _p(c_p)\) on the windows to \(\sum _e\prod _pV_p(e_p,c_p)\) times the configuration with \(o_p(e_p)\) on the windows.

2. Let \(R\) and \(R'\) be injective maps into the basis of a space. If \(M|x_0\rangle =\sum _af(a)|R(a)\rangle \) and \(G|R(a)\rangle =\sum _bg(a,b)|R'(b)\rangle \) for every \(a\), then \(GM|x_0\rangle =\sum _b\bigl(\sum _ag(a,b)f(a)\bigr)|R'(b)\rangle \).

For a map \(e\) from the \(m\) sites of a chain into the \(n\) sites of another chain and an operator \(X\) on \((\mathbb {C}^d)^{\otimes m}\), the operator \(X\) placed by \(e\) has the matrix element \(X_{x\circ e,\, y\circ e}\) at the pair of configurations \((x,y)\) if \(x\) and \(y\) agree at every site outside the range of \(e\), and \(0\) otherwise; for injective \(e\) it is \(X\) acting on the sites \(e(1),\dots ,e(m)\) and the identity on the other sites. Placement is defined in the same way for a map \(e\colon J\to I\) of any finite sets of sites; for injective \(e\), placing a product \(\bigotimes _{j\in J}m_j\) gives the product with \(m_j\) at the site \(e(j)\) and \(\mathbb 1\) off \(e(J)\). For an injective map \(f\), placing by \(f\) the operator \(X\) placed by \(e\) places \(X\) by \(f\circ e\). Operators placed by injective maps between chains with disjoint ranges commute. If \(e_k\), \(k\in K\), are injective maps with pairwise disjoint ranges and \(X_k\) are operators on the corresponding chains, the product of the operators \(X_k\) placed by \(e_k\) has the matrix element \(\prod _{k\in K}(X_k)_{x\circ e_k,\, y\circ e_k}\) at \((x,y)\) if \(x\) and \(y\) agree at every site outside all the ranges, and \(0\) otherwise.

Let \(d\ge 1\). For one-site operators \(m_i\) and vectors \(|v_i\rangle \) in \(\mathbb {C}^d\) on any finite set of sites, \((\bigotimes _im_i)(\bigotimes _i|v_i\rangle )=\bigotimes _im_i|v_i\rangle \), and \(\bigotimes _ic_i|v_i\rangle =(\prod _ic_i)\bigotimes _i|v_i\rangle \) for scalars \(c_i\). Every nonzero \(x\in \mathbb {C}^d\) equals \(c\, u|0\rangle \) for a unitary \(u\) and a scalar \(c\neq 0\), and every nonzero product vector is a scalar multiple of \((\bigotimes _iu_i)|0\cdots 0\rangle \) for one-site unitaries \(u_i\). A tensor product \(\bigotimes _iu_i\) of one-site unitaries is a local circuit of depth \(2\). Every nonzero vector prepared in depth \(T\) is a scalar multiple of \(U|0\cdots 0\rangle \) for a local circuit \(U\) of depth \(T+2\).

If a round \(R\) implements \(W\) on \(E\) and \(|v\rangle \in E\), every output of \(R\) followed by rounds \(R'\) is a multiple of an output of \(R'\) from \(W|v\rangle \). The empty sequence implements \(\mathbb 1\) on every set, and a sequence implementing \(W\) on \(E\) implements it on every subset of \(E\). If a sequence \(\mathcal R\) implements \(W\) on \(E\), \(W\) maps \(E\) into \(E'\), and a sequence \(\mathcal R'\) implements \(W'\) on \(E'\), then \(\mathcal R\) followed by \(\mathcal R'\) implements \(W'W\) on \(E\). If \(\mathcal R\) implements \(W\) on \(E\) and the product vector \(|\pi \rangle \neq 0\) lies in \(E\), then \(W|\pi \rangle \) is prepared with measurement rounds in the total depth of \(\mathcal R\).

A vector \(|v\rangle \) has \(|0\rangle \) at the sites of a set \(Z\) if \(v(x)=0\) for every configuration \(x\) with \(x_i\neq 0\) for some \(i\in Z\). A permutation \(\pi \) of the sites acts on vectors by \((P_\pi v)(x)=v(x\circ \pi )\). These two notions, the second and third items below, and the statement on scalars in the first item hold for any finite set of sites; the other statements are for the ring. Then:

1. the operator \(u\) at the site \(t\) maps \(|v\rangle \) to \(x\mapsto \sum _{j}u_{x_t,j}\, v(x[t\mapsto j])\), and keeps \(|0\rangle \) at the sites of \(Z\) if \(t\notin Z\); so does multiplication by a scalar;

2. \(P_\pi P_\sigma =P_{\pi \sigma }\), \(P_\pi (\bigotimes _{i}g_{i}) =(\bigotimes _{i}g_{\pi ^{-1}(i)})P_\pi \), and conjugating an operator \(X\) placed at the sites \(e_1,\dots ,e_m\) by \(P_\sigma \) places \(X\) at the sites \(\sigma (e_1),\dots ,\sigma (e_m)\): \(P_\sigma XP_{\sigma ^{-1}}\) is \(X\) at \(\sigma \circ e\);

3. if \(|v\rangle \) has \(|0\rangle \) at the sites of \(Z\), then \(P_\pi |v\rangle \) has \(|0\rangle \) at the sites of \(\pi (Z)\), and an operator acting on sites outside \(Z\), in particular a product of such operators, keeps \(|0\rangle \) at the sites of \(Z\);

4. gates on pairwise disjoint neighbouring pairs, given by a list, form a layer whose operator is the product of the gates along the list.

Let \(d\ge 1\), let \(r\le n\), and let \(e\) be an injective map from the bond levels \(\{ 0,\dots ,D'-1\} \), \(D'\ge 1\), to the configurations of \(r\) sites. Let \(Q_0,\dots ,Q_{n-1}\) be square matrices of size \(D'\) indexed by the physical index, with bonds \(b_0=1,b_1,\dots ,b_n\), such that \(Q_p^i\) vanishes on the rows \(\alpha \ge b_p\) and \(Q_p\) is isometric on its first \(b_{p+1}\) bond levels. Then there is a unitary \(U\) on the open chain of \(n\) sites such that, for every unitary \(Z\) acting on the last \(r\) sites, \(UZ\) is a product of at most \(\max (1,n-r)\) gates on windows of \(r+1\) sites (in particular, so is \(U\)), with

where \(e(x)\) occupies the last \(r\) sites. If \(r\ge 2\), there are \(K_0,K_1\), depending only on \(d\) and \(r\), such that \(U\) is a product of at most \(K_0+(n-r)K_1\) gates on two neighbouring sites.

For \(W\in \mathbb {C}^{m\times n}\), a tensor \(C\) with physical dimension \(n\), and \(L\ge 0\), blocking the rotated tensor over \(L\) sites is rotating the blocked tensor by the \(L\)-fold Kronecker power of \(W\):

For every tensor \(A\) and every \(L\), \(\mathcal{E}_{A^{[L]}}=\mathcal{E}_A^{\, L}\).

Let \(A\) be a canonical tensor with positive physical dimension and positive bond dimension \(D\). There are distinct normal representatives \((A_j)_j\) and a positive integer \(R\leq 3D^5\) such that, for every \(R\leq L\leq N\), every collection of positive operators \(H_i\) with the same kernels as the canonical range-\(L\) parent terms satisfies

The equality includes \(N=L=R\), and in particular holds at \(N=L=3D^5\) for the canonical parent Hamiltonian. Repeated copies of the normal representatives are allowed in \(A\).

Every normal tensor in the sense of [ 13 , Section 2.2 ] is injective after blocking by some finite length; its positive bond dimension follows from normality.

Let \(A\) be a normal tensor in the sense of [ 13 , Section 2.2 ] . Its bond dimension is necessarily positive, and there is a positive definite matrix \(\sigma \) such that the gauge obtained from \(\sigma \) is trace-preserving, primitive, and irreducible. No scalar rescaling is needed.

Let \(A\) and \(B\) be normalized MPS tensors. Every eigenvalue \(\mu \) of \(F_{AB}\) satisfies \(|\mu |\leq 1\).

If \(A\) and \(B\) are gauge equivalent and \(A\) is normal, then \(B\) is normal.

Let \(D{\gt}0\). If \(A\) is algebraically normal and left-canonical, then

Taking an interval inside an interval adds their starting positions. The blocked interval has transfer map \(E_{[a,a+n)}=E_a\circ \cdots \circ E_{a+n-1}\), in the original order.

Let \(A\) be a normal tensor satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). Let \(\rho {\gt}0\) satisfy \(\mathcal{E}_A(\rho )=\rho \) and \(\operatorname{Tr}\rho =1\). Suppose \(|\lambda _2|\) bounds the moduli of the transfer eigenvalues other than \(1\), and \(\xi =-1/\log |\lambda _2|{\gt}0\). There are constants \(C,c{\gt}0\) and \(N_0\geq 2\), depending on these tensor data, such that for every \(\eta {\gt}0\) and every \(N\geq N_0\), the periodic state is nonzero and the vector \(\psi _{N,\eta }\) in Definition 35.8.3.1 is a unit vector prepared from a product state by a local circuit of depth \(T\) satisfying

The constants and threshold are chosen before \(\eta \) and \(N\). This is the case \(\xi {\gt}0\) of the polynomial accuracy asserted after Lemma 1 of [ 219 ] , without a divisibility restriction; zero correlation length remains outside this construction [ 89 ] .

There is \(C=C(d,D)\) with the following property. Let \(B\) be normal and left canonical, with a faithful fixed density matrix \(\sigma \), and let \(\phi _N(B)=\zeta ^N\phi _N(A)\) for every \(N\), with \(\zeta \ne 0\). Let \(0{\lt}t{\lt}1\) bound the moduli of the eigenvalues other than \(1\) of its transfer map, and set \(\xi =-1/\log t\). For each prescribed \(a{\gt}\xi /2\), there is \(b\ge 1\) such that every \(N\ge 1\) with \(\phi _N(A)\ne 0\), and every integer \(q\ge a\log (N/\varepsilon )+b\), admit a unit approximation of error at most \(\varepsilon \) prepared in depth \(T\le C\min (N,q)\). In particular, \(q\) may be the ceiling of this threshold; no divisibility of \(N\) is required.

This refines the sufficient blocking scale in [ 219 ] ; it makes no assertion at \(a=\xi /2\) or at the spectral-bound endpoint \(t=0\).

Every translation-invariant MPS of constant bond dimension, short- or long-range correlated, is prepared with error \(\varepsilon \) in depth \(O(\log (N/\varepsilon ))\) by local circuits assisted by measurements, the measurements being used only for the preparation of the GHZ-type state [ 219 , paragraph “Long-range MPS using measurements” ] . Theorem 35.21.1.11 proves this for direct sums of normal blocks whose \(q\)-site states are orthogonal and for block lengths dividing \(N\). Theorem 35.21.2.8 proves it for direct sums of normal blocks in the canonical form [ 219 , eq. (S2) ] whose \(q\)-site states may overlap, with every multiplicity and every nonzero complex weight, and for every chain length \(N\geq 2\) at which \(|\phi _N(A)\rangle \neq 0\), with the mixed transfer maps of distinct blocks of spectral radius below one, the isometries of the direct sum with weights one, and the GHZ-type state prepared in depth \(O(q)\) rather than in constant depth [ 72 ] . The same paragraph claims the depth \(O(\log \log (N/\varepsilon ))\) when the tree-RG circuit is also assisted by measurements; for normal tensors and the chain lengths divisible by \(s2^{k+1}\) for some \(k\) with \(a\log (N/\varepsilon )+b\le s2^{k+1}\le 2(a\log (N/\varepsilon )+b)\) this is Theorem 35.20.5, and for tensors that are not normal it is not formalized.

There is \(C\), depending only on \(d\) and \(D\), such that the following holds. Let \(A\) be a tensor, let \(\omega \) be a unit vector on \(\mathbb {C}^D\otimes \mathbb {C}^D\), and cut the ring of \(N\ge 1\) sites into \(M\ge 1\) blocks of lengths \(3D\le \ell _k\le L\) for which every blocked tensor is injective. Then the state \(|\psi _{\ell ,\omega }\rangle \) of Definition 35.5.10 is prepared in depth at most \(CL\). In particular, for \(\sigma \ge 0\) with \(\operatorname{Tr}\sigma =1\), a block length \(q\ge 3D\) for which the \(q\)-site blocked tensor is injective, and \(M\ge 1\), the approximating state of Definition 35.3.8 on \(N=Mq\) sites is prepared in depth at most \(Cq\). This is the depth count behind [ 219 , eq. (1) ] ; it does not choose \(q\) in terms of \(N\) and \(\varepsilon \).

Let \(A\) blocked over \(s\) sites be injective, let \(\rho \ge 0\) with \(\operatorname{tr}\rho =1\), and let \(u\) be a unitary with \(u|0\rangle |0\rangle =|\omega \rangle \). For \(q=s2^{k+1}\) and \(M\ge 1\), the approximating state \(|\widetilde\phi _N\rangle =V^{\otimes M}|\Omega \rangle \) on \(N=Mq\) sites (Definition 35.3.8) is the state on registers of the tree network, with \(k+1\) isometry layers and unitary \(u\), of the tensor \(B\) obtained by blocking \(A\) over \(s\) sites.

Let the two-site blocked tensor of \(A\) be injective, let \(\rho \ge 0\) with \(\operatorname{tr}\rho =1\), and let \(u\) be a unitary with \(u|0\rangle |0\rangle =|\omega \rangle \). For \(q=2^{k+1}\) and \(M\ge 1\), the approximating state \(|\widetilde\phi _N\rangle =V^{\otimes M}|\Omega \rangle \) on \(N=Mq\) sites (Definition 35.3.8) is the state of the tree network of \(A\) with \(k+1\) isometry layers and unitary \(u\).

Let \(A^i=\bigoplus _{j=1}^b\mu _jA_j^i\) be the direct sum with complex weights \(\mu _j\) of normal blocks \(A_j\) in the gauge (3), with positive definite fixed points \(\rho _j\) of trace one, each placed on its own bond coordinates. Let \(\lambda _2\in \mathbb {C}\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\) and the moduli of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), let \(\xi \) be its correlation length, and let \(0{\lt}\gamma {\lt}1\). Form the approximating state \(\widetilde\phi _N\) of Definition 35.17.1 with the pairs of the \(\rho _j\) on the bond coordinates of their blocks and \(\beta _j=\lvert \mu _j\rvert ^N\). There are constants \(C,C'{\gt}0\) such that for all weights \(\mu \) not all zero, every block length \(q\) and every \(M\geq 1\), with \(N=qM\) and \(y=Me^{-\gamma q/\xi }\),

This is [ 219 , Lemma 1\('\)(ii) ] for blocks of multiplicity one, with \(\beta _j=\lvert \mu _j\rvert ^N\) in place of the coefficients \(\beta _j=\mu _j^N\) that [ 219 , eqs. (S6) and (S7) ] take from [ 219 , eq. (S4) ] , with \(\xi _{\mathrm{diag}}\) replaced by \(\max (\xi _{\mathrm{diag}},\xi _{\mathrm{off\text{-}diag}})\), and for \(0{\lt}\gamma {\lt}1\) in place of \(0{\lt}\gamma {\lt}1/2\). The constants do not depend on the weights, and the normalization \(\lvert \mu _j\rvert \leq 1\) with one weight of modulus one is not needed. With \(\beta _j=\mu _j^N\) the bound is false (Theorem 35.17.4.1) [ 142 ] .

With the notation of Definition 35.17.1, including the approximating state \(\widetilde\phi _N\) defined there, \(\xi _{\mathrm{diag}}=\max _j\xi _j\) the largest correlation length of the blocks \(A_j\), \(0{\lt}\gamma {\lt}1/2\), and \(q=o(N)\),

This is [ 219 , Lemma 1\('\)(ii) ] . As printed it is false: it fails both for a block of multiplicity two (Theorem 35.17.8, with the pair of that block on one of its copies; the source does not say where the pair sits, and the same computation, recorded in the gap note but not formalized, gives the failure for every placement) and for two blocks of multiplicity one whose states on \(q\) sites are not orthogonal (Theorem 35.17.9) [ 128 , 142 ] . For blocks of multiplicity one whose states on \(q\) sites are orthogonal it holds without the condition \(q=o(N)\) (Theorem 35.17.12). For blocks of multiplicity one with real weights \(0\leq w_j\leq 1\), the largest equal to one, whose states overlap, the approximating state with \(\beta _j=w_j^N\) converges at the rate obtained by replacing \(\xi _{\mathrm{diag}}\) with \(\max (\xi _{\mathrm{diag}},\xi _{\mathrm{off\text{-}diag}})\), the larger of the correlation lengths of the blocks and of their mixed transfer maps, again without the condition \(q=o(N)\) (Theorems 35.17.2.6 and 35.17.2.7).

Let \(A\) be normal in the gauge (3) with correlation length \(\xi \), let \(|\phi _N\rangle \) be the normalized vector (1), and let \(0{\lt}\gamma {\lt}1\). For \(N=qM\),

for \(q\ge 1\). Explicitly, there is a constant \(C{\gt}0\) depending only on \(A\), \(\rho \), and \(\gamma \) such that \(\varepsilon (\widetilde\phi _N,\phi _N)\le Cy\, e^{Cy}\) for all \(q\ge 0\) and \(M\ge 1\), where \(y=M\, e^{-2\gamma q/\xi }\). The rate \(e^{-\gamma q/\xi }\) for \(0{\lt}\gamma {\lt}1/2\) is [ 219 , Lemma 1, eq. (17), and Lemma 1\('\)(i) ] , and [ 223 , App. B4, eqs. (S52) and (S53) ] gives the rate \(e^{-q/\xi }\). The rate \(e^{-2\gamma q/\xi }\) for every \(0{\lt}\gamma {\lt}1\) and every normal tensor is new here. Numerically, the error decays as \(e^{-\gamma q/\xi }\) with \(\gamma \approx 2\) [ 219 , Supplemental Material, discussion of Fig. S1 ] , [ 243 , eq. (4) ] .

Let \(A\) be the direct sum with multiplicities of Definition 35.17.1.1, with normal blocks \(A_j\) in the gauge (3) and positive definite fixed points \(\rho _j\) of trace one, let \(0{\lt}\gamma {\lt}1\), and let \(\widetilde\phi _N\) be the approximating state of Definition 35.17.1 with the pairs of \(\rho _j\) on the copies \(k_j\) and the coefficients \(\beta '_j\) of Definition 35.17.4.3 in place of \(\beta _j\). Put \(y=Me^{-\gamma q/\xi }\).

1. Let \(\lambda _2\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\), with correlation length \(\xi \). There are \(C,C'{\gt}0\) such that for all nonzero complex weights, every choice of the copies \(k_j\), every block length \(q\geq 1\) at which the \(q\)-site states of distinct blocks are orthogonal and every \(M\geq 1\) with \(\beta \neq 0\), \(\varepsilon (\widetilde\phi _N,\phi _N)\leq Cy\, e^{Cy}\) and \(\varepsilon (\widetilde\phi _N,\phi _N)\leq C'y\).

2. Let \(\lambda _2\) moreover bound the moduli of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\). There are \(C,C'{\gt}0\) such that for all nonzero complex weights, every choice of the copies \(k_j\), every \(q\geq 1\) and every \(M\geq 1\) with \(\beta \neq 0\), \(\varepsilon (\widetilde\phi _N,\phi _N)\leq Cy\, e^{Cy}\) and \(\varepsilon (\widetilde\phi _N,\phi _N)\leq C'y\).

This is [ 219 , Lemma 1\('\)(ii) ] for the source’s construction with the coefficients \(\beta '_j\) in place of the coefficients \(\beta _j\) that [ 219 , eq. (S7) ] takes from [ 219 , eq. (S4) ] , for weights of any phase, in the scopes of Theorems 35.17.1.4 and 35.17.3.7. With the coefficients \(\beta _j\) the bound is false (Theorems 35.17.8 and 35.17.4.1) [ 26 ] .

Let \(A^i=\bigoplus _{j=1}^b\mu _jA_j^i\) with \(b\geq 1\), weights \(\mu _j{\gt}0\), and normal blocks \(A_j\) in the gauge (3) with fixed points \(\rho _j\), each placed on its own bond coordinates. Let \(\lambda _2\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\), so that its correlation length \(\xi \) bounds every \(\xi _j\), and let \(0{\lt}\gamma {\lt}1\). Form the approximating state \(\widetilde\phi _N\) of Definition 35.17.1 with the pairs of the \(\rho _j\) on the bond coordinates of their blocks and \(\beta _j=\mu _j^N\). There is \(C{\gt}0\) such that for all weights \(\mu _j{\gt}0\), every block length \(q\) at which the \(q\)-site states of distinct blocks are orthogonal, \(\sum _{i_1,\dots ,i_q}\overline{(A_j^{i_1}\cdots A_j^{i_q})_{ab}}\, (A_{j'}^{i_1}\cdots A_{j'}^{i_q})_{a'b'}=0\) for \(j\neq j'\) and all \(a,b,a',b'\), and every \(M\geq 1\), with \(N=qM\) and \(y=(N/q)\, e^{-\gamma q/\xi }\),

This is [ 219 , Lemma 1\('\)(ii) ] restricted to multiplicity one and to blocks whose \(q\)-site states are orthogonal, for \(0{\lt}\gamma {\lt}1\) in place of \(0{\lt}\gamma {\lt}1/2\). For multiplicity one the phase of \(\mu _j\) can be absorbed into \(A_j\), which stays normal with the same transfer map, so the condition \(\mu _j{\gt}0\) only fixes this presentation. The orthogonality is an added hypothesis: without it the statement fails (Theorem 35.17.9). Under it no condition \(q=o(N)\) is needed [ 142 ] .

Let \(A^0=\operatorname {diag}(1,3/5)\) and \(A^1=\operatorname {diag}(0,4/5)\), the canonical form with the normal blocks \(A_1=(1,0)\) and \(A_2=(3/5,4/5)\), each of multiplicity one and weight one, so that \(\beta =(1,1)\). The blocks are normal and in canonical form II and form a basis of normal tensors whose states become orthogonal as \(N\to \infty \), so \(A\) satisfies every hypothesis of [ 219 , Lemma 1\('\)(ii) ] . Put \(s=(3/5)^q\), the overlap of the states of the two blocks on \(q\) sites, \(u=\bigl(\sqrt{1+s}+\sqrt{1-s}\bigr)/2\) and \(v=\bigl(\sqrt{1+s}-\sqrt{1-s}\bigr)/2\). For every \(q\geq 1\) and \(M\geq 1\) the approximating state of Definition 35.17.1 satisfies

and for \(M\geq 3\) the error is at least \((9/25)^q/16\), while for every \(M\geq 1\) it is at most \(M(9/25)^q\), so the construction converges whenever \((N/q)(9/25)^q\to 0\). The transfer maps of both blocks have no eigenvalue other than one, so every \(\lambda _2\) bounds their subleading eigenvalues. If \(r=e^{-\gamma /\xi _{\mathrm{diag}}}{\lt}9/25\), then for every constant \(C\) there is \(q\geq 3\) such that, with \(M=q\), \(N=q^2\) and \(y=(N/q)e^{-\gamma q/\xi _{\mathrm{diag}}}\), \(\varepsilon (\widetilde\phi _N,\phi _N){\gt}Cy\, e^{Cy}\). The sequence \(q=M\) satisfies \(q=o(N)\), so the bound (89) fails even when every multiplicity is one [ 142 ] . What fails is the rate: for \(M\geq 3\) the error lies between \((9/25)^q/16\) and \(M(9/25)^q\), where \(9/25=\tau ^2\) is the square of the eigenvalue \(\tau =3/5\) of the mixed transfer map of the two blocks.

Let \(A^i=\bigoplus _{j=1}^bA_j^i\) with \(b\geq 1\) be the direct sum with weights one of normal blocks \(A_j\) in the gauge (3), with positive definite fixed points \(\rho _j\) of trace one, each placed on its own bond coordinates. Let \(\lambda _2\in \mathbb {C}\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\) and the moduli of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), so that, when \(\lvert \lambda _2\rvert {\lt}1\), its correlation length \(\xi \) bounds every \(\xi _j\) and every \(\xi _{jj'}\), and let \(0{\lt}\gamma {\lt}1\). Form the approximating state \(\widetilde\phi _N\) of Definition 35.17.1 with the pairs of the \(\rho _j\) on the bond coordinates of their blocks and \(\beta _j=1\). There are constants \(C,C'{\gt}0\) such that for every block length \(q\) and every \(M\geq 1\), with \(N=qM\) and \(y=Me^{-\gamma q/\xi }\), which is \((N/q)e^{-\gamma q/\xi }\) for \(q\geq 1\),

This is [ 219 , Lemma 1\('\)(ii) ] for blocks of multiplicity one and weight one, with \(\xi _{\mathrm{diag}}\) replaced by \(\max (\xi _{\mathrm{diag}},\xi _{\mathrm{off\text{-}diag}})\) and for \(0{\lt}\gamma {\lt}1\) in place of \(0{\lt}\gamma {\lt}1/2\). The states of distinct blocks need not be orthogonal, and no condition \(q=o(N)\) is needed. With \(\xi _{\mathrm{diag}}\) alone the bound is false (Theorem 35.17.9) [ 142 ] .

In the setting of Theorem 35.17.2.6, let \(0{\lt}\gamma {\lt}1\). There are constants \(C,C'{\gt}0\) such that for every block length \(q\) and every \(M\ge 1\), with \(N=qM\) and \(y=Me^{-2\gamma q/\xi }\), which is \((N/q)e^{-2\gamma q/\xi }\) for \(q\ge 1\),

This is [ 219 , Lemma 1\('\)(ii) ] for blocks of multiplicity one and weight one, with \(\xi _{\mathrm{diag}}\) replaced by \(\max (\xi _{\mathrm{diag}},\xi _{\mathrm{off\text{-}diag}})\) and the exponent \(\gamma q/\xi \) by \(2\gamma q/\xi \) [ 142 ] .

Let \(A^i=\bigoplus _{j=1}^bw_jA_j^i\) be the direct sum with real weights \(w_j\geq 0\), not all zero, of normal blocks \(A_j\) in the gauge (3), with positive definite fixed points \(\rho _j\) of trace one, each placed on its own bond coordinates. Let \(\lambda _2\in \mathbb {C}\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\) and the moduli of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), let \(\xi \) be its correlation length, and let \(0{\lt}\gamma {\lt}1\). Form the approximating state \(\widetilde\phi _N\) of Definition 35.17.1 with the pairs of the \(\rho _j\) on the bond coordinates of their blocks and \(\beta _j=w_j^N\). There are constants \(C,C'{\gt}0\) such that for all such weights, every block length \(q\) and every \(M\geq 1\), with \(N=qM\) and \(y=Me^{-\gamma q/\xi }\),

This is [ 219 , Lemma 1\('\)(ii) ] for blocks of multiplicity one with weights of possibly different moduli and with phases absorbed into the blocks, with \(\xi _{\mathrm{diag}}\) replaced by \(\max (\xi _{\mathrm{diag}},\xi _{\mathrm{off\text{-}diag}})\), for \(0{\lt}\gamma {\lt}1\) in place of \(0{\lt}\gamma {\lt}1/2\), with constants that do not depend on the weights, and without the normalization of the source that the largest weight is one [ 142 ] .

Let \(\mu \in \mathbb {C}\) with \(\lvert \mu \rvert =1\), and let \(A^0=\operatorname {diag}(1,0)\) and \(A^1=\operatorname {diag}(0,\mu )\), the direct sum \(A_1\oplus \mu A_2\) of the one-dimensional normal blocks \(A_1=(1,0)\) and \(A_2=(0,1)\), each of multiplicity one, with weights \(1\) and \(\mu \). After ordering its bond coordinates this is the canonical form of a basis of normal tensors in canonical form II with weights of modulus one, so \(A\) satisfies every hypothesis of [ 219 , Lemma 1\('\)(ii) ] . For every \(q\geq 1\) and \(M\geq 1\), with \(N=qM\), the approximating state of Definition 35.17.1, with the pairs of the fixed point \(1\) of each block, satisfies

More generally, for coefficients \(\alpha \) with \(\lvert \alpha _1\rvert ^2+\lvert \alpha _2\rvert ^2=1\) the overlap is \((\overline{\alpha _1}+\overline{\alpha _2})/\sqrt2\). For \(\mu =-1\) and odd \(q\) and \(M\) the approximating state with \(\beta =(1,\mu ^N)\) is orthogonal to \(|\phi _N\rangle \), so \(\varepsilon (\widetilde\phi _N,\phi _N)=1\). The transfer maps of both blocks have no eigenvalue other than one, so every \(\lambda _2\) bounds their subleading eigenvalues. For \(\mu =-1\), every \(\lambda _2\) with \(0{\lt}\lvert \lambda _2\rvert {\lt}1\), every \(\gamma {\gt}0\) and every constant \(C\) there is an odd \(M\) such that, with \(q=M\), \(N=M^2\) and \(y=(N/q)e^{-\gamma q/\xi _{\mathrm{diag}}}\), \(\varepsilon (\widetilde\phi _N,\phi _N){\gt}Cy\, e^{Cy}\). The sequence \(q=M\) satisfies \(q=o(N)\), so the bound (89) fails for orthogonal blocks of multiplicity one when a weight is not a nonnegative real number [ 142 ] .

Let \(A^0=\operatorname {diag}(1,1,0)\) and \(A^1=\operatorname {diag}(0,0,1)\), the canonical form with the normal block \(A_1=(1,0)\) of multiplicity two and weights \((1,1)\) and the normal block \(A_2=(0,1)\) of multiplicity one, so that \(\beta =(2,1)\) and \(|\phi _N\rangle \propto 2|0\cdots 0\rangle +|1\cdots 1\rangle \). The blocks are normal and in canonical form II, they form a basis of normal tensors whose states become orthogonal as \(N\to \infty \), and the weights satisfy \(\lvert \mu _{j,k}\rvert \leq 1\) with one of modulus one, so \(A\) satisfies every hypothesis of [ 219 , Lemma 1\('\)(ii) ] . Place the pair of the first block on its first copy. For every \(q\geq 1\) and \(M\geq 1\) the approximating state of Definition 35.17.1 satisfies

which does not depend on \(q\) and tends to \(1/\sqrt5\) as \(M\to \infty \). The transfer maps of both blocks have no eigenvalue other than one, so every \(\lambda _2\) with \(0{\lt}\lvert \lambda _2\rvert {\lt}1\) bounds their subleading eigenvalues. For every such \(\lambda _2\), every \(\gamma {\gt}0\) and every constant \(C\) there is \(M\geq 1\) such that, with \(q=M\), \(N=M^2\) and \(y=(N/q)e^{-\gamma q/\xi _{\mathrm{diag}}}\), \(\varepsilon (\widetilde\phi _N,\phi _N){\gt}Cy\, e^{Cy}\). The sequence \(q=M\) satisfies \(q=o(N)\), so the bound (89) fails for this tensor [ 128 ] .

With the notation of Theorem 35.17.1.2, let the blocks \(A_j\) be normal in the gauge (3) with fixed points \(\rho _j\), let \(\lambda _2\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\), with correlation length \(\xi \), and let \(0{\lt}\gamma {\lt}1\). There are constants \(C,C'{\gt}0\) such that for all nonzero complex weights, every block length \(q\) at which the \(q\)-site states of distinct blocks are orthogonal, \(B_j^\dagger B_{j'}=0\) for \(j\neq j'\), and every \(M\geq 1\) with \(N=qM\) and \((\beta _1,\dots ,\beta _b)\neq 0\), the corrected approximating state of Definition 35.17.1.3 satisfies, with \(y=(N/q)\, e^{-\gamma q/\xi }\),

This is [ 219 , Lemma 1\('\)(ii) ] for the corrected approximating state, with arbitrary multiplicities and complex weights, restricted to blocks whose \(q\)-site states are orthogonal, for \(0{\lt}\gamma {\lt}1\) in place of \(0{\lt}\gamma {\lt}1/2\), and without the condition \(q=o(N)\) [ 26 , 142 ] . When the states are orthogonal, the corrected state is the combination \(\sum _j\alpha _j|\phi _M(V_jP_{j,\infty })\rangle \) of the approximating states of the blocks.

With the notation of Theorem 35.17.1.2, let the blocks \(A_j\) be normal in the gauge (3) with positive definite fixed points \(\rho _j\) of trace one, let \(\lambda _2\in \mathbb {C}\) bound the moduli of the eigenvalues other than one of every transfer map \(\mathcal{E}_{A_j}\) and the moduli of all eigenvalues of the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), let \(\xi \) be its correlation length, and let \(0{\lt}\gamma {\lt}1\). There are \(C,C'{\gt}0\) such that for all nonzero complex weights, every block length \(q\) and every \(M\geq 1\) with \(N=qM\) and \((\beta _1,\dots ,\beta _b)\neq 0\), the corrected approximating state of Definition 35.17.1.3 satisfies, with \(y=Me^{-\gamma q/\xi }\),

This is [ 219 , Lemma 1\('\)(ii) ] for the corrected state, with \(\xi _{\mathrm{diag}}\) replaced by \(\max (\xi _{\mathrm{diag}},\xi _{\mathrm{off\text{-}diag}})\), and for \(0{\lt}\gamma {\lt}1\) in place of \(0{\lt}\gamma {\lt}1/2\). The constants do not depend on the weights: the sums \(\beta _j\) may cancel and \(\sum _j\lvert \beta _j\rvert ^2\) may be arbitrarily small, and the normalization \(\lvert \mu _{j,k}\rvert \leq 1\) of the source is not used [ 99 ] . The states of distinct blocks need not be orthogonal, and no condition \(q=o(N)\) is needed.

Let \(A\) be normal in the gauge (3) with correlation length \(\xi \), and let \(0{\lt}\gamma {\lt}1\). There is \(C{\gt}0\), depending only on the tensor, its normalized fixed state and the chosen spectral bound and rate, such that every partition of \(N\) sites into \(M\ge 1\) blocks of lengths \(\ell _j\ge q\) with injective blocked tensors satisfies \(\varepsilon (\psi _{\ell ,\omega },\phi _N)\le CM e^{-2\gamma q/\xi }\). Here \(\psi _{\ell ,\omega }\) is the state of Definition 35.5.10 constructed from the fixed-point pair, and \(\phi _N\) is the normalized target. This refines the first-order rate of [ 219 , Lemma 1\('\)(i), eq. (S11) ] .

Let \(A_1,\dots ,A_b\) be tensors of bond dimensions \(D_j\), and place their bond indices side by side, so that the bond index \(a\) of block \(j\) becomes the index \(\iota _j(a)\) among \(D_1+\cdots +D_b\geq 1\) indices; encode these indices injectively in \(r_1\geq 1\) sites. There is \(C\) such that for every block length \(q\geq 3r_1\) at which every blocked tensor \(B_j=V_jP_j\) is injective and \(B_j^\dagger B_{j'}=0\) for \(j\neq j'\), some unitary \(U\) on \(q\) sites, a product of at most \(Cq\) gates on neighbouring sites, satisfies \(U|\iota _j(l),0,\iota _j(r)\rangle =V_j|l,r\rangle \) for all \(j\), \(l\) and \(r\). The columns \(V_j|l,r\rangle \) are jointly orthonormal, so \(\sum _jD_j^2\leq d^q\).

Every tree contraction \(T\) satisfies \(T^\dagger T=\mathbb {1}\).

The actual block pairs satisfy \(\langle \omega _j|\omega _{j'}\rangle =\delta _{jj'}\). Consequently \(W|j\rangle =|\omega _j\rangle \) is an isometry.

Let \(\iota _{j,k}\) include the virtual coordinates of copy \(k\) of block \(j\) in the flattened canonical direct sum. Their ranges are disjoint for distinct pairs \((j,k)\), and the assembled tensor satisfies

Its sector coefficient at length \(N\) is exactly \(\beta _j=\sum _k\mu _{j,k}^N\).

\(=\sum _{j,k}\mu _{j,k}\)

Let \(A_j,B_j\) be equally labelled tensor families at one positive length \(N\), and suppose the periodic columns of each family are independent. Let \(J:\mathbb {C}^b\to \mathcal H_N\) be an isometry and \(U_A,U_B\) genuine local unitaries of depths \(T_A,T_B\). For \(X=A,B\), assume

Then \(U=U_BU_A^\dagger \) is a local circuit of depth \(T_A+T_B\) and

The coefficients in (5) satisfy

Reversing the right site order and transposing each matrix gives precisely the original left-to-right physical matrix product.

For matrices \(W_k\) from \(\mathbb {C}^m\) to \(\mathbb {C}^{n_k}\), one on each of \(M\) sites, and vectors \(x,y\) on \((\mathbb {C}^m)^{\otimes M}\),

For every chain and every cutting into blocks, \(\lVert \phi _N\rVert =\lVert \phi _{\mathrm{pos}}\rVert \); no injectivity is assumed. Suppose every blocked tensor \(B_k\) is injective on a set \(S_k\) of bond pairs, as in Definition 35.1.3.8. Then \(\langle (\bigotimes _kV_k)x,(\bigotimes _kV_k)y\rangle =\langle x,y\rangle \) for all vectors \(x,y\) on the \(M\) blocks such that \(\overline{x(a)}\, y(a)=0\) whenever \(a_k\notin S_k\) for some \(k\); if every \(B_k\) is injective, this holds for all \(x,y\). More generally, for matrices \(W_k\) with \(\sum _i\overline{(W_k)_{ia}}(W_k)_{ib}=1\) if \(a=b\in S_k\) and \(0\) otherwise, the same identity holds with \(W_k\) in place of \(V_k\). The state \(|\phi _{\mathrm{pos}}\rangle \) vanishes at every \(a\) with some \(a_k\notin S_k\). For products \(|\Omega \rangle \) and \(|\Omega '\rangle \) of pairs \(\omega ^k\) and \(\omega '^k\), \(\langle \Omega ,\Omega '\rangle =\prod _k\langle \omega ^k,\omega '^k\rangle \); in particular the norm of \(|\Omega \rangle \) is \(\prod _k\lVert \omega ^k\rVert \). For all vectors \(\omega ^k\),

Here a vector \(x/\lVert x\rVert \) with \(x=0\) is read as \(0\), so the identity holds for every chain, and it is an identity of overlaps with normalized states when \(\phi _{\mathrm{pos}}\ne 0\).

There is \(C\), depending only on \(d\) and \(D\), such that the following holds. Cut the ring of \(N\ge 1\) sites into \(M\ge 1\) blocks of lengths \(3D\le \ell _k\le L\), let \(A_1,\dots ,A_N\) be tensors of bond dimension \(D\) whose blocked tensor \(B_k\) is injective on a set \(S_k\) of bond pairs for every \(k\), and let \(\omega ^k\) be unit vectors on \(\mathbb {C}^D\otimes \mathbb {C}^D\) whose product \(|\Omega \rangle \) vanishes at every bond configuration \(c\) with some \(c_k\notin S_k\). Then \((\bigotimes _kV_k)|\Omega \rangle \) is prepared in depth at most \(CL\). In particular this holds when every \(B_k\) is injective, for all unit vectors \(\omega ^k\). A finite set has an enumeration listing the elements of a given subset first, and for \(d\le 1\) every vector on a ring of \(N\ge 1\) sites is prepared in depth \(0\).

Let \(\sigma \) be a density matrix and \(0\le \eta \le 1\). Suppose every site’s transfer map preserves trace and its normalized Choi matrix satisfies \(J(E_{A_i})\ge (\eta /D)(\sigma \otimes I)\). For the blocked tensor \(B\) and any two density matrices \(\rho _1,\rho _2\),

The sites may themselves be contiguous blocks, provided the hypothesis is imposed on their actual transfer maps.

Fix \(\sigma {\gt}0\) with \(\operatorname {Tr}\sigma =1\). There is \(K\), depending only on \(D\) and \(\sigma \), such that the following holds. Partition a chain of \(N\) sites into contiguous blocks of lengths \(\ell _j\le s\), with \(s\ge 1\), and let \(F_j\) be their transfer maps. Suppose each \(F_j\) preserves trace, fixes \(\sigma \), and satisfies \(J(F_j)\ge (\eta /D)(\sigma \otimes I)\) for one \(0\le \eta \le 1\). For the physical matrix \(B\) of the whole chain and its positive polar factor \(P_B\), the operator-norm bounds are

The constants are independent of the site tensors, the physical dimension, the chain length, and the partition. For fixed \(s\) and \(0{\lt}\eta {\lt}1\), these are uniform exponential bounds in physical length.

For every cutting of the ring into blocks, \(|\phi _N\rangle =(\bigotimes _{k=1}^MV_k)|\phi _{\mathrm{pos}}\rangle \).

For a chain \(A_0,\dots ,A_{n-1}\) with common bond dimension \(D\), let \(B\) be its blocked tensor. Then

The first site’s map is outermost. If every site map preserves trace, so does \(E_B\).

Fix two explicitly matched families \((A_j)_{j=1}^b\) and \((B_j)_{j=1}^b\) of normal sectors with \(b{\gt}0\) and common physical dimension \(d\ge 2\). Each sector is left canonical with a faithful stationary density of trace one. Within each family, the mixed transfer spectral radii of distinct sectors are strictly less than one. There exist \(N_0\ge 2\) and \(c{\gt}0\), fixed before length and accuracy, such that for every \(N\ge N_0\) the two fixed polar encoders are isometries, and for every \(0{\lt}\varepsilon \le 1\) one genuine nearest-neighbor unitary \(U\) satisfies

The same \(U\) satisfies, for every finite reference space \(\mathcal R\) and every logical-reference vector \(\xi \),

The endpoints depend only on the labelled families and \(N\).

Fix \(d\geq 1\), \(D\geq 1\), a register length \(r\geq 2\), and injective encodings of the \(D\) bond values and \(b\) sector labels in \(r\) physical sites. There is a constant \(C\) with the following property. Let \(A\) be a tensor, let \(S\) be a set of bond pairs, and let \(\omega _j\) be orthonormal pair vectors. Cut a positive ring into \(M\ge 1\) blocks with \(3r\leq \ell _k\leq L\). Assume the cyclic product of each pair vector is supported on \(S^M\) for this \(M\), and each blocked tensor is injective on the coordinate subspace spanned by \(S\). There is a local circuit \(U\) of depth at most \(CL\), simultaneously satisfying

where \(V_k\) is the blocked polar isometry, and \(|\operatorname {reg}(j)\rangle \) carries the encoded value \(j\) on every designated register and zero on every other physical site. Consequently, the same \(U\) implements every linear combination of (26), with arbitrary complex coefficients. This is the unitary stage following seed preparation in [ 219 , Long-range MPS using measurements ] .

Fix \(d\ge 2\), a tensor \(B\), a set \(S\) of virtual pairs, an eventual support-injectivity threshold \(L\), and a bound on the number of sector labels. There are \(s\ge \max (2,L)\) and \(C\), fixed before the ring and amplitudes, with the following property. Suppose

Let the pairs \(\omega _j\) be orthonormal and \(\sum _j|\alpha _j|^2=1\). Assume that, at the actual number \(M\) of blocks, each product of the cyclic pairs \(\omega _j\) is supported on \(S^M\). Then

is prepared in quantum depth at most \(C(h+1)\).

Let \(U_A,U_B\) be genuine local unitaries of depths \(T_A,T_B\) on the same positive ring with \(d\ge 1\). For arbitrary column matrices \(F_A,F_B,J\) of compatible sizes, \(U=U_BU_A^\dagger \) has depth \(T_A+T_B\) and

No cost for preparing \(J\) from a product state is included.

=

The physical boxes denote complete circuits, not elementary gates on \(N\) sites. The logical wire carries every superposition at once.

Every trace-preserving actual tensor chain with \(D{\gt}0\) admits density matrices \(\sigma _a\) with \(\sigma _N=\sigma _0\) such that

No common or faithful stationary density is assumed.

Let \(a,b,\chi _A,\chi _B\) be unit vectors in the same physical ring space \(\mathcal H_N=(\mathbb C^d)^{\otimes N}\). Let \(U_A,U_B,R\) be local circuits of depths \(T_A,T_B,T_R\) with \(R\chi _A=\chi _B\). Put \(a'=U_A\chi _A\), \(b'=U_B\chi _B\) and assume \(\varepsilon (a',a)\leq \varepsilon _A\) and \(\varepsilon (b',b)\leq \varepsilon _B\). Then

For an exactly common seed, take \(R=I\) and \(T_R=0\). All auxiliary sites, if present, are included in these full vectors with their specified reset states. No measurement is inverted.

=

The boxes in this diagram denote whole circuits on \(N\) sites, not free gates on growing physical blocks.

Fix normalized target families \(a_N,b_N\), constants \(c_A,c_B,c_R\), and \(N_0\). Suppose that for every \(0{\lt}\varepsilon \leq 1\) and positive \(N\geq N_0\) there are compatible coherent seeds and circuits as in Theorem 35.7.2, with endpoint errors at most \(\varepsilon /4\) and respective depths bounded by \(c_A\log (N/\varepsilon )\), \(c_B\log (N/\varepsilon )\) and \(c_R\log (N/\varepsilon )\). There is a conversion of error at most \(\varepsilon \) and depth at most \((c_A+c_B+c_R)\log (N/\varepsilon )\).

In bond coordinates the corrected state equals \(W^{\otimes M}\sum _j\alpha '_j|j\rangle ^{\otimes M}\). If \(M\geq 1\) and \((\beta _j)_j\neq 0\), its norm is one.

For three bonds the identity is

\(=\)

Let \(q\geq 1\), let \(V\) be the polar factor of the actual \(q\)-site blocked canonical tensor, and let \(L_j\) be its copy isometries. Write \(|\Omega _j\rangle \) for the product of the unembedded pairs of \(\sigma _j\) and \(K_{j,k_j}\) for the pair embedding on the chosen copy. Then

Thus the unnormalized physical vector is independent of the chosen copies. Let \(|\phi _N\rangle \) be the normalized periodic state of \(A\) on \(N=qM\) sites, put \(\alpha _j=\beta _j/(\sum _l|\beta _l|^2)^{1/2}\), and let \(|\widetilde\phi '_N\rangle \) and \(|\widetilde\phi _N\rangle \) be the normalizations of \(V^{\otimes M}\sum _j\alpha '_jK_{j,k_j}^{\otimes M}|\Omega _j\rangle \) and \(V^{\otimes M}\sum _j\alpha _jL_j^{\otimes M}|\Omega _j\rangle \). Then

A vanishing coefficient vector or physical vector is normalized to zero.

The local identity used in the proof is

\(=\frac{\mu _{j,k_j}^q}{c_j}\)

For unit vectors \(\psi ,\phi \) and unitary observables \(A,B\), write \(\operatorname {Cov}_\psi (A,B)= \langle \psi |AB|\psi \rangle - \langle \psi |A|\psi \rangle \langle \psi |B|\psi \rangle \). Then \(|\langle \psi |A|\psi \rangle |\le 1\), and

Both expectations and covariances are unchanged by a unit scalar multiplying the state.

Let \(A\) be normal and in the gauge (3), with correlation length \(\xi {\gt}0\), let \(L\ge 1\) be a length such that the products of \(L\) matrices of \(A\) span \(M_{D}(\mathbb {C})\) (one may take \(L=1\) when \(A\) is injective), and let \(|\phi _N\rangle \) be the normalized vectors (1). For operators \(X,Y\) on \((\mathbb {C}^d)^{\otimes L}\) let \(X_1\) act on the sites \(1,\dots ,L\), let \(Y_s\) act on the sites \(s,\dots ,s+L-1\), and put

There are Hermitian operators \(\mathcal O,\mathcal O'\) on \((\mathbb {C}^d)^{\otimes L}\) of norm one, a constant \(c{\gt}0\), an integer \(K\in \{ 1,2\} \) and an integer \(s_0\ge L+2\) such that for every \(s\ge s_0\) some \(s'\in \{ s,s+1,\dots ,s+K-1\} \) satisfies, for every \(N\ge 3s'\),

One may take \(K=1\) when \(\lambda _2\) is real and \(K=2\) otherwise. This is a Hermitian variant of the correlation estimate of [ 219 , Lemma 2 ] , with the separation \(s\) ranging over windows of \(K\) consecutive values and the connected correlator. The source’s own observables cannot be Hermitian when \(\lambda _2\) is not real; for them the bound holds in modulus at every separation, while the printed real inequality fails when \(\lambda _2\) is not real or is negative  [ 44 ] .

Let \(A\) be normal with correlation length \(\xi {\gt}0\). Here \(\xi =-1/\log \lvert \lambda _2/\lambda _1\rvert \), where \(\lambda _1\) is an eigenvalue of \(\mathcal{E}_A\) of largest modulus and \(\lambda _2\) is an eigenvalue of largest modulus among those different from \(\lambda _1\); in the normalization \(\lambda _1=1\) this is Definition 14.2.6. Let \(|\phi _N\rangle \) be the normalized vectors (1), and let \(|\psi _N\rangle \) be prepared in depth \(T_N\) with \(T_N=o(\log N)\). Then there is \(N_0\) such that \(\varepsilon (\phi _N,\psi _N){\gt}1/2\) for all \(N{\gt}N_0\). This is [ 219 , Theorem 1 ] .

Let \(A\) and \(\xi =-1/\log \lvert \lambda _2/\lambda _1\rvert {\gt}0\) be as in Theorem 35.8.2.24.

1. There is \(C\) such that for all \(N\ge 1\) and \(T\ge 0\), if a unit vector \(|\psi \rangle \) prepared in depth \(T\) satisfies \(\varepsilon (\phi _N,\psi )\le 1/2\), then

   \begin{align} \frac\xi 4\ln N & \le T+\frac\xi 4\ln (T+1)+C. \label{eq:ldp_depth_lower_bound_constant} \end{align}

2. For every \(0{\lt}a{\lt}\xi /4\) there is \(C_a\) such that under the same conditions \(a\ln N\le T+C_a\).

This is the explicit form of [ 219 , Theorem 1 ] that its proof gives: its final inequality compares \(N/\ln N\) with a multiple of \(e^{4T/\xi }\). Two constants of that inequality need repair  [ 119 ] ; the rate \(4/\xi \) does not change.

Let \(A\) be normal and in the gauge (3), let \(L\ge 1\) be a length such that the products of \(L\) matrices of \(A\) span \(M_{D}(\mathbb {C})\), and let \(\lambda _2\neq 1\) be an eigenvalue of \(\mathcal{E}_A\) of largest modulus among those different from \(1\), with \(\xi =-1/\log \lvert \lambda _2\rvert {\gt}0\).

1. There are \(B\ge 0\) and \(C{\gt}0\) such that for every \(T\ge 0\) and \(N\ge B(T+1)\), if a unit vector \(|\psi \rangle \) prepared in depth \(T\) satisfies \(\lvert \langle \phi _N | \psi \rangle \rvert \ge 1/2\), then \(N\le C(T+1)\, e^{4T/\xi }\).

2. If \(|\psi _N\rangle \) are unit vectors prepared in depth \(T_N\) with \(T_N=o(\log N)\), then \(\varepsilon (\phi _N,\psi _N){\gt}1/2\) for all sufficiently large \(N\).

3. There is \(C'\) such that for all \(N\ge 1\) and \(T\ge 0\), if a unit vector \(|\psi \rangle \) prepared in depth \(T\) satisfies \(\varepsilon (\phi _N,\psi )\le 1/2\), then \(\frac\xi 4\ln N\le T+\frac\xi 4\ln (T+1)+C'\).

Part (2) is Theorem 35.8.2.24 for a tensor in the gauge (3); Lemma 35.8.2.27 removes the restriction  [ 120 ] .

Let \(A\) be normal. There is a constant \(c\) depending only on \(A\) such that, for every \(N\ge 2\) and \(0{\lt}\varepsilon \le 1\) for which the periodic state \(|\phi _N(A)\rangle \) is nonzero, a state \(|\psi _N\rangle \) with \(\varepsilon (\psi _N,\phi _N)\le \varepsilon \) is prepared in depth at most \(c\log (N/\varepsilon )\). If \(D\ge 1\), there is \(N_0\) such that \(|\phi _N(A)\rangle \neq 0\) for every \(N\ge N_0\), so for \(N\ge N_0\) the condition can be dropped. This is [ 219 , eq. (1) ] ; the source normalizes \(|\phi _N\rangle =|\phi _N(A)\rangle /c_N\) with \(c_N{\gt}0\), and for some normal tensors \(|\phi _2(A)\rangle =0\) [ 91 ] .

There is \(C\), depending only on \(d\) and \(D\), such that the following holds. Let \(A\) be normal with \(D\ge 1\). There are \(a{\gt}0\) and \(b\ge 1\), depending only on \(A\), such that for \(0{\lt}\varepsilon \le 1\), every \(N\ge 1\) and every \(q\) dividing \(N\) with \(q\ge a\log (N/\varepsilon )+b\), a unit vector \(|\psi \rangle \) with \(\varepsilon (\psi ,\phi _N)\le \varepsilon \) is prepared in depth at most \(Cq\). If moreover \(N\ge 2\) and \(q\le 2(a\log (N/\varepsilon )+b)\), for instance \(q=\lceil a\log (N/\varepsilon )+b\rceil \) when it divides \(N\), the depth is at most \(c\log (N/\varepsilon )\) with \(c\) depending only on \(A\).

There is \(C\), depending only on \(d\) and \(D\), such that the following holds. Let \(B\) be a normal tensor in the gauge (3), with a positive definite fixed point \(\sigma \) of trace one, and let \(A\) be a tensor with \(|\phi _N(B)\rangle =\zeta ^N|\phi _N(A)\rangle \) for all \(N\) and some \(\zeta \neq 0\). Let \(0{\lt}t{\lt}1\) bound the moduli of the eigenvalues of \(\mathcal{E}_B\) other than \(1\), and let \(\xi =-1/\log t\). For every \(a{\gt}\xi /2\) there is \(b\ge 1\) such that for \(0{\lt}\varepsilon \le 1\), every \(N\ge 1\) and every \(q\) dividing \(N\) with \(q\ge a\log (N/\varepsilon )+b\), a unit vector \(|\psi \rangle \) with \(\varepsilon (\psi ,\phi _N(A))\le \varepsilon \) is prepared in depth at most \(Cq\). The block length \(q=\lceil 2\xi (1+\eta )\ln N\rceil \) of [ 219 , paragraph after Lemma 1 ] has slope \(2\xi \), and [ 223 , App. B4, eq. (S54) ] takes \(q=O(\xi \log (N/\varepsilon ))\). The slopes \(a{\gt}\xi /2\) are new here.

Assume the coordinate maps are injective with disjoint ranges. For every nonempty word \(w\), \(B^w=\sum _j\mu _j^{|w|}E_jA_j^wE_j^\dagger \). Thus, for \(N\ge 1\), \(|\phi _N(B)\rangle =\sum _j\mu _j^N|\phi _N(A_j)\rangle \). If \(C_N(B)\) and \(C_N(A_j)\) denote their blocked physical matrices, then \(C_N(B)=\sum _j\mu _j^NC_N(A_j)K_j^\dagger \). No orthogonality of physical sector states is assumed.

For a normal tensor with \(D\ge 1\), the same fixed dimension and layer bounds hold for all \(N\ge N_0\), for some fixed \(N_0\), without a separate nonvanishing hypothesis.

Let \(A\) be a normal tensor of physical dimension \(d\) and bond dimension \(D\). There are constants \(R\ge 1\) and \(C\ge 2\), chosen before \(\varepsilon \) and \(N\), such that for every \(\varepsilon {\gt}0\) and every positive length \(N\) with nonzero periodic state, a network with some \(1\le D'\le D\), physical leaf bound \(R\) and coarse depth \(h\) satisfies

Its internal tree dimensions are at most \(D^2\), and its physical leaf dimensions are at most \(d^R\). Thus the layer count is \(O(\log \log (N/\varepsilon ))\) in the large-size, small-error regime, with no divisibility restriction on \(N\).

Let \(A\) be a tensor, let \(\sigma \in M_{D}(\mathbb {C})\), and partition a ring of \(N\) sites into \(M\geq 1\) blocks of lengths \(\ell _k\) with \(\sum _k\ell _k=N\). Suppose that the positive polar tensor of every blocked tensor \(A^{[\ell _k]}\) equals \(P_\infty (\sigma )\). Then the state obtained from the pair of \(\sigma \) and the block polar isometries is exactly \(|\phi _N(A)\rangle \).

For injective coordinate maps with disjoint ranges and \(N\ge 1\), the complete canonical encoder satisfies

=

Let \(B\) be a tensor with \(\mathcal{E}_B(X)=\operatorname{tr}(X)\sigma \) for every \(X\). If \(\sigma \geq 0\), the positive polar tensor of \(B\) is exactly \(P_\infty (\sigma )\). If \(\sigma {\gt}0\), then \(B\) is injective. This is the exact fixed-point case of the transfer/polar identity in [ 219 ] .

If \(V_{A,N}\) is injective, then \(Q_N^\dagger Q_N=I_b\). If the blocked unweighted sum is injective on a bond-pair set \(S\), every row of \(Q_N\) outside \(S\) vanishes. For the isometry assertion, the coordinate maps are injective with disjoint ranges and \(N\ge 1\).

Fix \(d,D\ge 1\), \(r\ge 2\), and injective encodings of the \(D\) bond values and \(b\) sector labels in \(r\) sites. There is a constant \(C\) before the following choices. Let \(A_j\) have \(D_j{\gt}0\) and injective virtual placements with disjoint ranges. For any positive \(N\ge 3r\), assume its blocked unweighted sum is injective on the sector-pair set. Then a genuine local unitary \(U_N\) has depth at most \(CN\) and satisfies \(U_NJ_N=F_{A,N}\), where \(J_N\) is the common register encoder for the single block of length \(N\). The same unitary acts on every logical superposition.

There is \(C=C(d,D)\) with the following property. Let \(A\) be an inhomogeneous chain with bond dimensions at most \(D\) on \(N\ge 1\) sites, with nonzero periodic state. Let \(K,r{\gt}0\), let \(k,L\ge 0\) be integers, and choose

Suppose that whenever \(L\le q\le N\), there is a partition into blocks \(q\le \ell _j\le 2q\) whose positive-part state is approximable by unit pairs with error at most \(KN^ke^{-rq}\). Then \(q\ge 3D+1\) and some unit vector \(\psi \) satisfies \(\varepsilon (\psi ,\phi _N(A))\le \varepsilon \) and is prepared in depth \(T\le C\min (N,q)\).

This is a sufficient quantitative condition for the inhomogeneous scheme of  [ 219 ] ; qualitative convergence does not supply the assumed rate  [ 66 ] .

The tensor \(B\) is nonzero, unital and trace preserving, and is normal with blocking length two. Nevertheless, its canonical polar factor \(V=BP^+\) does not satisfy \(V^\dagger V=\mathbb {1}_4\). Thus the literal full-input pseudoinverse extension in [ 219 , footnote 3 to equations (13)–(15) ] fails even for a normal tensor with sufficient physical dimension. The support-domain correction is recorded in [ 129 ] .

With \(W\), \(X^{\otimes M}\) and \(\chi _M\) as in Definition 35.17.3, formed with \(\alpha =\alpha ^{(N)}\) and the pairs of Definition 35.17.2, where each \(\sigma _j\) is positive semidefinite of unit trace, \(W\) and \(W^{\otimes M}\) are isometries and \(|\Omega '\rangle =W^{\otimes M}|\chi _M\rangle \), up to the regrouping of the legs \(R_kL_{k+1}\). If moreover \(M\geq 1\) and \((\beta _1,\dots ,\beta _b)\neq 0\), then \(\langle \Omega ' | \Omega ' \rangle =1\). This is the preparation of [ 219 , eq. (19) ] . The same holds for the state of Definition 35.17.3 with \(\alpha =\alpha ^{(N)}\) for any orthonormal pairs \(\omega _j\); the identity \(|\Omega '\rangle =W^{\otimes M}|\chi _M\rangle \) holds for any pairs.

Let \(e\) be an idempotent of a normed algebra over \(\mathbb {C}\), and let \(\operatorname{Tr}\) be linear with \(\operatorname{Tr}(xy)=\operatorname{Tr}(yx)\), \(\lvert \operatorname{Tr}x\rvert \le K\lVert x\rVert \) and \(\operatorname{Tr}e=1\). There is \(C{\gt}0\) such that every \(T\) with \(eTe=\alpha e\) for a real \(\alpha \le 1\), \(\lVert T-e\rVert \le \delta \) and \(1-\alpha \le \delta ^2\) satisfies, for \(M\ge 2\) and \(CM\delta ^2{\lt}1\),

Let \(e\) be an idempotent of a normed ring, \(f=1-e\), and let \(Z\) satisfy \(eZe=0\) and \(\lVert fZe\rVert ,\lVert eZf\rVert ,\lVert fZf\rVert \le z\le 1/2\). Let \(\operatorname{Tr}\) be additive with \(\operatorname{Tr}(xy)=\operatorname{Tr}(yx)\) and \(\lvert \operatorname{Tr}x\rvert \le K\lVert x\rVert \). Then for \(M\ge 2\),

Let \(A_j,A_k\) be normal left-canonical blocks of positive bond dimensions. If their dimensions agree, assume they are not gauge-phase equivalent. Every eigenvalue of their mixed transfer map has modulus less than one.

Let \(D^2\le d^{3D}\). There is \(C\), depending only on \(d\) and \(D\), such that the following holds. Cut the ring of \(N\ge 1\) sites into \(M\ge 1\) blocks of lengths \(3D\le \ell _k\le L\), let \(A_1,\dots ,A_N\) be a chain with bond dimensions at most \(D\), and let \(\omega ^k\) be unit vectors on \(\mathbb {C}^{b_k}\otimes \mathbb {C}^{b_k}\), as in Definition 35.10.5. Then there are isometries \(W_k\) with \(W_k\Pi _k=V_k\), for the polar decompositions \(B_k=V_kP_k\) of the blocked tensors of the padded chain, such that \(|\psi \rangle =(\bigotimes _kW_k)|\Omega \rangle \) is a unit vector prepared in depth at most \(CL\), and

that is, \(\varepsilon (\psi ,\phi _N)=\varepsilon (\Omega ,\phi _{\mathrm{pos}})\) for the normalized states when \(\phi _{\mathrm{pos}}\ne 0\); with the reading of \(x/\lVert x\rVert \) for \(x=0\) as in Theorem 35.10.4, the identity of the overlaps holds for every chain. For every \(d\), with \(C\) depending only on \(d\) and \(D\): if the chain is approximated by pairs with error \(\delta \) on the given blocks, some unit vector prepared in depth at most \(CL\) has error at most \(\delta \) against the normalized state \(|\phi _N\rangle \). No injectivity of the blocked tensors is assumed. This is a scope-restricted variant, for one ring, of the claim of the paragraph “Inhomogeneous short-range correlated MPS” of  [ 219 ] , which states total depth \(O(\log (N/\varepsilon ))\) with error \(\varepsilon (\Omega ,\phi _{\mathrm{pos}})\) for matrix product states with bond dimension at most \(D\). The lower bound \(3D\le \ell _k\) is absent from that statement  [ 66 ] , and the unrestricted claim is not formalized. The hypothesis \(D^2\le d^{3D}\) follows from the source’s requirement \(d^q\ge D^2\) for blocks of \(q\ge 3D\ge 1\) sites. The bound \(CL\) gives the stated depth when \(L=O(\log (N/\varepsilon ))\); the source’s definition fixes blocks of length \(q=O(\log N)\). The choice of \(L\) in terms of \(N\) and the error is not made here.

Fix \(d,D\ge 1\), a faithful density matrix \(\sigma \), and \(0{\lt}\eta \le 1\). Suppose every site map of every chain in a family preserves trace, fixes \(\sigma \), and satisfies the normalized Choi bound \(J(E_{A_i})\ge (\eta /D)(\sigma \otimes I)\). Every positive-length periodic target satisfies \(\| \phi _N\| ^2\ge \eta {\gt}0\). Physical preparation to error \(0{\lt}\varepsilon \le 1\) has depth at most \(C\log (N/\varepsilon )\), with one constant for the family.

Under the uniform ordered-mixing hypothesis of Theorem 35.12.4, but without a nonzero-target assumption, there exist \(N_0\) and \(C\) such that for every \(N\ge \max (1,N_0)\) and \(0{\lt}\varepsilon \le 1\), the original normalized periodic state has error at most \(\varepsilon \) against a unit vector prepared in depth at most \(C\log (N/\varepsilon )\).

Fix \(d\ge 1\) and \(D\). There is \(C\) such that the normalized state of every nonzero inhomogeneous chain of bond dimension \(D\) on \(N\ge 1\) sites is prepared exactly by a local circuit of depth at most \(CN\). No injectivity or correlation-decay assumption is required. This is the sequential circuit of  [ 219 ] , applied to one block covering the ring, including the isometric extension for a non-injective tensor.

There is \(C_0{\gt}0\), depending only on \(D\) and \(\sigma \), with the following property. Let \(B_j\) be any family of tensors with common bond dimension \(D\), let \(P_j\) be their positive polar factors, and suppose \(\| T_{B_j}-T_\sigma \| \le \delta \) for every \(j\), where \(\delta \ge 0\). For \(M\ge 1\), set \(\tau _j=T_{P_j,P_\infty }=\sum _u\overline{P_\infty ^u}\otimes P_j^u\). Then

There is \(C_1{\gt}0\), depending only on \(D,\sigma \), such that the following holds. Partition an inhomogeneous ring into \(M\ge 1\) actual contiguous blocks. Suppose the transfer matrices of each block and of the whole ring differ from \(T_\sigma \) by at most \(\delta \ge 0\). With \(\Omega =\bigotimes _j\omega _\sigma \) and \(c_N=\| \phi _N\| =\| \phi _{\mathrm{pos}}\| \), one has

The algebraic convention is \(0^{-1}=0\); a normalized quantum state requires \(c_N{\gt}0\).

Fix \(d,D\ge 1\), \(K,r{\gt}0\), and \(\sigma {\gt}0\) of trace one. Let \(A_N\) be a family of inhomogeneous chains with nonzero periodic states. Suppose every positive contiguous interval \(I\) in the ordering \(0,\ldots ,N-1\) satisfies

There is \(C\) such that for every \(N\ge 1\) and \(0{\lt}\varepsilon \le 1\) a unit vector \(\psi \) can be prepared by physical local gates in depth at most \(C\log (N/\varepsilon )\), with \(1-|\langle \psi ,\phi _N/\| \phi _N\| \rangle |\le \varepsilon \).

Fix a physical dimension \(d\ge 1\), a bond bound \(D\), and \(c{\gt}0\). There are constants \(C\) and \(N_0\), depending only on \(d,D,c\), with the following property. For every \(N\ge 1\), let \(A^{(N)}\) be a chain with bond dimensions at most \(D\), and cut it into \(M_N\ge 1\) blocks satisfying \(c\log N\le \ell _{N,k}\le L_N\). Suppose its positive-part state is nonzero and approximable by unit pairs with error at most \(\delta _N\). Then there are unit vectors \(\psi _N\) prepared by local circuits of depths \(T_N\le CL_N\) with \(0\le \varepsilon (\psi _N,\phi _N)\le \delta _N\). For \(N{\lt}N_0\) one can take \(\psi _N=\phi _N/\lVert \phi _N\rVert \) exactly. In particular, \(\delta _N\to 0\) implies \(\varepsilon (\psi _N,\phi _N)\to 0\), and \(L_N=O(\log N)\) implies \(T_N=O(\log N)\). No injectivity is required.

The positive lower bound on block lengths is an additional hypothesis: the source’s upper bound \(q_N=O(\log N)\) alone does not imply it. Convergence of the errors supplies no accuracy-dependent rate, so this statement does not establish \(O(\log (N/\varepsilon ))\) for arbitrary requested accuracy  [ 66 ] .

Fix \(d\ge 1\) and a family \(A^{(N)}\) of inhomogeneous chains with bond dimensions at most \(D\). Suppose there are constants \(K,r{\gt}0\) and an integer \(k\ge 0\), independent of \(N\), such that at every scale \(1\le q\le N\) there is a partition into blocks of lengths between \(q\) and \(2q\) with a nonzero positive-part state approximable by unit pairs with error at most \(KN^k e^{-rq}\). Then there is \(C\) such that for every \(N\ge 1\) and \(0{\lt}\varepsilon \le 1\), a unit vector \(\psi \) can be prepared with depth \(T\le C\log (N/\varepsilon )\) and error \(\varepsilon (\psi ,\phi _N)\le \varepsilon \).

This is a conditional variant of the accuracy-dependent claim in the inhomogeneous paragraph of  [ 219 ] . The uniform rate is an additional assumption: convergence of the approximation errors alone does not establish it  [ 66 ] .

Fix \(N_0\ge 0\). Suppose a family of nonzero targets at lengths \(N\ge \max (1,N_0)\) has pair error at most \(KN^ke^{-rq}\) at every blocking scale \(1\le q\le N\), on some partition with \(q\le \ell _j\le 2q\). Here \(K,r{\gt}0\) and \(k\ge 0\) are uniform in \(N,q\). Then its normalized states at these same lengths can be approximated to error \(0{\lt}\varepsilon \le 1\) in depth \(C\log (N/\varepsilon )\).

For \(d\geq 2\), every bond dimension \(D\) and every number \(b\) of labels, there are \(C\) and \(L_0\) such that the following holds. Let \(A\) be a tensor of bond dimension \(D\), \(S\) a set of bond pairs, \(\omega _1,\dots ,\omega _b\) orthonormal pairs whose products vanish on every configuration of bond pairs leaving \(S\), and \(\alpha \) a unit vector. Cut the ring of \(N\) sites into \(M\geq 1\) blocks of lengths \(L_0\leq \ell _k\leq L\) at which every blocked tensor of \(A\) is injective on \(S\). Then \(\sum _j\alpha _j\bigl(\bigotimes _kV^{(\ell _k)}\bigr)\bigotimes _k|\omega _j\rangle \) is prepared with measurements and a circuit in depth at most \(CL\). A nonzero vector prepared by a local circuit of depth \(T\) is prepared with measurements and a circuit in depth \(T\).

Let \(x=e^{-\gamma /\xi }{\gt}0\), so that \(e^{-\gamma q/\xi }=x^q\) and \(e^{-2\gamma /\xi }=x^2\). Let \(L_1,\dots ,L_b\) be natural numbers, \(C\geq 1+\sum _jx^{-L_j}\), \(M\geq 1\), and \(y=Mx^q\). If \(Cy{\lt}1\), then \(x{\lt}1\) and \(L_j\leq q\) for every \(j\). Hence, if \(\varepsilon \leq 1\) and \(\varepsilon \leq Cy\, e^{Cy}\) holds whenever \(x{\lt}1\), \(L_j\leq q\) for every \(j\), and \(Cy{\lt}1\), then \(\varepsilon \leq Cy\, e^{Cy}\).

Let \(D^2\le d^{3D}\). There is \(C\), depending only on \(d\) and \(D\), such that the following holds. Cut the ring of \(N\ge 1\) sites into \(M\ge 1\) blocks of lengths \(3D\le \ell _k\le L\), let \(A_1,\dots ,A_N\) be tensors of bond dimension \(D\), and let \(\omega ^k\) be unit vectors on \(\mathbb {C}^D\otimes \mathbb {C}^D\). Then there are isometries \(W_k\) with \(W_k\Pi _k=V_k\) for the polar decompositions \(B_k=V_kP_k\) of the blocked tensors such that \((\bigotimes _kW_k)|\Omega \rangle \) is prepared in depth at most \(CL\). No injectivity is assumed.

For matrices \(W_k\) from \(\mathbb {C}^{D^2}\) to the sites of block \(k\), let \((\bigotimes _kW_k)x\) have the coefficient \(\sum _a\prod _k\langle s_k|W_k|a_k\rangle x(a)\) at the configuration \(s\) read block by block; for \(W_k=V_k\) it is the vector of Definition 35.10.2. If every \(W_k\) is an isometry, \(\bigotimes _kW_k\) preserves inner products and norms. If \(W_k\Pi _k=V_k\) for every \(k\), then \(|\phi _N\rangle =(\bigotimes _kW_k)|\phi _{\mathrm{pos}}\rangle \), and if moreover every \(W_k\) is an isometry, then for all vectors \(\omega ^k\)

with the reading of \(x/\lVert x\rVert \) for \(x=0\) as in Theorem 35.10.4.

Encode the bond indices injectively in \(D\ge 1\) sites. There is \(C\) such that for every \(q\ge 3D\) and all tensors \(A_1,\dots ,A_q\) of bond dimension \(D\), one on each site, with blocked tensor \(B=VP\) and \(V^\dagger V=\Pi \), there are a unitary \(U\) on the \(q\) sites, a product of at most \(Cq\) gates on neighbouring sites, and an isometry \(W\) from \(\mathbb {C}^{D^2}\) to the \(q\) sites with \(W\Pi =V\) and \(U|l,0,r\rangle =W|l,r\rangle \) for all \(l\) and \(r\). No injectivity is assumed.

In Definition 35.17.2, let each \(\sigma _j\) be positive semidefinite with \(\operatorname {Tr}\sigma _j=1\). For every choice of the copies \(k_j\) the embedded pairs satisfy \(\langle \omega _j | \omega _{j'} \rangle =\delta _{jj'}\). The chosen coordinates of block \(j\) carry the block \(\mu _{j,k_j}A_j\) of \(A\), and embedding the pair embeds its product state. Moreover, for \(M\geq 1\), in the bond space \(\mathbb {C}^{D_j}\otimes \mathbb {C}^{D_j}\) of block \(j\) alone, the tensor \(\bigoplus _k\mu _{j,k}^qP_{j,\infty }\) generates on \(M\) sites the state \(\bigl(\sum _k\mu _{j,k}^{qM}\bigr)|\Omega _j\rangle =\beta _j|\Omega _j\rangle \) with \(N=qM\) and \(|\Omega _j\rangle \) formed from the unembedded pair. The orthogonality is asserted after [ 219 , eq. (S7) ] , where the reason given is that the blocks form a basis of normal tensors; here it follows from the placement of the pairs alone. The identification of these pairs with the fixed points of the positive parts of the blocked tensor is not proved here and can fail when a block has multiplicity greater than one [ 182 ] .

Let \(A\) be normal with \(D\ge 1\). There are \(s\ge 1\), \(C\), \(a{\gt}0\) and \(b\ge 1\), depending only on \(A\), with the following property. Let \(\varepsilon {\gt}0\), \(N\ge 1\) and \(k\ge 0\) be such that \(s2^{k+1}\) divides \(N\) and \(s2^{k+1}\ge a\log (N/\varepsilon )+b\). Then a unit vector \(|\psi \rangle \) with \(\varepsilon (\psi ,\phi _N)\le \varepsilon \) is prepared with measurement rounds in depth at most \(C(k+1)\). If moreover \(s2^{k+1}\le 2(a\log (N/\varepsilon )+b)\), then \(k\le \log _2(a\log (N/\varepsilon )+b)\), so the depth is at most \(C(\log _2(a\log (N/\varepsilon )+b)+1)=O(\log \log (N/\varepsilon ))\). This is the depth of the paragraph “Tree-RG circuit with measurements” of  [ 219 ] for normal tensors and the chain lengths divisible by \(s2^{k+1}\) for some \(k\) in this window, for instance \(N=Ms2^K\) with \(K\) large; the other chain lengths are treated in Theorem 35.21.9.

Let \(A\) be normal. There is a constant \(c\) depending only on \(A\) such that, for every \(N\ge 2\) and \(0{\lt}\varepsilon \le 1\) for which the periodic state \(|\phi _N(A)\rangle \) is nonzero, a unit vector \(|\psi \rangle \) with \(\varepsilon (\psi ,\phi _N)\le \varepsilon \) is prepared with measurement rounds in depth at most \(c\log (\log (N/\varepsilon )+1)\), which is \(O(\log \log (N/\varepsilon ))\). No outcome is post-selected. If \(D\ge 1\), there is \(N_0\) such that the condition \(|\phi _N(A)\rangle \neq 0\) holds for every \(N\ge N_0\). This is the depth of the paragraph “Tree-RG circuit with measurements” of  [ 219 ] for normal tensors; the source normalizes \(|\phi _N\rangle \), which presupposes \(|\phi _N(A)\rangle \neq 0\) [ 91 ] .

Let \(A\) be the direct sum with multiplicities of Definition 35.17.1.1, with \(b\geq 1\) blocks, every block \(A_j\) normal in the gauge (3) with a positive definite fixed point \(\sigma _j\) of trace one. There are \(a{\gt}0\), \(b'\geq 1\) and \(c\), depending only on \(A\), with the following property. Let \(N\geq 2\), \(0{\lt}\varepsilon \leq 1\), and let \(q\) be a block length dividing \(N\) with \(a\log (N/\varepsilon )+b'\leq q\leq 2(a\log (N/\varepsilon )+b')\), at which \(B_j^\dagger B_{j'}=0\) for \(j\neq j'\), such that the weights \(\beta _j=\sum _k\mu _{j,k}^N\) are not all zero. Then a unit vector \(|\psi \rangle \) with \(\varepsilon (\psi ,\phi _N)\leq \varepsilon \) is prepared with measurements and a circuit in depth at most \(c\log (N/\varepsilon )\). This is the depth \(O(\log (N/\varepsilon ))\) of the paragraph “Long-range MPS using measurements” of [ 219 ] , with measurements used only for the GHZ-type state, for the corrected approximating state, restricted to blocks whose \(q\)-site states are orthogonal and to block lengths dividing \(N\) [ 72 ] . Because \(q\) must divide \(N\) and lie in the window above, the statement applies only to chain lengths with a divisor in the window and is vacuous for the others, for instance for every large prime \(N\).

For every site-dependent chain of positive bond dimension \(D\) with at least one physical site, choose any central site. Let \(B=VP\) be its blocked polar decomposition, without an injectivity assumption. There exist \(s\le D^2\) and an isometry \(J\colon \mathbb {C}^s\to \mathbb {C}^{D^2}\) with \(JJ^\dagger =\Pi \) such that \(VJ\) has a mixed sequential factorization. In particular,

Both outer bonds have dimension one, every side bond is at most \(D^2\), and every inward site map is isometric on its bond space. Either endpoint may be the centre, and zero support is allowed. This is the support-domain correction of footnotes 3 and 4 to equations (13)–(15) of  [ 219 ] . For a non-injective block, the printed full-input identity fails because \(V^\dagger V=\Pi \ne \mathbb {1}_{D^2}\); a normal counterexample is proved in Theorem 35.4.10; the witnesses and correction are recorded in  [ 129 ] .

If the matrix product \(V\) of (5) is an isometry, it has a mixed sequential factorization about its chosen central site.

Let \(A_1,\ldots ,A_q\) have positive bond dimension \(D\), let \(q\ge 1\), and suppose the blocked tensor \(B\) is injective. Its polar isometry \(V=BP^{-1}\) has a mixed sequential factorization about any chosen physical site, including either endpoint. This is the injective case of the two-sided construction in [ 219 , footnote 4 to eqs. (13)–(15) ] . For non-injective \(B\), footnote 3 uses the pseudoinverse and \(V^\dagger V\) is the polar support projector, rather than the identity; the full-input isometry assertion therefore fails. The corrected support-restricted factorization is given below in Theorem 35.4.11 [ 129 ] .

Right multiplication by an input coordinate map \(J\) commutes with attaching the input to the central tensor. For any blocked map \(B\), including a non-injective one, its polar partial isometry \(V\) has such a central input factor and satisfies \(V^\dagger V=\Pi \), where \(\Pi \) is the polar support projector. Splitting the physical configurations into left, central, and reversed-right parts preserves this identity.

Let

be the canonical repeated-block form, with normal inequivalent blocks in the left-canonical gauge, and positive definite transfer fixed points \(\sigma _j\) of trace one. The copy weights are arbitrary nonzero complex numbers under the existing convention that zero padding is omitted. There is \(c\), fixed before \(N\) and \(\varepsilon \), such that every \(N\ge 2\) with \(\phi _N(A)\ne 0\) and every \(0{\lt}\varepsilon \le 1\) admit a unit vector \(\psi \) satisfying

The protocol prepares \(\psi \) in measurement rounds of quantum depth \(T\). Finite-block physical ranges may overlap. Individual sums \(\beta _j(N)=\sum _k\mu _{j,k}^N\) may vanish.

Let \(A\) be a normal tensor with \(D\ge 1\). There are \(s_0\ge 1\), \(\xi {\gt}0\) and \(C{\gt}0\) such that for every \(s\ge s_0\), every \(\varepsilon {\gt}0\) and every \(M\ge 1\) there are \(k\ge 0\) and a binary finite-range network with sites of dimension \(d^s\) and \(k+1\) isometry layers with \(k\le \log _2\max (1,\xi \log (CN/\varepsilon ))\), \(N=Ms2^{k+1}\), whose state on registers \(|\psi \rangle \) satisfies \(\varepsilon (\psi ,\phi _N)\le \varepsilon \) against the normalized state \(|\phi _N\rangle \) of \(A\). This is a scope restriction, to the chain lengths \(N=Ms2^{k+1}\) with \(k\) as above, of the containment of normal translation-invariant matrix product states in finite-range networks with \(O(\log \log N)\) layers, within the approximation error, of  [ 219 , paragraph “Connection to MERA” ] . Theorem 35.22.9 gives an unequal-block network for every positive length with nonzero periodic state, with a physical leaf bound fixed before \(N\) and \(\varepsilon \); see the resolved note  [ 154 ] .

Let \(\psi ,v\) lie in a complex inner-product space with \(\lVert \psi \rVert \le 1\). Suppose \(|1-|\langle \psi ,v\rangle ||\le a\) and \(|\lVert v\rVert ^2-1|\le b\). Define \(\widehat v=v/\lVert v\rVert \) when \(v\ne 0\) and \(\widehat v=0\) when \(v=0\). Then \(1-|\langle \psi ,\widehat v\rangle |\le a+b\).

Let \(A\) act on \(S\). If configurations \(\sigma ,\tau \) differ at a site outside \(S\), then \(\langle \sigma |A|\tau \rangle =0\). If \(A,B\) act on disjoint sets, then \(\langle \sigma |AB|\sigma \rangle = \langle \sigma |A|\sigma \rangle \langle \sigma |B|\sigma \rangle \).

For an arbitrary tensor \(B\), virtual vector \(v\in \mathbb {C}^{D^2}\), and one block of \(N\) sites, choose \(\omega (l,r)=v(r,l)\). The blocked polar state with pair \(\omega \) is exactly \(S_Nv\), where \(S_N\) is the polar partial isometry of \(C_N(B)\).

Let \(\mathcal A\) be a normed complex algebra, let \(e^2=e\), and let \(\tau :\mathcal A\to \mathbb C\) be linear with \(\tau (e)=1\), \(\tau (ab)=\tau (ba)\) and \(|\tau (a)|\le K\lVert a\rVert \) for \(K\ge 0\). There is \(C{\gt}0\), depending only on these data, with the following property. For any sequences \(T_j\in \mathcal A\), \(\alpha _j\in \mathbb C\), real \(\delta \), and integer \(M\ge 2\), suppose

If \(CM\delta ^2{\lt}1\), then \(\left|\tau (\prod _{j=0}^{M-1}T_j)-1\right| \le CM\delta ^2e^{CM\delta ^2}\).

Let \(e\) be an idempotent in a normed ring and put \(f=1-e\). Let \(\tau \) be an additive complex-valued map satisfying \(\tau (ab)=\tau (ba)\) and \(|\tau (a)|\le K\lVert a\rVert \), where \(K\ge 0\). Suppose that a sequence \(Z_j\) satisfies \(eZ_je=0\) and \(\lVert fZ_je\rVert ,\lVert eZ_jf\rVert ,\lVert fZ_jf\rVert \le z\le 1/2\). For every \(M\ge 2\),

All products are ordered with increasing index from left to right.

There is \(C=C(d,D)\) such that for each faithful density matrix \(\sigma \) and \(K,r{\gt}0\), there is \(b\ge 1\) with the following property. Every nonzero chain satisfying the ordered mixing hypothesis of Theorem 35.13.3 has a unit approximation of error at most \(\varepsilon \), prepared in depth \(T\le C\min (N,q)\) with \(q=\lceil r^{-1}\log (N/\varepsilon )+b\rceil \ge 3D+1\).

Fix a faithful density matrix \(\sigma \) on \(\mathbb C^D\). There is \(C_p{\gt}0\), independent of the physical dimension, chain and ring length, with the following property. Suppose every positive contiguous block of a nonzero chain has ordered transfer error at most \(Ke^{-r\ell }\) relative to \(X\mapsto \operatorname {Tr}(X)\sigma \), with \(K,r{\gt}0\). For every \(1\le q\le N\), the remainder-absorbing partition has \(q\le \ell _j{\lt}2q\) and normalized pair error at most \(C_pKN e^{-rq}\).

In a normed ring, let \(e^2=e\), \(\lVert e\rVert ,\lVert 1\rVert \le c\), and \(\lVert X_j-e\rVert \le \delta \) for every \(j\). Then for every \(M\ge 0\), \(\lVert \prod _{j=0}^{M-1}X_j-e^M\rVert \le c((1+c\delta )^M-1)\).

In any ring, for a sequence \(X_j\), an element \(e\), and \(M\ge 0\),

For every family of bond dimensions \(D_1,\dots ,D_b\) there are \(C\) and \(L_0\) such that the following holds. Let \(A_j\) be tensors of bond dimensions \(D_j\), \(\omega _j\) unit vectors on \(\mathbb {C}^{D_j}\otimes \mathbb {C}^{D_j}\) and \(\alpha \) a unit vector in \(\mathbb {C}^b\). Cut the ring of \(N\) sites into \(M\geq 1\) blocks of lengths \(L_0\leq \ell _k\leq L\) such that every blocked tensor of every \(A_j\) is injective and \(B_j^\dagger B_{j'}=0\) for \(j\neq j'\) on every block. Then \(\sum _j\alpha _j|\psi _{\ell ,\omega _j}(A_j)\rangle \), with the states of Definition 35.5.10, is prepared with measurements and a circuit in depth at most \(CL\). This is the preparation of \(V^{\otimes N/q}W^{\otimes N/q}|\chi _{N/q}\rangle \) in the paragraph “Long-range MPS using measurements” of [ 219 ] , for blocks with orthogonal states.

Let \(A\) be the direct sum with multiplicities of Definition 35.17.1.1, with nonzero complex weights, every block \(A_j\) normal in the gauge (3) with a positive definite fixed point \(\rho _j\) of trace one, and let the mixed transfer maps \(\mathcal{E}_{jj'}\), \(j\neq j'\), have all eigenvalues of modulus below one. There is \(c\), depending only on \(A\), such that for every \(N\geq 2\) and \(0{\lt}\varepsilon \leq 1\) at which \(|\phi _N(A)\rangle \neq 0\), a unit vector \(|\psi \rangle \) with \(\varepsilon (\psi ,\phi _N)\leq \varepsilon \) is prepared with measurements and a circuit in depth at most \(c\log (N/\varepsilon )\). There is \(N_0\) such that \(|\phi _N(A)\rangle \neq 0\) for every \(N\geq N_0\) at which the weights \(\beta _j=\sum _k\mu _{j,k}^N\) are not all zero, so the same holds for these \(N\). This is the depth \(O(\log (N/\varepsilon ))\) of the paragraph “Long-range MPS using measurements” of [ 219 ] , with measurements used only for the GHZ-type state, for blocks whose \(q\)-site states may overlap, every multiplicity, every nonzero complex weight and every chain length, under the readings recorded in [ 72 , 142 , 91 ] .

For the tensor of Theorem 35.17.9, put \(s=(3/5)^q\), let \(G_q\) be the Gram matrix on its diagonal bond pairs, and let \(J|j\rangle =|j,j\rangle \). For every \(q\ge 1\), \(G_q\) is positive definite and the positive polar matrix is \(P_q=J\sqrt{G_q}J^\dagger \). In operator norm,

For every \(C\ge 0\) and every threshold \(q_0\), there is \(q\ge \max (1,q_0)\) such that \(\lVert P_q-JJ^\dagger \rVert {\gt}Cs^2\). Thus this matrix difference is not \(O((3/5)^{2q})\). The quadratic normalized-state error of Theorem 35.17.5.4 concerns a different quantity.

Let \(0{\lt}\gamma {\lt}1\). There is \(C{\gt}0\) such that the following holds for all weights \(\beta \in \mathbb {C}^b\setminus \{ 0\} \) and every cut of a ring of \(N\) sites into \(M\geq 1\) blocks of lengths \(\ell _k\geq q\) at which every blocked map \(\mathcal B^{(\ell _k)}\) is injective on the bond pairs of the blocks. With \(\alpha _j=\beta _j/(\sum _l\lvert \beta _l\rvert ^2)^{1/2}\) and \(|\phi \rangle =\sum _j\beta _j|\phi _N(A_j)\rangle \),

Here \(-\gamma q/\xi \) is read as \(0\) when \(\xi =0\), that is when \(\lambda _2=0\); the bound then gives no decay in \(q\). The constant does not depend on the weights. For the canonical form \(A^i=\bigoplus _j\operatorname {diag}(\mu _{j,1},\dots ,\mu _{j,m_j})\otimes A_j^i\) of [ 219 , eq. (S2) ] , \(|\phi _N(A)\rangle \) is \(|\phi \rangle \) with \(\beta _j=\sum _k\mu _{j,k}^N\), so this bounds the error for every multiplicity and every complex weight. This is [ 219 , Lemma 1\('\)(ii) ] for the state of the paragraph “Long-range MPS using measurements”, with blocks of unequal lengths, with \(\xi _{\mathrm{diag}}\) replaced by \(\max (\xi _{\mathrm{diag}},\xi _{\mathrm{off\text{-}diag}})\) [ 142 ] , and with the isometries of the direct sum with weights one in place of that of the blocked map of \(A\) [ 72 ] . Unlike Theorem 35.17.3.7, it has no factor \(\min (1,\sum _j\lvert \beta _j\rvert ^2)^{-1/2}\). The source states the bound for \(0{\lt}\gamma {\lt}1/2\); the range \(\gamma {\lt}1\) is a project result.

For any subchain with positive bond dimension \(D\), its coefficient matrix \(T_{\sigma ,(\alpha ,\beta )}=(A_1^{\sigma _1}\cdots A_n^{\sigma _n})_{\alpha \beta }\) factors as \(T=ES\). Here \(E_{\sigma ,a}=(Q_1^{\sigma _1}\cdots Q_n^{\sigma _n})_{1a}\), the outer bond has dimension one, every bond has dimension at most \(D^2\), and each site map \(Q_p\) is an isometry from its inward bond to its outward bond and physical site. In particular, \(E^\dagger E=\mathbb {1}\).

Let \(S,S'\subseteq \{ 0,\dots ,D-1\} \) be disjoint, and let \(\omega ,\omega '\in \mathbb {C}^D\otimes \mathbb {C}^D\) satisfy \(\omega _{(a,c)}=0\) for \(a\notin S\) and \(\omega '_{(a,c)}=0\) for \(a\notin S'\). Then \(\langle \omega | \omega ' \rangle =0\). The embedded pairs of distinct blocks in Definition 35.17.2 are supported in this way, so this gives the off-diagonal half of the orthogonality \(\langle \omega _j | \omega _{j'} \rangle =\delta _{jj'}\), which Theorem 35.17.4 completes.

In the setting of Theorem 35.5.1.2, suppose \(\operatorname{Tr}\sigma =\operatorname{Tr}\mathcal{E}_B(\sigma )=1\). Put \(\iota (X)=(X^i\sqrt\sigma )_i\), with the norm induced by the coordinate inner product. Then \(\operatorname {Re}\alpha \le 1\) and \(1-\operatorname {Re}\alpha =\lVert \iota (P)-\iota (P_\infty )\rVert ^2/2\).

Let \(B\) be a tensor, \(P\) its positive polar tensor, and \(\sigma \ge 0\) a matrix on its virtual space. Let \(P_\infty \) be the fixed-point tensor associated with \(\sigma \), and put \(\alpha =\operatorname{Tr}\sum _iP^i\sigma (P_\infty ^i)^\dagger \). Then \(\alpha \) is real and nonnegative.

Let \(V\) be an injective column matrix, \(F\) its polar isometry, and \(W\) any isometry on the same coordinate spaces. Set \(\eta =\| V^\dagger V-I\| \) and \(\zeta =\| W^\dagger V-I\| \). Then

Let \(M=WQ\) with \(Q\ge 0\), and suppose \(W^\dagger W=E\) for the orthogonal projector \(E\) onto the range of \(Q\). Then \(Q=(M^\dagger M)^{1/2}\), \(E\) is the support projector of \(M\), and \(W\) is the partial isometry of the polar decomposition of \(M\). In particular, every positive semidefinite \(Q\) with \(Q^2=M^\dagger M\) is the positive part of \(M\), and every orthogonal projector with the range of that positive part is the support projector.

Let \(A\) be a normal tensor in the gauge \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), with a positive definite fixed matrix \(\rho \) satisfying \(\mathcal{E}_A(\rho )=\rho \) and \(\operatorname{Tr}\rho =1\). Let \(\lambda _2\) bound the moduli of the transfer eigenvalues other than \(1\), assume \(\xi =-1/\log |\lambda _2|{\gt}0\), and fix \(0{\lt}\gamma {\lt}2\). There is a constant \(C_\gamma {\gt}0\), independent of \(\eta ,q,M\), such that for every \(\eta {\gt}0\) and every uniform decomposition \(N=Mq\geq 2\),

This is a polynomial-accuracy corollary of the exponential approximation estimate of  [ 219 ] . The uniform-blocking and positive-correlation-length scope is recorded in  [ 89 ] . The coefficient \(2\xi \) printed after Lemma 1 of  [ 219 ] is the case \(\gamma =1/2\); every coefficient larger than \(\xi /2\) is admissible.

Let \(A\) be a normal tensor in the gauge \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), with a positive definite fixed matrix \(\rho \) satisfying \(\mathcal{E}_A(\rho )=\rho \) and \(\operatorname{Tr}\rho =1\). Let \(\lambda _2\) bound the moduli of the transfer eigenvalues other than \(1\), and assume \(\xi =-1/\log |\lambda _2|{\gt}0\). There is a constant \(C{\gt}0\) such that for every \(\eta {\gt}0\) and every uniform decomposition \(N=Mq\geq 2\) with

the fixed-point approximation satisfies

The constant is independent of \(\eta ,q,M\). This is the uniform-blocking, positive-correlation-length case of the paragraph following Lemma 1 of  [ 219 ] ; the other chain lengths and zero correlation length are discussed in  [ 89 ] .

Let \(K,r{\gt}0\), \(N\ge 1\), and let \(k\) be a nonnegative integer. If \(a\ge \max (k,1)/r\), \(b\ge \max (\log K,0)/r\), and \(q\ge a\log (N/\varepsilon )+b\), then \(KN^ke^{-rq}\le \varepsilon \).

Let \(P_q=(\mathcal B_q^\dagger \mathcal B_q)^{1/2}\). Then \(P_q\to P_\infty \) as \(q\to \infty \), where \(P_\infty =(\sqrt\rho )^{\mathsf T}\otimes \mathbb {1}\) [ 219 , eq. (8) ] . More generally, for any tensors \(B_q\) whose transfer maps satisfy \(\mathcal{E}_{B_q}(X)\to \operatorname{Tr}(X)\, \rho \) for every \(X\), where \(\rho \ge 0\), the positive parts of the physical matrices of \(B_q\) converge to \((\sqrt\rho )^{\mathsf T}\otimes \mathbb {1}\), and the corresponding tensors converge to \(P_\infty \).

Let \(A\) be normal in the gauge (3) with correlation length \(\xi \), and let \(0{\lt}\gamma {\lt}1\). There is \(C{\gt}0\) such that for all \(q\) and all \(M\ge 2\) with \(Cy{\lt}1\), where \(y=M\, e^{-2\gamma q/\xi }\),

For \(M=1\) there is \(K\) with \(\lVert \phi _1(P_q)\rVert ^2 -\lvert \langle \phi _1(P_\infty )|\phi _1(P_q)\rangle \rvert ^2\le Ke^{-2\gamma q/\xi }\) for all \(q\). These bounds are of second order in \(e^{-\gamma q/\xi }\). Lemma 35.5.1, the estimate of [ 227 ] quoted as [ 219 , eq. (S9) ] , is of first order, and so is the bound of [ 223 , App. B4, eq. (S52) ] , which is linear in the residual. The second-order bounds are new here.

Let \(A\) be normal in the gauge (3), let \(\lambda _2\in \mathbb {C}\) satisfy \(\lvert \mu \rvert \le \lvert \lambda _2\rvert \) for every eigenvalue \(\mu \ne 1\) of \(\mathcal{E}_A\), let \(\xi \) be the correlation length at \(\lambda _2\) of Definition 14.2.6, and let \(0{\lt}\gamma {\lt}1\). Write \(r=e^{-\gamma /\xi }\), where \(-\gamma /\xi \) is read as \(0\) when \(\xi =0\); thus \(r=\lvert \lambda _2\rvert ^{\gamma }\) when \(\lambda _2\ne 0\), and \(r=1\) when \(\lambda _2=0\). Then \(a{\lt}r\) for every real number \(a{\lt}1\) with \(a\le \lvert \lambda _2\rvert \). There are constants \(C,K,K'\) depending only on \(A\), \(\rho \), \(\lambda _2\), and \(\gamma \) such that \(\lVert \mathcal{E}_A^{\, n}(X)-\operatorname{Tr}(X)\rho \rVert \le Cr^n\lVert X\rVert \) for all \(n\) and \(X\), and

for all \(q\). In [ 227 , Supplemental Material, eq. (26) ] the bound on \(P_q-P_\infty \) comes from Theorem 35.3.4, which halves the exponent. This gives the rate of [ 227 , eq. (34) ] and the range \(0{\lt}\gamma {\lt}1/2\) of [ 219 , Lemma 1 and eq. (S9) ] . The Lipschitz bound at the positive definite limit \(P_\infty ^2\) gives the range \(0{\lt}\gamma {\lt}1\). This step is [ 223 , App. B3, eqs. (S27)–(S32) ] for diagonalizable transfer matrices, and eqs. (S33)–(S38) there extend it to every normal tensor at each correlation length larger than \(\xi \).

Let \(A\) be as in Theorem 35.21.2.8, and let \(|\phi _N(A)\rangle \neq 0\) for all large \(N\). There is \(C\) such that the sequence of normalized states \(|\phi _N\rangle \) is asymptotically prepared with measurements in depth \(C\log N\): unit vectors \(|\psi _N\rangle \), prepared with measurements and a circuit in depth at most \(C\log N\), satisfy \(\lVert |\psi _N\rangle \langle \psi _N|-|\phi _N\rangle \langle \phi _N|\rVert _1\to 0\).

This is the direction from the trivial sequence to the sequence of MPS of the theorem of [ 227 ] stating that in one dimension every translation-invariant MPS of fixed bond dimension is in the same phase as the trivial state, for the equivalence of sequences of states of its paragraph “Phases of matter”; the depth \(C\log N\) is polylogarithmic. The converse direction, and the reduction of an arbitrary translation-invariant tensor to the canonical form assumed here, are not formalized [ 172 ] .

For a nonempty interval \([a,b)\), write \(B^{\boldsymbol s}=A_a^{s_a}\cdots A_{b-1}^{s_{b-1}}\), a matrix of size \(D_a\times D_b\). Let \(J_i:\mathbb {C}^{D_i}\to \mathbb {C}^D\) be the coordinate isometry, and put \(P_i=J_iJ_i^\dagger \). Then

The empty product on an actual bond pads to \(P_a\), whereas the empty product of ambient matrices is \(I_D\).

For consecutive blocks, \(Q_{a:b}Q_{b:c}=Q_{a:c}\), and the matrix trace of a whole-ring reference is one. Every contiguous reference product has norm bounded in terms of \(D\) alone. If every blocked transfer and the whole-ring transfer have error at most \(\delta \) from their corner references, the normalized positive-part state has pair error at most \(C_DM\sqrt\delta \). The pair leaving a block uses the density at the next block’s incoming bond.

Give every map between actual matrix spaces its Hilbert–Schmidt induced operator norm, namely the \(L^2\) operator norm of its matrix in the matrix-unit bases. Padding changes transfer-matrix errors by at most a constant depending only on \(D\). The Gram reference and its positive square root are

Consequently the positive-polar error is bounded by a dimension-dependent constant times \(\| F_{a:b}-\operatorname {Tr}(\cdot )\sigma _a\| ^{1/2}\). Moreover \(\sqrt{\widehat\sigma _i}=J_i\sqrt{\sigma _i}J_i^\dagger \), so the pair obtained from the padded reference is precisely the padding of the pair on the actual bond.

Fix \(d{\gt}0\) and \(D\), and let the preceding data vary with the ring length \(N\). Suppose \(K,r{\gt}0\) are independent of \(N\) and every nonempty block \([a,b)\) of every partition of the ordered ring satisfies

There are \(N_0\) and \(C\), chosen before \(N\) and \(\varepsilon \), such that for positive ring lengths \(N\ge N_0\) and \(0{\lt}\varepsilon \le 1\), the normalized ring state admits a unit-vector approximation with error at most \(\varepsilon \) prepared by a ring circuit of depth at most \(C\log (N/\varepsilon )\).

Let \(d{\gt}0\), and fix \(D\), \(s\ge 1\) and \(\eta {\gt}0\). Suppose a family of original varying-bond rings satisfies the preceding positive-dimension, channel and local minorization conditions with these same constants. There are \(N_0\ge s\) and \(C\), chosen before \(N\) and \(\varepsilon \), such that every \(N\ge N_0\) and \(0{\lt}\varepsilon \le 1\) admits a unit vector of physical ring preparation depth at most \(C\log (N/\varepsilon )\) and normalized overlap error at most \(\varepsilon \) against the original MPS.

Fix \(s\ge 1\) and \(\eta \le 1\). Suppose that every available length-\(s\) window admits a density \(\tau _a\), possibly singular, such that its normalized, output-first Choi matrix satisfies

There are cyclic densities \(\sigma _i\) with \(F_{a,b}(\sigma _b)=\sigma _a\). Every nonempty partition block satisfies

Neither compatible references nor an interval convergence rate are hypotheses.

The common register encoder satisfies \(J^\dagger J=I_b\).

Let \(G_0,\dots ,G_{m-1}\) be lists of register gates, the gates of each list with pairwise disjoint stretches, such that no register of a gate of \(G_{j'}\) lies in the interior of a gate of \(G_j\) for \(j'{\lt}j\), and every matrix of a gate a product of at most \(K\) unitaries on neighbouring sites. Then measurement rounds of total depth \(m(4s+K+2)\) implement the product of the gates of \(G_{m-1},\dots ,G_0\), the list \(G_0\) applied first, on the vectors with \(|0\rangle \) at the interiors of all the gates.

Let \(A\) be normal in the gauge (3), and let \(0{\lt}t{\lt}1\) bound the moduli of the eigenvalues of \(\mathcal{E}_A\) other than \(1\), with \(\xi =-1/\log t\). There is \(C{\gt}0\) such that for every \(s\ge 1\) with \(A\) blocked over \(s\) sites injective, every unitary \(u\) with \(u|0\rangle |0\rangle =|\omega \rangle \), every \(\varepsilon {\gt}0\), and every \(k\ge 0\) and \(M\ge 1\) with \(N=Ms2^{k+1}\) and

the state on registers \(|\psi \rangle \) of the tree network, with \(k+1\) isometry layers, of \(A\) blocked over \(s\) sites satisfies \(\varepsilon (\psi ,\phi _N)\le \varepsilon \). Moreover, for every such \(s\) and \(u\), every \(\varepsilon {\gt}0\) and every \(M\ge 1\) there is \(k\ge 0\) satisfying (61) with \(k\le \log _2\max (1,x)\).

For every \(s\ge 1\) there is \(C\), depending only on \(d\) and \(s\), with the following property. Let \(A\) be a tensor whose blocked tensor over \(s\) sites is injective, and let \(\sigma \ge 0\) with \(\operatorname{Tr}\sigma =1\). For every \(k\ge 0\) and \(M\ge 1\), the approximating state of Definition 35.3.8 with blocks of \(q=s2^{k+1}\) sites, on \(N=Mq\) sites, is prepared with measurement rounds in depth at most \(C(k+1)\). No outcome is post-selected.

Let \(A\) be a tensor whose blocked tensor \(\mathcal B\) over \(s\) sites has an injective two-site blocked tensor, let \(k\ge 0\) and \(q=s2^{k+1}\), and let \(\mathcal B_q=VP\) be the polar decomposition of \(A\) blocked over \(q\) sites. Let a register of \(s\) sites carry the bond legs \(\mathbb {C}^D\) through an injective map of \(\{ 0,\dots ,D-1\} \) into its configurations, and \(\mathbb {C}^{D^2}\) through an injective map of \(\{ 0,\dots ,D^2-1\} \) into them. Every layer of the tree circuit (12) of \(\mathcal B\) is an isometry from one register to two, and extends to a unitary on the two registers of each sub-block of Theorem 35.20.1. The tree \(T\) of these unitaries satisfies

for every configuration \(\tau \) of the block and all bond indices \(l,r\), where \(|l,0,\dots ,0,r\rangle \) carries \(l\) on the first and \(r\) on the last register.

Let \(k\ge 0\), \(q=s2^{k+1}\), and cut the ring of \(N=Mq\) sites into \(M\ge 1\) blocks of \(q\) sites, each grouped into \(2^{k+1}\) registers of \(s\) consecutive sites. For \(0\le j\le k\) and \(0\le p{\lt}2^j\) the sub-block \(p\) of depth \(j\) consists of the registers \(p2^{k+1-j},\dots ,(p+1)2^{k+1-j}-1\), and a unitary \(X_{j,p}\) on \(2s\) sites acts on its first and its last register. Let \(T\) be the product of the depths \(k,\dots ,0\) of a block, the depth \(0\) applied first. If every \(X_{j,p}\) is a product of at most \(K\) unitaries on neighbouring sites, measurement rounds of total depth \((k+1)(4s+K+2)\) implement \(T\) on every block, \(T^{\otimes M}\), on the vectors with \(|0\rangle \) at every site of every block other than its first and its last \(s\) sites.

There are \(C\) and \(L_0\), depending only on \(d\) and the bond dimensions \(D_1,\dots ,D_b\), such that the following holds. Let \(A\) be the direct sum with multiplicities of Definition 35.17.1.1, let \(\sigma _j\geq 0\) with \(\operatorname{Tr}\sigma _j=1\), let \(q\geq L_0\) be a block length at which every blocked tensor of every block is injective and \(B_j^\dagger B_{j'}=0\) for \(j\neq j'\), let \(M\geq 1\), and let \(\alpha \) be a unit vector. Then the corrected approximating state \(V^{\otimes M}\sum _j\alpha _jL_j^{\otimes M}|\Omega _j\rangle \) of Definition 35.17.1.3, on \(N=Mq\) sites, equals \(\sum _j\alpha _j|\psi _{q,\omega _j}(A_j)\rangle \) with \(\omega _j\) the pair of \(\sigma _j\), is a unit vector, and is prepared with measurements and a circuit in depth at most \(Cq\). This is the paragraph “Long-range MPS using measurements” of [ 219 ] for the corrected approximating state, restricted to blocks whose \(q\)-site states are orthogonal [ 142 , 26 ] .

Let \(A\) be the direct sum with multiplicities of Definition 35.17.1.1. Its periodic state on \(N\geq 1\) sites is \(\sum _j\beta _j|\phi _N(A_j)\rangle \) with \(\beta _j=\sum _k\mu _{j,k}^N\). For \(q\geq 1\) the blocked map of \(A\) is \(B=\sum _jc_jB_jL_j^\dagger \), with \(B_j\) the blocked map of \(A_j\), and \(L_j^\dagger L_{j'}=\delta _{jj'}\mathbf1\). If \(B_j^\dagger B_{j'}=0\) for \(j\neq j'\), then, with \(B_j=V_jP_j\),

On the copies of block \(j\) the positive part is therefore of rank one in the copy index, where [ 219 , eq. (S5) ] prints \(\operatorname {diag}(\mu _{j,1}^q,\dots ,\mu _{j,m_j}^q)\otimes P_j\) [ 26 ] .

Fix two endpoint families, \(N_0\ge 2\), \(L,C\in \mathbb N\), \(K\ge 1\), and \(r{\gt}0\). Suppose that at every \(N\ge N_0\) and \(L\le q\le N\) with \(Ne^{-rq}\le 1\), some genuine local circuit of depth at most \(Cq\) converts the encoders with error at most \(K\sqrt{Ne^{-rq}}\). Suppose also that an exact conversion of depth at most \(CN\) exists for every such \(N\). Then there is \(c{\gt}0\), fixed before \(N\) and \(\varepsilon \), such that for all \(N\ge N_0\) and \(0{\lt}\varepsilon \le 1\) one conversion has depth at most \(c\log (N/\varepsilon )\) and operator-norm error at most \(\varepsilon \).

For arbitrary coordinate maps \(\iota ,\kappa \), \((E_\iota ^\dagger E_\kappa )_{a,c}=\mathbf1_{\iota (a)=\kappa (c)}\). In particular, injective maps give \(E_j^\dagger E_j=I\), and disjoint ranges give \(E_i^\dagger E_j=0\) when \(i\ne j\). Hence \(E_j^\dagger (E_jX)=X\) and \(E_i^\dagger (E_jX)=0\) in those cases. The pair embeddings satisfy \((K_j)_{(x,y),(a,c)}=\mathbf1_{x=\iota _j(a),\, y=\iota _j(c)}\), with the analogous isometry and orthogonality identities.

Under the preceding canonical and common-spectrum hypotheses, embed the sectors with injective coordinate maps of disjoint ranges. Let \(W_A\) be their simultaneous blocked-polar encoder, with pairs from the embedded fixed densities. There is \(K_Z{\gt}0\), chosen before the partition, such that for every partition into \(M\ge 1\) supported- injective blocks of lengths \(\ell _k\ge q\ge 1\), with \(y=Mx^q\),

The cross matrix is complex, so this estimate retains relative logical phases.

Let canonical data for a tensor of physical and bond dimensions \(d,D\ge 1\) supply its retained normal representatives. For \(3D^5\le L\le N\), the range of the polar encoder of those actual representatives equals the kernel of the range-\(L\) parent Hamiltonian. No further gauge or column rescaling is identified implicitly.

For every length, \(\operatorname {ran}F_{A,N}\) is the span of the original periodic sector vectors. If those vectors are linearly independent, \(F_{A,N}^\dagger F_{A,N}=I_b\).

The entries used in coherent estimates are exactly

Let \(A_j\) be finitely many normal left-canonical sectors with faithful stationary densities \(\sigma _j\) of trace one. Suppose \(|\lambda _2|{\lt}1\) bounds all nontrivial diagonal-transfer eigenvalues and every mixed-transfer eigenvalue between distinct sectors. For \(0{\lt}\gamma {\lt}1\), put \(x=\exp (-\gamma /\xi (\lambda _2))\). There is \(K_G\ge 0\), independent of \(N\), such that

Suppose \(V_{A,N}\) is injective, \(W\) is an isometric block encoder, and for \(K_G,K_Z,r\ge 0\),

If \(N\ge 1\), \(q\le N\), \(M\le N\), and \(z=Ne^{-rq}\le 1\), then

For injective coordinate maps with disjoint ranges and \(N\ge 1\), \(C_NT=V_{A,N}\). At a diagonal coordinate of sector \(j\), \(T_{(\iota _j(a),\iota _j(a)),i}=\delta _{ij}\). Every row of \(T\) outside the sector-pair set vanishes.

Let \(\psi _N=(2|0^N\rangle +|1^N\rangle )/\sqrt5\) and \(\phi _N=(|0^N\rangle +|1^N\rangle )/\sqrt2\). Every depth-\(T\) nearest-neighbour unitary circuit \(U\) on the ring, with \(N{\gt}4T+4\), satisfies

These are the normalized periodic vectors of the tensors in Theorem 35.7.1.4. Conversely, overlap error strictly below \(9/5000\) forces \(N\le 4T+4\).

Define

On every positive chain, their normalized periodic vectors are \((2|0^N\rangle +|1^N\rangle )/\sqrt5\) and \((|0^N\rangle +|1^N\rangle )/\sqrt2\). At every interaction length the local boundary-generated spaces and canonical parent projectors agree. Consequently their periodic-chain constraint spaces and canonical parent Hamiltonians agree for every chain length.

Let \(A_1,\dots ,A_q\), \(q\ge 1\), be tensors of bond dimension \(D\), one on each site, suppose their blocked tensor \(B\) with \(B^{i_1\cdots i_q}=A_1^{i_1}\cdots A_q^{i_q}\) is injective, and let \(B=VP\) be its polar decomposition as in Theorem 35.1.3.3. For equal tensors \(A_p=A\), \(B\) is the blocked tensor \(\mathcal B_q\). There are integers \(D'_1=D^2\), \(D'_{q+1}=1\), and \(D'_k\le D^2\), and matrices \(W_k^i\in \mathbb {C}^{D'_{k+1}\times D'_k}\) for \(k=1,\dots ,q\) and \(i=1,\dots ,d\), with

for every \(k\), configuration \(i_1\cdots i_q\), and index \(x\), where \(\mathbb {C}^{D'_1}\) is identified with \(\mathbb {C}^D\otimes \mathbb {C}^D\). This is the matrix product form of [ 219 , eqs. (13)–(15) ] , where the isometry carrying the input is written \(\widetilde C=V_1\). The block length \(q\ge 1\) is implicit in the source, which draws at least one site; the empty block \(q=0\) is excluded, see  [ 105 ] .

Let \(A_1,\dots ,A_q\), \(q\ge 1\), be tensors of bond dimension \(D\), one on each site, let \(G\) be a \(D^2\times D^2\) matrix, and let \(V\colon \mathbb {C}^D\otimes \mathbb {C}^D\to (\mathbb {C}^d)^{\otimes q}\) be an isometry with matrix elements \(\langle i_1\cdots i_q|V|x\rangle =\sum _{\alpha ,\beta }(A_1^{i_1}\cdots A_q^{i_q})_{\alpha \beta }\, G_{(\alpha ,\beta ),x}\). Then there are integers \(b_0=1\), \(b_q=D^2\) and \(b_p\le D^2\), and families of \(b_{p-1}\times b_p\) matrices \(X_p^i\), \(p=1,\dots ,q\), with \(\sum _i(X_p^i)^\dagger X_p^i=\mathbb {1}_{b_p}\) for every \(p\), such that \(\langle i_1\cdots i_q|V|x\rangle =(X_1^{i_1}X_2^{i_2}\cdots X_q^{i_q})_{1x}\).

Let \(A_1,\dots ,A_q\), \(q\ge 1\), be tensors of bond dimension \(D\), one on each site, let their blocked tensor \(B=VP\) be injective on a set \(S\) of bond pairs, and let \(\pi \) enumerate the bond pairs with the elements of \(S\) first, so that \(\pi (x)\in S\) exactly for \(x{\lt}\lvert S\rvert \). There are integers \(b_0=1\), \(b_q=\lvert S\rvert \) and \(b_p\le D^2\), and families of \(b_{p-1}\times b_p\) matrices \(X_p^i\) with \(\sum _i(X_p^i)^\dagger X_p^i=\mathbb {1}_{b_p}\) for every \(p\), such that \(\langle i_1\cdots i_q|V|\pi (x)\rangle =(X_1^{i_1}\cdots X_q^{i_q})_{1x}\) for every \(x{\lt}\lvert S\rvert \). For \(S\) the set of all pairs this is Theorem 35.4.12.

Let \(A_1,\dots ,A_q\), \(q\ge 1\), be tensors of bond dimension \(D\), one on each site, with blocked tensor \(B=VP\) and \(V^\dagger V=\Pi \); no injectivity is assumed. Let \(r\le D^2\) and let \(T\) be a unitary on \(\mathbb {C}^{D^2}\) with \(\Pi T=T\Lambda \), where \(\Lambda \) is the diagonal matrix with \(\Lambda _{yy}=1\) for \(y{\lt}r\) and \(\Lambda _{yy}=0\) otherwise. Then \(VT|x\rangle =0\) for every \(x\ge r\), and there are integers \(b_0=1\), \(b_q=r\) and \(b_p\le D^2\), and families of \(b_{p-1}\times b_p\) matrices \(X_p^i\) with \(\sum _i(X_p^i)^\dagger X_p^i=\mathbb {1}_{b_p}\) for every \(p\), such that \(\langle i_1\cdots i_q|VT|x\rangle =(X_1^{i_1}\cdots X_q^{i_q})_{1x}\) for every \(x{\lt}r\). This is the case \(q\ge 1\) of the footnote of  [ 219 , paragraph “The sequential-RG circuit” ] : the derivation of [ 219 , eqs. (13)–(15) ] remains valid for non-injective \(B\), with \(P^{-1}\) read as a pseudo-inverse. The source states no lower bound on \(q\); at \(q=0\) the conclusion fails for every \(D\), see  [ 105 ] .

There is \(C\), depending only on \(d\) and \(D\), such that for every tensor \(A\) and every \(N\ge 3D\) for which the \(N\)-site blocked tensor is injective and \(|\phi _N(A)\rangle \neq 0\), the normalized state \(|\phi _N\rangle \) is prepared exactly in depth at most \(CN\). For \(d\ge 1\) and every \(n_0\) there is \(K\) such that every unit vector on a ring of \(N\) sites with \(2\le N\le n_0\) is prepared in depth at most \(K\).

Fix \(d\ge 1\) and \(r_1\ge 2\). There is \(C\) such that, for every cyclic partition into \(M\ge 1\) blocks of lengths \(\ell _k\ge 3r_1\) and every unit vector \(\alpha \) on \((\mathbb {C}^d)^{\otimes r_1}\), the GHZ-type state of Definition 35.21.1.1 is prepared by measurement rounds of total quantum depth at most \(C\). There is no upper bound on \(\ell _k\). Every site outside the specified registers ends in \(|0\rangle \). The statement includes \(M=1\) and every combination of even and odd block lengths.