Dependencies

On the open chain of \(n\) sites, a gate on a window of \(m\) sites is a unitary acting on the sites \(p\) with \(a\le p{\lt}a+m\), for some \(a\). Near the right end of the chain fewer than \(m\) of these sites exist, so such a gate acts on at most \(m\) consecutive sites. An operator is a product of at most \(K\) gates on windows of \(m\) sites if it is a product of at most \(K\) such gates. These are the gates of the sequential circuit of  [ 219 , paragraph “The sequential-RG circuit” ] , each of which acts on the bond register and one further site.

Let \(|0\rangle \in \mathbb {C}^D\) be the first standard basis vector. For \(v,r\in \mathbb {C}^D\) let \(|0\rangle v^{\mathsf T}\) be the matrix with first row \(v\) and all other rows zero, and \(r\langle 0|\) the matrix with first column \(r\) and all other columns zero.

Given \(b_0,\dots ,b_N\le D\) with \(b_0=b_N=1\) and square matrices \(Q_k^i\), the varying-bond open chain with dimensions \(b_k\) has as site matrices the upper-left \(b_k\times b_{k+1}\) corners of \(Q_k^i\).

For \(\psi \in (\mathbb {C}^d)^{\otimes N}\) and \(k\ge 0\), the cut matrix of \(\psi \) at \(k\) is the matrix with rows indexed by the configurations \(\sigma \) of \(k\) sites, columns indexed by the configurations \(\tau \) of \(N-k\) sites, and entries \(\psi (\sigma \tau )\); for \(k{\gt}N\) only the first \(N\) letters of \(\sigma \) enter. The cut rank \(\operatorname {rank}_k\psi \) is its rank. For \(k\le N\) it is the rank of the reduced density operator of the first \(k\) sites.

Let \(d\ge 1\). A vector \(\psi \) is deterministically generated with a \(D\)-dimensional ancilla if there are unitaries \(U_0,\dots ,U_{N-1}\) on \(\mathbb {C}^D\otimes \mathbb {C}^d\) and unit vectors \(|\varphi _I\rangle ,|\varphi _F\rangle \in \mathbb {C}^D\) such that the ancilla component of the joint state at every configuration \(\sigma \) equals \(\psi (\sigma )|\varphi _F\rangle \); that is, the final joint state is \(|\varphi _F\rangle \otimes |\psi \rangle \) and the ancilla decouples in the last step without measurement.

A family \(Q^i\in M_{D}(\mathbb {C})\), \(i=0,\dots ,d-1\), is isometric on its first \(b\) columns if for all \(\beta ,\beta '{\lt}b\)

For \(b=D\) this is \(\sum _i(Q^i)^\dagger Q^i=\mathbb {1}_D\).

Let \(d\ge 1\) and \(N\ge 2\). Starting from \(|0\rangle ^{\otimes N}\), an operation \(U^{[k]}\) acts on sites \(k\) and \(k+1\), for \(k=1,\dots ,N-1\) in turn. The scheme is deterministic if every \(U^{[k]}\) is unitary and probabilistic if the operations are arbitrary [ 228 , Section 5.2 ] . Here sites carry the source’s labels \(1,\dots ,N\); source site \(j\) is ket position \(N-j\) in the labelling of this chapter. After \(U^{[k]}\) the scheme is described by the vector \(\chi _k\) on the first \(k+1\) sites, defined by \(\chi _1=U^{[1]}|0,0\rangle \) and

and the generated state is \(\chi _{N-1}\).

A vector \(\psi \) has an open-boundary representation with bond dimension at most \(D\) if \(\psi =c_A\) for some varying-bond open chain \(A\) with common bound \(D\), with \(c_A\) as in (1).

A vector \(\psi \in (\mathbb {C}^d)^{\otimes N}\) is probabilistically generated with a \(D\)-dimensional ancilla if it satisfies (1) for some matrices \(A_k^i\in M_{D}(\mathbb {C})\) and vectors \(|\varphi _I\rangle ,|\varphi _F\rangle \in \mathbb {C}^D\). The vector is the unnormalized post-measurement vector, as in the source’s “up to normalization”.

The reverse of a varying-bond open chain \(A\) of length \(N\) has dimensions \(D_{N-k}\) and site matrices \((A_{N-1-p}^i)^T\).

For a varying-bond open chain \(A\) of length \(N\), a site \(q\) and \(c\in \mathbb {C}\), the chain obtained by scaling the site \(q\) by \(c\) has the bond dimensions of \(A\), the site matrices \(c\, A_q^i\) at the site \(q\), and the site matrices \(A_p^i\) at every other site \(p\).

Let \(|\varphi _I\rangle ,|\varphi _F\rangle \in \mathbb {C}^D\) and let \(A_k^i\in M_{D}(\mathbb {C})\) for \(k=0,\dots ,N-1\) and \(i=0,\dots ,d-1\). Following [ 228 , Section 5.1 ] , the state produced by the operations \(\sum _{i,\alpha ,\beta }(A_k^i)_{\alpha \beta }|\alpha ,i\rangle \! \langle \beta ,0|\) followed by projecting the ancilla onto \(|\varphi _F\rangle \) is

Let \(d\ge 1\). A unitary \(U\) on \(\mathbb {C}^D\otimes \mathbb {C}^d\) acting on the site state \(|0\rangle \) induces the matrices \((A^i)_{\alpha \beta }=\langle \alpha ,i|U|\beta ,0\rangle \). For unitaries \(U_0,\dots ,U_{N-1}\), with \(U_k\) applied at step \(k+1\) and acting on site \(N-1-k\), and an initial ancilla state \(|\varphi _I\rangle \), the ancilla component of the joint state at configuration \(\sigma \) is \(A_0^{\sigma _0}\cdots A_{N-1}^{\sigma _{N-1}}|\varphi _I\rangle \in \mathbb {C}^D\), where \(A_k\) is induced by \(U_{N-1-k}\)  [ 228 , Section 5.1 ] .

A vector \(v\in \mathbb {C}^D\) is supported below \(b\) if \(v_\beta =0\) for \(\beta \ge b\). A matrix \(M\in M_{D}(\mathbb {C})\) is row-supported below \(a\) if \(M_{\alpha \beta }=0\) for \(\alpha \ge a\).

Let \(d\ge 1\) and let the ancilla be \(\mathbb {C}^D\otimes \mathbb {C}^d\), the second factor being a tag, started in a unit vector \(|\varphi _I\rangle \). In every step an arbitrary unitary \(W^{[k]}\) acts on the ancilla, followed by the fixed interaction \(T\) of the ancilla with the current site, which interchanges the tag with the site on site inputs \(|0\rangle \):

and the ancilla decouples after the last step, that is, the final joint state is \(|\varphi _F\rangle \otimes |\psi \rangle \) for a unit vector \(|\varphi _F\rangle \in \mathbb {C}^D\otimes \mathbb {C}^d\). For \(d=2\) this is the third scheme of  [ 228 , Section 5.1 ] ; the extension to sites \(\mathbb {C}^d\) is the remark of  [ 238 ] that the ideas and proofs apply to any multi-qudit state. The step with ancilla unitary \(W\) leaving the site in \(|i\rangle \) is the matrix \(S_i(W)\) on \(\mathbb {C}^D\otimes \mathbb {C}^d\) with entries

After the steps with unitaries \(W^{[1]},\dots ,W^{[N]}\), the ancilla component of the joint state at the configuration \(i_N\cdots i_1\) is \(S_{i_N}(W^{[N]})\cdots S_{i_1}(W^{[1]})|\varphi _I\rangle \).

Fix a chain length \(N\) and dimensions \(D_0,\ldots ,D_N\leq D\) with \(D_0=D_N=1\). A varying-bond open chain consists of matrices \(A_k^i\in \mathbb C^{D_k\times D_{k+1}}\) for \(k=0,\ldots ,N-1\) and \(i=0,\ldots ,d-1\). Intermediate dimensions may vanish. The number \(D\) is a common upper bound and need not be the least such bound.

This is the open-boundary representation of [ 228 , Section II.B ] . The source takes \(D=\max _k D_k\); allowing an arbitrary common upper bound does not change the rectangular chain.

For a physical configuration \(\sigma =(\sigma _0,\ldots ,\sigma _{N-1})\), define

Since \(D_0=D_N=1\), this is the scalar matrix product in [ 228 , Section II.B ] . If an intermediate dimension vanishes, the virtual-path set is empty. For \(N=0\), there is one empty path and its empty product is \(1\).

For each site \(k\), embed \(A_k^i\) into the upper-left \(D_k\times D_{k+1}\) corner of a \(D\times D\) matrix \(\widetilde A_k^i\) and set every other entry to zero. The matrices \(\widetilde A_k^i\) form a square site-dependent closed chain.

This square padding embeds the rectangular open chain into square matrices, whereas [ 228 , Theorem 3 ] places the rectangular matrices into the off-diagonal blocks of an \(ND\times ND\) cyclic matrix.

On the open chain of \(n\) sites the following hold.

1. A unitary on \(l\le m\) sites placed on consecutive sites is a gate on a window of \(m\) sites, and if \(n\le m\) every unitary on the chain is one such gate.

2. Products of at most \(K\) and \(K'\) gates on windows of \(m\) sites multiply to a product of at most \(K+K'\) such gates, and placing the chain as a block of consecutive sites of a longer chain keeps a product of at most \(K\) such gates.

3. If \(n\ge 2\), a gate on a window of two sites is a gate on two neighbouring sites.

4. For \(m\ge 2\) there is \(C\), depending only on \(d\) and \(m\), such that for every \(n\ge 2\) a product of at most \(K\) gates on windows of \(m\) sites is a product of at most \(KC\) gates on two neighbouring sites.

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.

If \(U\) is an operation on \(\mathbb {C}^D\otimes \mathbb {C}^d\) and \(W\) a matrix on \(\mathbb {C}^D\), the matrices induced by \(U(W\otimes \mathbb {1})\) are \(A^iW\), where \(A^i\) are those induced by \(U\). Consequently, replacing the first operation \(U_0\) by \(U_0(W\otimes \mathbb {1})\) turns the ancilla component of the joint state started in \(|\varphi \rangle \) into the one started in \(W|\varphi \rangle \).

Let \(b_0,\dots ,b_{n+1}\le D\) with \(b_0=1\), and let \(Q_0,\dots ,Q_n\) be families with \(Q_p^i\) row-supported below \(b_p\), vanishing on the columns \(\beta \ge b_{p+1}\), and isometric on the first \(b_{p+1}\) columns. For every \(r'\in \mathbb {C}^D\) there is a varying-bond open chain \(A\) with common bound \(D\) such that \(c_A(\sigma )=(Q^\sigma r')_0\) for every \(\sigma \), \(D_k=b_k\) for \(k\le n\), and \(\sum _i(A_p^i)^\dagger A_p^i=\mathbb {1}\) for \(p{\lt}n\). If moreover \(r'\) is supported below \(b_{n+1}\) and \(\| r'\| =1\), then also \(\sum _i(A_n^i)^\dagger A_n^i=1\).

If \(M'_{\alpha \beta }=M_{\alpha \beta }\) for all \(\beta {\lt}b\) and \(w\) is supported below \(b\), then \(M'w=Mw\). Consequently, if \(Q_p^i\) is row-supported below \(b_p\), the families \(Q'_p\) agree with \(Q_p\) on the columns \(\beta {\lt}b_{p+1}\), and \(v\) is supported below \(b_n\), then \(Q'^\sigma v=Q^\sigma v\) for every \(\sigma \).

The common bound \(D\) of a varying-bond open chain is positive.

Let \(d\ge 1\) and let \(A\) be a varying-bond open chain of length \(n+1\) with \(\sum _i(A_p^i)^\dagger A_p^i=\mathbb {1}\) for every site \(p{\lt}n\). Then \(D_k\le d^k\) for every \(k\le n+1\).

Let \(D{\gt}0\). The matrix \(|0\rangle v^{\mathsf T}\) is row-supported below one; \(|0\rangle \) is real and of norm one; \(\langle 0|w=w_0\) and \((M|0\rangle )_\alpha =M_{\alpha 0}\); the first coordinate of \(|0\rangle v^{\mathsf T}Mw\) is \(v^{\mathsf T}Mw\); the first column of \(M\, r\langle 0|\) is \(Mr\) and its other columns vanish; a vector supported below one equals \(w_0|0\rangle \); and \(\| x|0\rangle \| ^2=|x|^2\).

For a varying-bond open chain with square padding \(\widetilde A\), and every configuration \(\sigma \) of any length \(N\ge 0\), \(c_A(\sigma )=(\widetilde A^\sigma )_{00}\).

The reverse of \(A\) has coefficient \(\sigma \mapsto c_A(\sigma _{N-1},\dots ,\sigma _0)\).

If every \(Q_k^i\) is row-supported below \(b_k\) and vanishes on the columns \(\beta \ge b_{k+1}\), then padding the cut-down chain returns \(Q\). If \(Q_k^i\) is row-supported below \(b_k\) and \(Q_k\) is isometric on its first \(b_{k+1}\) columns, then the cut-down site matrices \(A_k^i\) satisfy \(\sum _i(A_k^i)^\dagger A_k^i=\mathbb {1}_{b_{k+1}}\).

Let \(N\ge 1\), let \(\psi \ne 0\), and let \(\operatorname {rank}_k\psi \le D\) for every \(k\le N\). Then \(\psi =c_A\) for a varying-bond open chain \(A\) with common bound \(D\) and \(D_k=\operatorname {rank}_k\psi \) for every \(k\le N\), such that \(\sum _i(A_p^i)^\dagger A_p^i=\mathbb {1}\) for every site \(p{\lt}N-1\). If moreover \(\sum _\sigma |\psi (\sigma )|^2=1\), this holds at the site \(N-1\) as well.

For every \(k\), \(\operatorname {rank}_k\psi \le d^k\) and \(\operatorname {rank}_k\psi \le d^{N-k}\); if \(d\ge 1\), then \(\operatorname {rank}_k\psi \le d^{\lfloor N/2\rfloor }\). If \(\psi \ne 0\), then \(\operatorname {rank}_k\psi \ge 1\) for every \(k\le N\), and \(\operatorname {rank}_0\psi =\operatorname {rank}_N\psi =1\).

Every varying-bond open chain \(A\) of length \(N\) satisfies \(\operatorname {rank}_k c_A\le D_k\) for every \(k\le N\).

Let \(k\le N\). For square matrices \(M_0,\dots ,M_{N-1}\) indexed by the physical index, \(M^{\sigma \tau }=(M_0^{\sigma _0}\cdots M_{k-1}^{\sigma _{k-1}})(M_k^{\tau _0}\cdots M_{N-1}^{\tau _{N-k-1}})\). Every configuration \(c\) of \(N\) sites is \(\sigma \tau \) with \(\sigma =(c_0,\dots ,c_{k-1})\) and \(\tau =(c_k,\dots ,c_{N-1})\).

Let \(d\ge 1\). A vector \(\psi \) with \(\sum _\sigma |\psi (\sigma )|^2=1\) is deterministically generated with a \(D\)-dimensional ancilla if and only if it is probabilistically generated with a \(D\)-dimensional ancilla.

Let \(d\ge 1\) and \(N\ge 1\). A vector \(\psi \) is deterministically generated with a \(D\)-dimensional ancilla if and only if \(\sum _\sigma |\psi (\sigma )|^2=1\) and \(\psi \) has an open-boundary representation with bond dimension at most \(D\)  [ 228 , Section 5, Theorem “Sequential generation with ancilla” ] .

Every deterministically generated vector \(\psi \) satisfies \(\sum _\sigma |\psi (\sigma )|^2=1\).

Every deterministically generated vector is probabilistically generated with an ancilla of the same dimension.

For square matrices \(Q_0,\dots ,Q_n\) and a configuration \(\sigma \) of \(n+1\) sites, \(Q^\sigma =(Q_0^{\sigma _0}\cdots Q_{n-1}^{\sigma _{n-1}})\, Q_n^{\sigma _n}\).

Let \(d\ge 1\) and \(N\ge 1\). Every \(\psi \in (\mathbb {C}^d)^{\otimes N}\) has an open-boundary representation with bond dimension at most \(d^{\lfloor N/2\rfloor }\), and is probabilistically generated with an ancilla of dimension \(d^{\lfloor N/2\rfloor }\). If moreover \(\sum _\sigma |\psi (\sigma )|^2=1\), then \(\psi \) is deterministically generated with an ancilla of dimension \(d^{\lfloor N/2\rfloor }\) and generated by a deterministic transition scheme with ancilla \(\mathbb {C}^{d^{\lfloor N/2\rfloor }}\otimes \mathbb {C}^d\).

Let \(N\ge 1\). A vector \(\psi \) is probabilistically generated with a \(D\)-dimensional ancilla if and only if \(\operatorname {rank}_k\psi \le D\) for every \(k\le N\). If \(d\ge 1\), it is deterministically generated with a \(D\)-dimensional ancilla if and only if moreover \(\sum _\sigma |\psi (\sigma )|^2=1\).

If a nonzero vector \(\psi \) is probabilistically generated with a \(D\)-dimensional ancilla, then \(c\, \psi \) has an open-boundary representation with bond dimension at most \(D\) for some \(c\ne 0\).

Every varying-bond open chain \(A\) of length \(n+1\) has a varying-bond open chain \(A'\) with the same common bound, \(c_{A'}=c_A\), and \(D'_k\le D_k\) for every \(k\), such that \(\sum _i(A'^i_p)^\dagger A'^i_p=\mathbb {1}\) for every site \(p{\lt}n\).

If moreover \(\sum _\sigma |c_A(\sigma )|^2=1\), the chain \(A'\) of Theorem 34.2.19 can be chosen with \(\sum _i(A'^i_p)^\dagger A'^i_p=\mathbb {1}\) at every site \(p\le n\).

Let \(N\ge 1\). The least dimension of an ancilla that probabilistically generates \(\psi \) is \(\max _{k\le N}\operatorname {rank}_k\psi \). If \(d\ge 1\) and \(\sum _\sigma |\psi (\sigma )|^2=1\), the least dimension of an ancilla that deterministically generates \(\psi \) is the same number.

Let \(d\ge 1\) and \(N\ge 2\). The states generated by sequential schemes without an ancilla, either deterministically or probabilistically (the latter up to normalization), are exactly the states with an open-boundary representation of bond dimension at most \(d\), normalized in the deterministic case  [ 228 , Section 5, Theorem “Sequential generation without ancilla” ] . The condition \(N\ge 2\) is necessary because the two-site gates require at least two sites  [ 107 ] .

Let \(d\ge 1\) and \(N\ge 2\), and let each \(U^{[k]}\) act on the whole chain \((\mathbb {C}^d)^{\otimes N}\) as \(U^{[k]}\) on sites \(k\) and \(k+1\) and as the identity on the other sites. For \(k=1,\dots ,N-1\),

In particular, the operations \(U^{[1]},\dots ,U^{[N-1]}\) applied in turn to \(|0\rangle ^{\otimes N}\) generate the state \(\chi _{N-1}\).

Let \(d\ge 1\) and \(N\ge 2\), and let \(\widehat U\) be the operation \(U\) on \(\mathbb {C}^d\otimes \mathbb {C}^d\) followed by the interchange of its two output factors; \(\widehat{\widehat U}=U\), and \(\widehat U\) is unitary if \(U\) is. The state generated by \(U^{[1]},\dots ,U^{[N-1]}\) without an ancilla is the joint state of the scheme with an ancilla \(\mathbb {C}^d\) for the unitaries \(\widehat U^{[k]}\), started in \(|0\rangle \), read out as the site at ket position \(0\):

with \(A_p\) induced by \(\widehat U^{[N-1-p]}\), so \((A_p^i)_{\beta \alpha }=\langle i,\beta |U^{[N-1-p]}|\alpha ,0\rangle \) [ 228 , Section 5.2 ] .

If \(Q\) is isometric on its first \(b\) columns and \(v\) is supported below \(b\), then \(\sum _i\| Q^iv\| ^2=\| v\| ^2\). Consequently, if \(Q_p^i\) is row-supported below \(b_p\), \(Q_p\) is isometric on its first \(b_{p+1}\) columns, and \(v\) is supported below \(b_n\), then \(\sum _\sigma \| Q^\sigma v\| ^2=\| v\| ^2\).

Scaling one site of a varying-bond open chain by \(c\) multiplies its coefficient by \(c\). Consequently, for \(N\ge 1\), if \(\psi \) has an open-boundary representation with bond dimension at most \(D\), then so has \(c\, \psi \) for every \(c\in \mathbb {C}\), and for \(N\ge 1\) and \(D\ge 1\) the zero vector has an open-boundary representation with bond dimension at most \(D\).

Let \(N\ge 1\) and \(D\ge 1\). A vector \(\psi \) has an open-boundary representation with bond dimension at most \(D\) if and only if \(\operatorname {rank}_k\psi \le D\) for every \(k\le N\). The backward direction is the statement that any state for which \(\max _m\operatorname {rank}(\rho _m)\le D\) can be written as a matrix product state of bond dimension \(D\)  [ 228 , Section 3.1 ] .

Let \(d\ge 1\). If \(\sum _\sigma |\psi (\sigma )|^2=1\), \(c\ne 0\), and \(c\, \psi \) has an open-boundary representation with bond dimension at most \(D\), then \(\psi \) is deterministically generated with a \(D\)-dimensional ancilla.

If \(c\ne 0\) and \(c\, \psi \) has an open-boundary representation with bond dimension at most \(D\), then \(\psi \) is probabilistically generated with a \(D\)-dimensional ancilla.

Let \(Y^i\in M_{D}(\mathbb {C})\) be row-supported below \(a\) for every \(i\). Then there are \(b\le D\), matrices \(Q^i\) and a matrix \(R\) with \(Y^i=Q^iR\) for every \(i\), such that each \(Q^i\) is row-supported below \(a\) and vanishes on the columns \(\beta \ge b\), the family \(Q\) is isometric on its first \(b\) columns, and \(R\) is row-supported below \(b\). Moreover \(b\le c\) whenever every \(Y^i\) vanishes on the columns \(\gamma \ge c\).

Let \(N\ge 1\) and \(D\ge 1\). A vector \(\psi \) is probabilistically generated with a \(D\)-dimensional ancilla if and only if it has an open-boundary representation with bond dimension at most \(D\)  [ 228 , Section 5, Theorem “Sequential generation with ancilla” ] .

If \(A^0,\dots ,A^{d-1}\) are \(a\times b\) matrices with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}_b\), then \(b\le ad\). In particular, a site \(p\) of a varying-bond open chain with \(\sum _i(A_p^i)^\dagger A_p^i=\mathbb {1}\) satisfies \(D_{p+1}\le d\, D_p\).

Let \(d\ge 1\) and \(N=n+2\). The vectors \(\psi (\sigma )=\bigl(A_0^{\sigma _1}\cdots A_n^{\sigma _{n+1}} |\varphi _I\rangle \bigr)_{\sigma _0}\), where \(A_p\) is induced by an operation \(U_{n-p}\) on \(\mathbb {C}^d\otimes \mathbb {C}^d\) and \(|\varphi _I\rangle \in \mathbb {C}^d\), are exactly the vectors with an open-boundary representation of bond dimension at most \(d\). With unitary operations and a unit vector \(|\varphi _I\rangle \) they are exactly the normalized such vectors.

Let \(d\ge 1\). The chain \(A'\) of Theorem 34.2.19 can be chosen with \(D'_k\le \min (D_k,d^k)\) for every \(k\); in particular \(D'_k\le \min (D,d^k)\).

Let \(d\ge 1\) and let \(A\) be a varying-bond open chain of length \(N=n+1\). There is a varying-bond open chain \(A'\) with the same common bound, \(c_{A'}=c_A\), \(\sum _i(A'^i_p)^\dagger A'^i_p=\mathbb {1}\) for every site \(p{\lt}n\), and

for every \(k\le N\).

Let \(d\ge 1\), and let \(\psi \) be a unit vector of \((\mathbb {C}^d)^{\otimes N}\) with an open-boundary representation of bond dimension \(D\le d^k\). Then \(\psi =U|0\rangle ^{\otimes N}\) for a unitary \(U\) that is a product of at most \(\max (1,N-k)\) gates, each acting on at most \(k+1\) consecutive sites of the open chain (with \(N-k\) read as \(0\) when \(N{\lt}k\)). In particular, if \(D\le d\) and \(N\ge 2\), then \(U\) is a product of at most \(N-1\) gates on two neighbouring sites, as in the scheme without an ancilla of Theorem 34.8.6. This is the sequential preparation of  [ 238 ] and  [ 219 , paragraph “The sequential-RG circuit” ] , with each isometry embedded into a unitary on consecutive sites.

The matrices induced by a unitary on \(\mathbb {C}^D\otimes \mathbb {C}^d\) satisfy \(\sum _i(A^i)^\dagger A^i=\mathbb {1}_D\).

If \(M\) is row-supported below \(a\), then so is \(MN\) for every \(N\), and \(Mv\) is supported below \(a\) for every \(v\). Consequently, if \(Q_p^i\) is row-supported below \(b_p\) for \(p=0,\dots ,n-1\) and \(v\) is supported below \(b_n\), then \(Q^\sigma v\) is supported below \(b_0\).

Let \(a\le D\), let \(R_{-1}\in M_{D}(\mathbb {C})\) be row-supported below \(a\), let \(P_0,\dots ,P_{n-1}\) be families of \(D\times D\) matrices, and let \(r\in \mathbb {C}^D\). Then there are \(b_0=a\) and \(b_1,\dots ,b_n\le D\), families \(Q_0,\dots ,Q_{n-1}\) such that \(Q_p^i\) is row-supported below \(b_p\), vanishes on the columns \(\beta \ge b_{p+1}\), and \(Q_p\) is isometric on its first \(b_{p+1}\) columns, and a vector \(r'\) supported below \(b_n\), such that for every configuration \(\sigma \)

Moreover \(b_{p+1}\le c\) whenever every \(P_p^i\) vanishes on the columns \(\gamma \ge c\).

Let \(a\le D\), let \(R_{-1}\in M_{D}(\mathbb {C})\) be row-supported below \(a\), and let \(P_0,\dots ,P_{n-1}\) be families of \(D\times D\) matrices. Then there are \(b_0=a\) and \(b_1,\dots ,b_n\le D\), families \(Q_0,\dots ,Q_{n-1}\) such that \(Q_p^i\) is row-supported below \(b_p\), vanishes on the columns \(\beta \ge b_{p+1}\), and \(Q_p\) is isometric on its first \(b_{p+1}\) columns, and a matrix \(R\) row-supported below \(b_n\), such that for every configuration \(\sigma \)

Moreover \(b_{p+1}\le c\) whenever every \(P_p^i\) vanishes on the columns \(\gamma \ge c\). If \(Q\) is isometric on its first \(b\) columns and \(v,w\) are supported below \(b\), then \(\sum _i\langle Q^iv,Q^iw\rangle =\langle v,w\rangle \). Consequently, if \(Q_p^i\) is row-supported below \(b_p\), \(Q_p\) is isometric on its first \(b_{p+1}\) columns, and \(v,w\) are supported below \(b_n\), then

Let \(A\) be a varying-bond open chain of length \(n+1\) with dimensions \(D_k\) and common bound \(D\), and let \(c\in \mathbb {C}\). There are \(b_0=1\) and \(b_k\le D_k\), families \(Q_0,\dots ,Q_n\) as in Theorem 34.2.9, and a vector \(r'\) supported below \(b_{n+1}\), such that \(Q^\sigma r'=c\, c_A(\sigma )|0\rangle \) for every \(\sigma \) and \(\| r'\| ^2=\sum _\sigma |c\, c_A(\sigma )|^2\).

Let \(d\ge 1\) and \(N\ge 1\). A vector \(\psi \) is generated by a deterministic transition scheme with ancilla \(\mathbb {C}^D\otimes \mathbb {C}^d\) if and only if \(\sum _\sigma |\psi (\sigma )|^2=1\) and \(\psi \) has an open-boundary representation with bond dimension at most \(D\). For \(d=2\) this is the statement of  [ 228 , Section 5, Theorem “Sequential generation with ancilla” ] for the third scheme.

Let \(d\ge 1\). A vector is generated by a deterministic transition scheme with ancilla \(\mathbb {C}^D\otimes \mathbb {C}^d\) if and only if it is deterministically generated with a \(D\)-dimensional ancilla.

For \(d=2\), an interaction \(T\) satisfies (4) if and only if \(T|\varphi \rangle |1\rangle |0\rangle =|\varphi \rangle |0\rangle |1\rangle \) and \(T|\varphi \rangle |0\rangle |0\rangle =|\varphi \rangle |0\rangle |0\rangle \) for every \(|\varphi \rangle \in \mathbb {C}^D\), the interaction of  [ 228 , Section 5.1 ] .

Let \(T\) be an interaction on \(\mathbb {C}^D\otimes \mathbb {C}^d\otimes \mathbb {C}^d\) satisfying (4), and let \(W\) be a matrix on \(\mathbb {C}^D\otimes \mathbb {C}^d\). Then for all \(x,y\) and \(i\),

Let \(d\ge 1\) and let \(Q\) be isometric on its first \(b\) columns. Then there is a unitary \(U\) on \(\mathbb {C}^D\otimes \mathbb {C}^d\) with \(\langle \alpha ,i|U|\beta ,0\rangle =Q^i_{\alpha \beta }\) for all \(i\), \(\alpha \) and \(\beta {\lt}b\).

At \(N=0\), the open-chain coefficient is \(1\).

Let \(N\geq 1\). For every physical configuration \(\sigma \),

No positivity assumption is imposed on the intermediate dimensions. The endpoint conditions already imply \(D\geq 1\) whenever such a chain exists. The equality concerns only this fixed length.