Dependencies

For fixed \(H,B,X\), the function \(t\mapsto C_B(X,t)\) is continuous.

For \(A\in \mathcal A_X\), one has \(\| [\tau _t(A),B]\| \leq C_B(X,t)\| A\| \).

The restricted commutator norm for \(H\) at time \(-t\) equals that for \(-H\) at time \(t\).

For operators \(H,B\) and a set of sites \(X\), define

Let \(H=\sum _j h_j\) be a finite sum, where each \(h_j\) is Hermitian and belongs to \(\mathcal A_{T_j}\). For arbitrary \(B,X\) and \(t\geq 0\),

No bound on the ground energy or positivity assumption on \(h_j\) is required, and several terms may share a support. This is the recursion of Hastings–Koma, Appendix A, (A.12)–(A.14).

For arbitrary \(H,B,X\), one has \(C_B(X,0)\leq 2\| B\| \).

If \(B\in \mathcal A_Y\) and \(X\cap Y=\varnothing \), then \(C_B(X,0)=0\).

Let \(f_i\geq 0\), for \(i\) in a finite set, be continuous on \([0,T]\) and satisfy \(f_i(t)\leq b_i+\sum _j a_{ij}\int _0^tf_j(s)\, ds\) with \(a_{ij}\geq 0\). If positive weights \(w_i\) satisfy \(\sum _ja_{ij}w_j\leq vw_i\) and \(b_i\leq cw_i\), then \(f_i(t)\leq cw_ie^{vt}\) for \(0\leq t\leq T\). If moreover \(b_i=0\) and \(v{\gt}0\), then \(f_i(t)\leq cw_i(e^{vt}-1)\).

A finite adaptive local protocol is a tree formed from onsite channels, nearest-neighbor channel layers, and onsite instruments with a continuation chosen according to the complete communicated outcome tuple. A depth bound \(T\) bounds the number of nearest-neighbor layers on every branch. Onsite channels and instruments have zero cost, and one layer adds one to the depth. A larger depth bound also bounds the same protocol. An operator \(\rho \) is adaptively converted into \(\sigma \) in depth \(T\) when \(\Psi (\rho )=\sigma \) for such a protocol \(\Psi \).

Write \(\mathcal H_W=(\mathbb {C}^d)^{\otimes W}\), with \(d\geq 1\). Restricting a configuration of \(W+A\) wires to its two blocks identifies \(\mathcal H_{W+A}\) with \(\mathcal H_W\otimes \mathcal H_A\). In these coordinates, the initialization and discard maps are \(\mathcal I_A(X)=X\otimes Z_A\) and \(\mathcal D_A(Y)=\operatorname{tr}_A Y\), where \(Z_A=|0^A\rangle \! \langle 0^A|\).

A chain operator \(\rho \) is approximately converted into \(\sigma \) in depth \(T\) with error \(\varepsilon \) if there is a local channel protocol \(\Psi \) of depth at most \(T\) with \(\lVert \Psi (\rho )-\sigma \rVert _1\le \varepsilon \). Here \(\lVert A\rVert _1\) is the sum of the singular values of \(A\).

Let \(\rho _N\) and \(\sigma _N\) be families of operators on chains of \(N\) sites, with fixed local dimensions that may differ between the families. The family \(\rho _N\) is asymptotically locally convertible into \(\sigma _N\) if there are an integer \(p\ge 0\), a constant \(C\ge 0\), and errors \(\varepsilon _N\to 0\) such that, for all sufficiently large \(N\), there is an approximate conversion with error \(\varepsilon _N\) and depth \(T_N\le C(1+\log (N+1))^p\). Two families are locally phase equivalent if they are asymptotically locally convertible in both directions.

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 \(\sigma _N=|\psi _N\rangle \langle \psi _N|\).

Consider a ring of \(Mk\) sites of local dimension \(d\). Grouping the sites \(jk,\dots ,jk+k-1\) into one site \(j\) of local dimension \(d^k\) gives a ring of \(M\) sites. On operators, it is the isomorphism \(\beta _k\colon M_d^{\otimes Mk}\to M_{d^k}^{\otimes M}\) that relabels the computational basis; it maps \(\bigotimes _i m_i\) to \(\bigotimes _j\bigl(m_{jk}\otimes \dots \otimes m_{jk+k-1}\bigr)\). A matrix \(\rho \) on the \(Mk\) sites is converted in depth \(T\) after blocking \(k\) sites into a matrix \(\sigma \) on the \(M\) blocked sites if \(\beta _k(\rho )\) is converted in depth \(T\) into \(\sigma \). This is the blocking of \(q\) neighbouring sites in the proof of the classification of matrix product states in [ 227 ] . A layer of two-site channels on the blocked ring costs one unit of depth; the source realizes each gate on the blocked ring by fewer than \(2q^2\) nearest-neighbour gates on the original ring, a compilation not carried out here [ 32 ] . The blocking factor \(k\) is part of the relation, and it is positive: blocking the whole ring into one site would make every conversion between density matrices possible in depth \(0\).

Every input, output, and intermediate local dimension is positive and at most \(B\). The constructors are the existing onsite channel and native pair-channel layer operations. Forgetting the certificates gives an ordinary native protocol; increasing \(B\) preserves them, and composition adds depths at a common budget. Every finite native protocol with positive input dimension admits some finite budget. This last assertion does not give a budget uniform in system size.

An owner-preserving injection between environment wires extends by the identity on all original wires. Thus circuits on \(W+A\) and \(W+B\) wires embed into the common layout on \(W+(A+B)\) wires, using disjoint environment blocks and the same original ports. The configuration split for the common layout orders the factors as \((\mathcal H_W\otimes \mathcal H_A)\otimes \mathcal H_B\).

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.

For a bond geometry \((B_b)_{b\in \beta }\) on \(I\) and \(X\subseteq I\), the one-step neighbourhood of \(X\) is \(X\) together with every bond that meets \(X\), and the light cone of radius \(r\) of \(X\) is its \(r\)-fold one-step neighbourhood. On the ring, the sites within ring distance \(r\) of \(X\) are the sites \(i+m\) with \(i\in X\) and \(\lvert m\rvert \le r\), indices modulo \(N\), and two sets of sites are at distance larger than \(s\) if no site of one is within ring distance \(s\) of a site of the other. For a graph, the bonds are its edges, and the distance of two vertices is the least number of edges of a path joining them, as in [ 227 ] .

Fix a bond geometry, for instance the pairs \(\{ k,k+1\} \) of the ring. A channel on the bond \(B\) is a map \(\rho \mapsto \sum _jK_j\rho K_j^\dagger \) on density matrices of all the sites, where each \(K_j\) is an operator on all the sites acting trivially off \(B\) and \(\sum _jK_j^\dagger K_j=\mathbb 1\). A channel layer is a composition of channels on disjoint bonds, and a local channel circuit of depth \(T\) is a composition \(\Phi =\Phi _T\circ \dots \circ \Phi _1\) of \(T\) channel layers. Its dual \(\Phi ^\dagger =\Phi _1^\dagger \circ \dots \circ \Phi _T^\dagger \) replaces each channel on a bond by \(A\mapsto \sum _jK_j^\dagger AK_j\). A density matrix is prepared in depth \(T\) by local channels if it equals \(\Phi (\bigotimes _i\sigma _i)\) for a local channel circuit \(\Phi \) of depth \(T\) and density matrices \(\sigma _i\) on single sites.

A local channel protocol of depth \(T\) is a composition \(\Psi =\Phi _T\circ L_T\circ \Phi _{T-1}\circ \dots \circ L_1\circ \Phi _0\) of channel layers \(L_t\) and maps \(\Phi _t\), each a composition of finitely many onsite channels, which may change the local dimension; \(\Phi _0\) contains at least one onsite channel, and \(\Phi _t\) for \(t\ge 1\) may be the identity. A matrix \(\rho \) on the chain is converted in depth \(T\) into a matrix \(\sigma \), possibly of another local dimension, if \(\sigma =\Psi (\rho )\) for a local channel protocol \(\Psi \) of depth at most \(T\). This is the channel form of the circuits \(V'=U_\ell V_\ell \cdots U_1V_1U_0\) of [ 227 ] , with ancillas attached to each site, local operations between the layers and the ancillas traced out at the end, without measurements.

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.

For a measurement round with circuit \(U\), outcome projectors \(P_m\), and conditional product unitaries \(V_m\), its average channel is \(\rho \mapsto \sum _m V_mP_mU\rho U^{\dagger }P_mV_m^{\dagger }\).

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.

An onsite channel from sites of dimension \(d\) to sites of dimension \(e\) is given, at each site \(i\), by operators \(K_{ij}\colon \mathbb {C}^d\to \mathbb {C}^e\) with \(\sum _jK_{ij}^\dagger K_{ij}=\mathbb 1\). It is the channel \(\rho \mapsto \sum _JK_J\rho K_J^\dagger \) on any finite set of sites, such as the ring, with \(K_J=\bigotimes _iK_{iJ(i)}\) for every choice \(J\) of one index per site; its dual is \(A\mapsto \sum _JK_J^\dagger AK_J\). Write \(\Phi _i(\sigma )=\sum _jK_{ij}\sigma K_{ij}^\dagger \) for its action at site \(i\). Attaching an ancilla of dimension \(a\) in the state \(\tau =\sum _kw_kw_k^\dagger \) with \(\sum _k\| w_k\| ^2=1\) is the onsite channel \(X\mapsto X\otimes \tau \) from dimension \(d\) to dimension \(da\), and discarding it is the onsite channel given by the partial trace over the second factor of \(\mathbb {C}^d\otimes \mathbb {C}^a\). Ancillas attached to each site and local operations on a site and its ancillas are the free operations of [ 227 ] .

An onsite instrument has local Kraus operators \(K_{i,j}\) satisfying \(\sum _jK_{i,j}^{\dagger }K_{i,j}=I\). For the communicated tuple of outcomes \(J\), let \(K_J=\bigotimes _iK_{i,J(i)}\) and let \(\Psi _J\) be the selected continuation. Averaging the outcomes gives the map

Each summand uses an unnormalized post-measurement operator.

Suppose \(q{\gt}0\), let \(S\subseteq I\), and let \(p\in (\mathbb {Z}/q\mathbb {Z}\times \mathbb {Z}/q\mathbb {Z})^I\). For the on-site Weyl operators \(W(p_x)\), put \(M_x=W(p_x)\) when \(x\in S\) and \(M_x=\mathbb 1\) otherwise. Define

In particular, \(P_{\varnothing }(p)=\mathbb 1\).

An embedding of two layouts is an injection of wires that preserves their spatial owners and sends each designated physical port to the corresponding port in the larger layout. Given an assignment of owners to \(A\) new wires, appending them to a layout gives such an embedding of its \(W\) original wires. None of the new wires is a communication port.

Every wire carries a \(d\)-level qudit. A layout assigns its wires to spatial sites and selects one physical port at each site. Free operations are unitaries supported inside one spatial site, wire permutations preserving each spatial site, and normalized onsite Kraus channels. A counted layer is an existing nearest-neighbor unitary layer placed on the physical ports, with identity on every memory wire. Products add physical depth bounds; a larger upper bound remains valid. Reversing a unitary circuit preserves its depth. Every resulting channel protocol is CPTP on all input operators.

Let \(\zeta =e^{2\pi i/d}\). The Fourier matrix is \(F_{ab}=d^{-1/2}\zeta ^{ab}\) for \(a,b\in \mathbb Z_d\), and for \(z,x\in \mathbb Z_d\) the generalized Pauli matrix \(X^xZ^z\) maps \(|q\rangle \) to \(\zeta ^{zq}|q+x\rangle \).

A register gate is a matrix \(X\) on \(2s\) sites together with a site \(a\) and \(L\ge 0\) with \(2L+2s\le N\). It acts on the two registers \(a,\dots ,a+s-1\) and \(a+2L+s,\dots ,a+2L+2s-1\), the first register on the first \(s\) sites of \(X\). Its stretch is \(a,\dots ,a+2L+2s-1\), its interior the \(2L\) sites \(a+s,\dots ,a+s+2L-1\) strictly between the registers, and, for \(0\le t\le s\), its zone \(t\) the \(2L\) sites \(a+t,\dots ,a+t+2L-1\). The gate acts on the ring as \(X\) on its two registers, which is unitary when \(X\) is.

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 \(W|v\rangle \).

A selected register of \(m\) original wires, together with all \(A\) fresh wires, is included in the enlarged layout in this order. For a fixed spatial site, its site register enumerates exactly the original wires owned by that site.

The map \(E_K\) on the operators of \((\mathbb {C}^q)^{\otimes I}\) defined above.

For \(y\in I\), define

This is the on-site unitary supremum of [ 247 ] , with every test unitary acting as the identity on the spectator.

Let \(I\) be a finite set of sites and let \(d,e\geq 0\) be integers. At each site identify \((a,b)\in \{ 0,\dots ,d-1\} \times \{ 0,\dots ,e-1\} \) with \(ea+b\in \{ 0,\dots ,de-1\} \). Applied at every site, this gives a bijection \(p\) from pairs of configurations to configurations of local dimension \(de\).

For \(a,b\in J\), let \(B_{ab}\) be the physical matrix whose entries are \((B_{ab})_{\sigma ,\tau }=B_{(\sigma ,a),(\tau ,b)}\). Define \(E_K^{\mathcal K}\) by applying \(E_K\) to each such block:

In particular, when \(q{\gt}0\), set \(Q=I\setminus K\) and write \(u\sqcup \rho \) for the configuration equal to \(u\) on \(K\) and to \(\rho \) on \(Q\). Then

Both spectator indices remain unchanged.

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.

The expectation \(E_K\) onto a set of sites \(K\) preserves adjoints and the order, maps \([0,I]\) into \([0,I]\), and fixes every operator commuting with all operators acting outside \(K\); such an operator acts on \(K\). In particular \(E_K\) fixes \(|Y|\) for every Hermitian \(Y\) acting on \(K\), and \(Y^{1/2}\) acts on \(K\) for every \(Y\ge 0\) acting on \(K\).

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 \(f\).

Every matrix is converted in depth \(0\) after blocking into its blocked form, and a conversion in depth \(T\) after blocking is one in every larger depth. Let \(\rho \) be converted in depth \(T\) after blocking \(k\) sites into \(\sigma \). A conversion in depth \(T'\) of \(\sigma \) on the blocked ring gives a conversion of \(\rho \) in depth \(T+T'\) after blocking, \(\sigma =\Psi (\rho )\) for a channel \(\Psi \), and \(\sigma \) is a density matrix when \(\rho \) is. If \(d\ge 1\), there is a map \(\Psi ^\dagger \) with \(\operatorname{Tr}(\sigma A)=\operatorname{Tr}(\rho \, \Psi ^\dagger (A))\) that maps operators acting on a set \(X\) of blocks to operators acting on the sites within distance \(kT\) of the sites of \(X\), and that is multiplicative on operators acting on sets of blocks whose neighbourhoods of radius \(T\) are disjoint. If \(\rho =\bigotimes _i\sigma _i\) with \(\operatorname{Tr}\sigma _i=1\), and \(Z\), \(Z'\) act on sets of blocks at distance larger than \(2T\), then \(\operatorname{Tr}(\sigma ZZ')=\operatorname{Tr}(\sigma Z)\operatorname{Tr}(\sigma Z')\).

Blocking and unblocking are trace preserving and completely positive. If \(Z\) acts on the sites of a set \(Y\) of blocks, then \(\beta _k(Z)\) acts on \(Y\); conversely, if \(W\) acts on a set \(X\) of blocks and \(d\ge 1\), then \(\beta _k^{-1}(W)\) acts on the sites of the blocks in \(X\). The sites of the blocks within distance \(r\) of \(X\) lie within distance \(kr\) of the sites of \(X\).

Let \(\Phi \) be a local channel circuit of depth \(T\) and let \(Z\) act on a set of sites \(X\). Then \(\Phi ^\dagger (Z)\) acts on the light cone of radius \(T\) of \(X\); on the ring, on the sites within ring distance \(T\) of \(X\).

Let \(\rho \) be prepared in depth \(T\) by local channels, 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 \(\operatorname{Tr}(\rho ZZ') =\operatorname{Tr}(\rho Z)\operatorname{Tr}(\rho Z')\).

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.

The product data initializer equals the physical encoder composed with preparation of the zero state on its complete complementary port/scratch reference. Preparing that reference commutes with an arbitrary logical channel. The coordinate identity includes every off-diagonal matrix unit, not only classical configurations or product states.

Let \(U\) be a local circuit of depth \(T\) on a bond geometry and let \(Z\) act on a set of sites \(X\). Then \(U^\dagger ZU\) acts on the light cone of radius \(T\) of \(X\); on the ring, on the sites within ring distance \(T\) of \(X\). This is the finite light cone of [ 219 , Theorem 1 ] .

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.

On the ring of \(N\) sites, for every \(N\ge 1\), the light cone of radius \(r\) of \(X\) is the set of sites within ring distance \(r\) of \(X\). On the open chain it is contained in the set of sites within distance \(r\) of \(X\) along the chain, and for the edges of a graph it is contained in the set of vertices at graph distance at most \(r\) from \(X\). In each case, sets of sites at distance larger than \(2r\) have disjoint light cones of radius \(r\). For every bond geometry, the light cone of radius \(s\) of the light cone of radius \(r\) of \(X\) is the light cone of radius \(r+s\) of \(X\).

Let \(\Phi \) be a local channel circuit. Then

for all operators \(\rho \) and \(A\) on the sites, and \(\Phi \) is trace preserving and completely positive. A local circuit \(U\) of depth \(T\) is the local channel circuit \(\rho \mapsto U\rho U^\dagger \) of depth \(T\) with one operator per channel.

Let \(\sigma \) be converted in depth \(T\) from a product \(\bigotimes _i\sigma _i\) of one-site matrices of trace one, and let \(Z\), \(Z'\) act on sets of sites at distance larger than \(2T\). Then \(\operatorname{Tr}(\sigma ZZ')=\operatorname{Tr}(\sigma Z)\operatorname{Tr}(\sigma Z')\). For pure states prepared from a product state by local unitary circuits this is the necessary condition for preparation by quantum circuits of [ 227 ] .

A local channel protocol is trace preserving and completely positive. Attaching ancillas, running a local channel circuit of depth \(T\) and discarding is a local channel protocol of depth \(T\). Every density matrix is converted into itself in depth \(0\), a conversion in depth \(T\) is a conversion in every larger depth, a conversion of a density matrix is a density matrix, and a density matrix prepared in depth \(T\) by local channels is converted in depth \(T\) from a product density.

If \(\rho \) is converted in depth \(T_1\) into \(\sigma \) and \(\sigma \) is converted in depth \(T_2\) into \(\tau \), then \(\rho \) is converted in depth \(T_1+T_2\) into \(\tau \).

Let \(H=\sum _jh_j\) with \(h_j\in \mathcal A_{T_j}\), and let \(C\) be a set of sites such that each \(T_j\) is contained in \(C\) or disjoint from \(C\). If \(A\in \mathcal A_C\) and \(B\in \mathcal A_D\) with \(C\cap D=\varnothing \), then \([\tau _t(A),B]=0\) for every \(t\).

The outcome projections satisfy \(\sum _mP_m=\mathbb 1\), and a product of unitaries on the sites is unitary; a unitary matrix maps only the zero vector to zero. Hence, for a protocol with \(|\pi \rangle \neq 0\), some outcome has \(P_mU|\pi \rangle \neq 0\), then \(V_mP_mU|\pi \rangle \neq 0\), and every vector that the protocol prepares is nonzero.

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 \(T\).

Identify each \(d^k\)-level native site with its \(k\) local qudit digits. The induced placement of channels on all data wires preserves identity and composition, with every port and scratch register retained as an explicit identity reference. A native operator on selected spatial sites becomes exactly the corresponding operator on those sites’ physical data digits. The basis regrouping is a coordinate equality, not an extra free gate.

Let \(\Phi \) be an onsite channel. If \(Z\) acts on a set of sites \(X\), then \(\Phi ^\dagger (Z)\) acts on \(X\). If moreover \(Z'\) acts on a set of sites disjoint from \(X\), then \(\Phi ^\dagger (ZZ')=\Phi ^\dagger (Z)\Phi ^\dagger (Z')\). The dual maps \(\bigotimes _im_i\) to \(\bigotimes _i\Phi _i^\dagger (m_i)\): for any operators \(K_{ij}\colon \mathbb {C}^d\to \mathbb {C}^e\), the map \(\rho \mapsto \sum _JK_J\rho K_J^\dagger \) with \(K_J=\bigotimes _iK_{iJ(i)}\) sends \(\bigotimes _im_i\) to \(\bigotimes _i\sum _jK_{ij}m_iK_{ij}^\dagger \).

An onsite channel is the concrete ordered product of its single-site factors. Each factor is the representation-independent lift of an actual one-site CPTP map, with identity on the complementary sites. The equality holds on arbitrary correlated chain inputs.

An onsite channel \(\Phi \) is trace preserving and completely positive, it maps \(\bigotimes _i\sigma _i\) to \(\bigotimes _i\Phi _i(\sigma _i)\), it satisfies \(\operatorname{Tr}(\Phi (\rho )A)=\operatorname{Tr}(\rho \, \Phi ^\dagger (A))\), and discarding an attached ancilla gives back the original operator.

The product of the transpositions of a list of hops fixes every site that is not a site of the list. Let valid lists of hops \(\mathcal L_g\) have their sites in sets \(S_g\) that are pairwise disjoint. Then the concatenation of the lists is valid, and the product of its transpositions agrees on \(S_g\) with that of \(\mathcal L_g\) and fixes every site outside all the \(S_g\).

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 \(\sigma \) be a permutation of the configurations of a finite set of sites, such as the ring, that changes only the sites of a set \(X\), the new values at these sites depending only on the old values there. Its permutation matrix \(P_\sigma \), \((P_\sigma v)(x)=v(\sigma x)\), is a unitary acting on \(X\). Examples are the shift \(x\mapsto x+f(x)e_t\) at a site \(t\in X\), with values in \(\mathbb Z_d\) and \(f\) reading only the sites of \(X\setminus \{ t\} \). If \(\sigma _k\), \(k\in K\), are such permutations for pairwise disjoint pairs \(\{ k,k+1\} \), the layer of the gates \(P_{\sigma _k}\) acts as \(v\mapsto v\circ \rho \), where \(\rho (x)\) agrees with \(\sigma _k(x)\) on the pair \(\{ k,k+1\} \) and with \(x\) off all the pairs; here the sites form the ring. For shifts \(\sigma _k\colon x\mapsto x+f_k(x)e_{t_k}\) this reads \(\rho (x)_{t_k}=x_{t_k}+f_k(x)\), and \(\rho (x)_i=x_i\) at every site \(i\) that is none of the \(t_k\). Finally, for permutations \(\tau _i\) of the labels of the sites \(i\) of the ring, the product \(\bigotimes _iP_{\tau _i}\) acts as \(v\mapsto v\circ \tau \), where \(\tau (x)_i=\tau _i(x_i)\).

Transport a native encoding or completed decoder through the sitewise digit identification and the data/complement split. CPTP is preserved, decoding after encoding remains the identity, and an exact native code intertwiner becomes an exact physical-channel intertwiner. The complementary port/scratch system is arbitrary and may be correlated with the logical input.

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.

Use \(1+2k\) wires per spatial site: one physical port, \(k\) data qudits and \(k\) scratch qudits. On a matching, select a data digit at each left endpoint and a scratch digit at each right endpoint. The common routing moves the left data register into the right scratch register and leaves the right data register fixed. Thus both data registers of each nontrivial selected bond lie at its right spatial site after routing. Reversing the routing returns their outputs to their original sites and restores all other wires.

A matching of ring bonds defines an involutive site permutation: each selected bond exchanges its endpoints and every other site is fixed. The actual layer containing these SWAP gates has exactly the associated permutation matrix. Permutation matrices preserve products and inverses. The self-loop of a one-site ring acts as the identity.

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 \(T+2\).

Finite CPTP maps at individual sites assemble into an onsite channel, and assembly preserves composition. Product isometric encodings have product completed local decoders whose composition is the identity on all chain operators, also with any finite reference. Extending an existing onsite channel between site-consistent input and output codes intertwines with its original chain map, including entangled chain inputs.

A local channel protocol \(\Psi \) of depth \(T\) has a dual \(\Psi ^\dagger \) with \(\operatorname{Tr}(\Psi (\rho )A)=\operatorname{Tr}(\rho \, \Psi ^\dagger (A))\) that maps operators acting on a set of sites \(X\) to operators acting on the sites within distance \(T\) of \(X\), and that satisfies

whenever \(Z\) and \(Z'\) act on sets whose neighbourhoods of radius \(T\) are disjoint. This is the light cone of a depth-\(T\) circuit in the proof of [ 219 , Theorem 1 ] , here with channels and onsite operations.

For vectors \(|u\rangle ,|v\rangle \) of a finite-dimensional Hilbert space, \(\lVert |u\rangle \langle v|\rVert _1\leq \lVert u\rVert \, \lVert v\rVert \). For unit vectors \(|\psi \rangle ,|\varphi \rangle \),

The number \(\zeta \) is a primitive \(d\)-th root of unity, and \(F\) and \(X^xZ^z\) are unitary. The entries are \((X^xZ^z)_{pq}=\zeta ^{zq}\) if \(p-x=q\) and \(0\) otherwise.

Let \(g_1,\dots ,g_n\) be register gates with pairwise disjoint stretches. For \(0\le t{\lt}s\), the round \(t\) teleports, for every gate, the content of the site \(a+t\) to \(a+2L+t\) along a chain of \(L\) hops, in one round of depth \(2\). The rounds \(t=s-1,\dots ,0\), in this order, implement the product \(P\) of the transpositions of all the chains on the vectors with \(|0\rangle \) at the interiors of all the gates, and their outputs have \(|0\rangle \) at the zones \(0\). The rounds teleporting back, \(t=0,\dots ,s-1\), in this order, implement \(P^{-1}\) on the vectors with \(|0\rangle \) at the zones \(0\). The rounds of each direction have total depth \(2s\).

Splitting a configuration into a selected register and its complementary wires identifies a placed operator with \(A\otimes I\). Under the same reindexing, placing a Kraus family indexed by any finite set gives precisely its local channel tensored with the identity on the complement. A further bijection of the complementary configuration space alone preserves this formula. A wire-sum decomposition gives such a bijection by enumerating exactly the wires outside the selected range; its complementary coordinates do not depend on selected values. Consequently the placed channel is independent of its chosen Kraus representation, and the placement identity holds on arbitrary correlated inputs.

An injection \(e\) of local basis labels into register words defines an isometry \(E\). The encoding is \(X\mapsto EXE^\dagger \). For a fixed density matrix \(\rho \), the completed decoder is

It is CPTP and satisfies \(D(EXE^\dagger )=X\) for every operator \(X\). The same identity holds after tensoring with the identity on any finite reference. Encoding the two endpoints separately gives the tensor product of their local encoding matrices. If \(m\le B\) and \(d\ge 2\), then \(k=\lceil \log _d B\rceil \) qudits suffice. A channel \(\Phi \) extends to the full input register by \(\widehat\Phi =\mathcal E_{\rm out}\Phi D_{\rm in}\); this extension is CPTP and agrees with \(\Phi \) on encoded inputs.

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 \(\mathcal R\).

Let \(q\geq 1\), and for \(p=(a,b)\) let \(W(p)=X^aZ^b\) be the Weyl operators on \(\mathbb {C}^q\). Then \(E_K(B)\) is the uniform average of \(W_pBW_p^\dagger \) over all products \(W_p=\bigotimes _{x\notin K}W(p_x)\) of Weyl operators outside \(K\).

If \(\| [B,U]\| \leq \varepsilon _y\) for every site \(y\notin K\) and every unitary \(U\in \mathcal A_{\{ y\} }\), then \(\| B-E_K(B)\| \leq \sum _{y\notin K}\varepsilon _y\).

For \(q\geq 1\), the map \(E_K\) takes values in \(\mathcal A_K\), fixes every operator in \(\mathcal A_K\), is unital, completely positive and trace preserving, and is a contraction in operator norm.

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.

A hop consists of three distinct sites \(c,e,f\) such that \(\{ e,f\} \) and \(\{ c,e\} \) are neighbouring pairs. Its first gate, on \(\{ e,f\} \), applies \(F\) at \(e\) and then \(|a\rangle _e|b\rangle _f\mapsto |a\rangle _e|b+a\rangle _f\); its second gate, on \(\{ c,e\} \), applies \(|a\rangle _c|b\rangle _e\mapsto |a\rangle _c|b-a\rangle _e\) and then \(F\) at \(c\). Both are unitary and act on their pairs. Let \(|v\rangle \) have \(|0\rangle \) at \(e\) and \(f\), and let \(z_c,z_e\in \mathbb Z_d\). Applying the two gates and then the projection onto \(|z_c\rangle \) at \(c\) and \(|z_e\rangle \) at \(e\) gives

where \(s_z|q\rangle =|q+z\rangle \) and \((c\, f)\) is the transposition of the sites \(c\) and \(f\). So the content of \(c\) is found at \(f\), up to the unitary \(X^{z_e}Z^{z_c}\), and \(c\) and \(e\) carry the outcomes. The operator acts on the sites \(c,e,f\).

1. Let \(\mathcal T\) and \(\mathcal B\) be valid lists of hops and \(G\) a layer of gates, and let \(P_{\mathcal T}\), \(P_{\mathcal B}\) be the products of the transpositions of their hops. If \(|v\rangle \) has \(|0\rangle \) at the sites \(e,f\) of \(\mathcal T\) and \(GP_{\mathcal T}|v\rangle \) has \(|0\rangle \) at the sites \(e,f\) of \(\mathcal B\), then every output of the teleportation round of \(\mathcal T\), followed by the teleportation round of \(\mathcal B\) after \(G\), is a multiple of \(P_{\mathcal B}GP_{\mathcal T}|v\rangle \). The two rounds have depth \(2\) and \(3\).

2. Let \(2L{\lt}N\) and let \(a\) be a site. The forward chain of the hops \((a+2j,\, a+2j+1,\, a+2j+2)\), \(j{\lt}L\), and the backward chain of the hops \((a+2j+2,\, a+2j+1,\, a+2j)\), \(j{\lt}L\), are valid. The product of the transpositions of the forward chain maps \(a\) to \(a+2L\) and fixes the sites \(a+j\) with \(2L{\lt}j{\lt}N\); that of the backward chain is its inverse. The sites \(e,f\) of the forward chain lie among \(a+1,\dots ,a+2L\), those of the backward chain lie among \(a,\dots ,a+2L-1\), and all sites of both chains among \(a,\dots ,a+2L\). If \(|v\rangle \) has \(|0\rangle \) at the sites \(a+1,\dots ,a+2L\), then after the forward chain the sites \(a,\dots ,a+2L-1\) carry \(|0\rangle \).

Grouping all digits of one spatial site preserves placement of native operators on selected sites. Independent channels on the complete physical site blocks therefore have actual zero-depth physical-port implementations. Their Kraus operators are supported within the named spatial sites; no free global regrouping operation is assumed.

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.

Every finite adaptive protocol is trace preserving and completely positive. Two protocols with depth bounds \(T_1,T_2\) compose with depth bound \(T_1+T_2\). Deterministic local channel protocols are adaptive protocols with the same depth bound. Adaptive conversion is reflexive in depth zero, monotone in its depth bound, transitive with additive depth, and preserves density matrices.

Let \(\overline U,\overline V\) be the two placed operators in the common layout and set \(C=\overline V\overline U\). Then

The reduction identity also holds with arbitrary initial environment matrices in place of \(Z_A,Z_B\). If \(U,V\) have certified physical-port depths at most \(T,S\), then \(C\) has certified depth at most \(T+S\).

In the original/environment coordinates, placement of a matrix \(U\) on the original wires is \(U\otimes I_A\). For an arbitrary matrix \(V\) on all \(W+A\) wires, the map \(\mathcal D_A\circ \operatorname {Ad}_V\circ \mathcal I_A\) is exactly the partial-trace formula in these coordinates, with \(\operatorname {Ad}_V(X)=VXV^\dagger \). In particular, placement of \(U\) followed by this reduction gives \(\operatorname {Ad}_U\).

Every operator is converted into itself with depth and error zero. Increasing the depth or error bound preserves approximate conversion. Approximate conversion with error zero is equivalent to exact conversion with the same depth bound.

Let \(\rho \) and \(\sigma \) be Hermitian. If \(\rho \) is approximately converted into \(\sigma \) with depth \(T_1\) and error \(\varepsilon _1\), and \(\sigma \) into \(\tau \) with depth \(T_2\) and error \(\varepsilon _2\), then \(\rho \) is approximately converted into \(\tau \) with depth \(T_1+T_2\) and error \(\varepsilon _1+\varepsilon _2\).

Let the native input and output dimensions both equal the physical qudit dimension \(d\geq 2\), let \(N\geq 2\), and suppose every intermediate local dimension is positive and at most one bound \(B\). For \(k=\lceil \log _d B\rceil \), an actual physical-port protocol satisfies

Here \(J_{\mathrm{port}}\) places the input on its original physical ports and initializes all designated data/scratch memory in a product zero state. The output occupies the same ports, and that designated memory is reset to zero and uncorrelated with it. The identity holds on arbitrary operators and remains exact with an external reference.

Fix \(d\ge 2\), \(B\), and a local input dimension \(m\le B\), and choose a fallback density matrix on a pair of local inputs. Set \(k=\lceil \log _d B\rceil \), independently of the chain length. For every ring of \(N\ge 2\) sites and every matching of CPTP pair channels, there are normalized Kraus maps on the two \(k\)-qudit data registers that agree with the prescribed code extensions on the entire register space, and intertwine exactly with the original pair channels on encoded inputs. Their actual placed matching channel has a physical-port protocol of depth at most \(2k\). A product of \(L\) such certified layers has depth at most \(2kL\). For \(B=1\) the register width and this bound are zero.

A fixed-register operation word with \(T\) pair layers has physical depth at most \(2kT\). In particular, fix \(d\geq 2\), \(N\geq 2\), and one positive intermediate-dimension bound \(B\). Set \(k=\lceil \log _d B\rceil \). Every bounded native protocol \(\Psi \) of depth \(T\) has an actual physical-port realization \(\Gamma \) of depth at most \(2kT\), with

where \(R\) is the complete port/scratch reference and the physical encodings are constructed from independent site codes. This equality holds on all input-reference operators. A uniform family overhead requires fixing \(d,B\) before \(N\).

Let \(d\geq 2\), \(N\geq 2\), and let \(\Psi \) be a depth-\(T\) native protocol whose input and output dimensions equal \(d\) and whose positive intermediate local dimensions are all at most \(B\). With \(k=\lceil \log _d B\rceil \), an actual unitary circuit on the physical ports, designated memory, and fresh local environments satisfies

Both the designated data/scratch memory and the added environments start in product-zero states. After the environment discard, the output returns to its original ports and the designated memory resets to zero. The identity holds for arbitrary input operators. Uniform family overhead requires \(B\) and \(d\) to be fixed independently of \(N\).

For \(d\ge 1\) and \(n\ge 2\) there is \(K\) such that every unitary on \((\mathbb {C}^d)^{\otimes n}\) is a product of at most \(K\) unitaries, each acting on two neighbouring sites of the open chain of \(n\) sites. This is the step that the unitaries with constant support “can be further expressed with a low-depth circuit of local gates” [ 219 , caption of Fig. 1 ] .

For chain operators \(A,B\), the trace norm satisfies \(\lVert A\rVert _1\ge 0\), \(\lVert 0\rVert _1=0\), \(\lVert A\rVert _1=0\) if and only if \(A=0\), and \(\lVert A+B\rVert _1\le \lVert A\rVert _1+\lVert B\rVert _1\).

On a ring with \(N\geq 2\) and positive input dimension, extract actual two-site Kraus matrices from every supported native gate. Injectivity of placement recovers their local completeness relations. Independent basis codes and completed product decoders produce a native channel layer on the new alphabet, with the same matching and an exact global code-intertwining identity.

Let \(E_S\) be the product code of a selected register and \(E\) the product code of the complete chain, possibly with different local input and output dimensions. If \(\widehat\Phi E_S=E_S\Phi \) as linear maps on operators, then the placed channels satisfy \(\operatorname {Lift}(\widehat\Phi )E=E\operatorname {Lift}(\Phi )\) on every chain operator. The remaining sites may carry arbitrary correlations with the selected register.

For any common alphabet size \(q\geq B\), a dimension-bounded native protocol of depth \(T\) compiles into an explicit word of onsite channels and native pair-channel layers, all on \(q\) levels. Exactly \(T\) entries are pair layers. The word map intertwines with independent input and output codes on every chain operator. The completed decoders use definite local density matrices; no simulator identity is an input assumption.

A CPTP map on \(m\) \(d\)-level qudits has a unitary dilation with \(2m\) fresh \(d\)-level environment qudits, initialized to zero. The original \(m\)-qudit system is retained. If the selected wires all belong to one spatial site, assigning every fresh wire to that site makes this an actual zero-intersite-depth circuit. Consequently every normalized Kraus channel supported within one site’s wires has such a local appended-environment realization, independently of its Kraus count.

Let \(M\geq 1\), \(b\geq 1\) and \(\alpha \in \mathbb C^b\) nonzero, with the labels \(\{ 0,\dots ,b-1\} =\mathbb Z_b\). On the ring of \(2M\) sites of dimension \(b\), numbered \(0,\dots ,2M-1\), place the system qudit \(n\in \{ 0,\dots ,M-1\} \) at the site \(2n\) and its ancilla at the site \(2n+1\). The GHZ-type state \(|\chi _M\rangle =\sum _{i\in \mathbb Z_b}\alpha _i|i\rangle ^{\otimes M}\) of Definition 33.3.14 on the system qudits, with every ancilla in \(|0\rangle \), is prepared with measurements in depth \(2\). In the protocol below every outcome gives exactly this state. This is the construction of [ 227 , Example 1 ] , a preparation by \(\mathrm{QCcc}_2\), for qudits, and the preparation of \(|\chi _M\rangle \) in constant depth stated in the paragraph “Long-range MPS using measurements” of [ 219 ] .

In the setting of Theorem 33.9.10, with \(q\geq 1\), assume that at most \(K_g(d+1)^k\) sites lie at distance exactly \(d\) from any site. Let \(N_l(a)=\{ x:d_G(a,x)\leq l\} \). Then

with \(v=2v_Rb_0e^{2R}\) and \(C=2e^RK_g\, 2^kk!\, e^{1/2}/(1-e^{-1/2})\).

In this setting let \(A\in \mathcal A_{X_i}\), let \(y\) be a site, and let \(B\in \mathcal A_{\{ y\} }\). For every \(\mu \geq 0\), every integer \(0\leq n\leq d_G(a_i,y)\) and every real \(t\),

Consequently there are positive constants \(C,v,c\), depending only on \(R,v_R,b_0\), such that \(\| [\tau _t(A),B]\| \leq C\| A\| \| B\| e^{v|t|-cd_G(a_i,y)}\) whenever \(d_G(a_i,y){\lt}\infty \), and \([\tau _t(A),B]=0\) otherwise. The constants do not depend on the number of sites, on \(q\), or on the graph.

In the setting of Theorem 33.9.10, if \(d_G(a_i,y)=\infty \) then \([\tau _t(A),B]=0\) for every \(t\).

Local channel phase equivalence is an equivalence relation on eventually Hermitian families, and in particular on density-matrix families.

A distant gate is a two-site unitary \(u\) on the sites \(a\) and \(a+2L+1\), where \(2L+1{\lt}N\); its stretch is \(a,a+1,\dots ,a+2L+1\) and its interior the \(2L\) sites \(a+1,\dots ,a+2L\). Let \(g_1,\dots ,g_n\) be distant gates with pairwise disjoint stretches. Two measurement rounds of total depth \(5\) implement the product of the gates \(g_1,\dots ,g_n\) on the vectors with \(|0\rangle \) at the interiors of all the gates: the first teleports the content of every site \(a\) along its forward chain to \(a+2L\); the second applies every \(u\) to the neighbouring pair \(\{ a+2L,a+2L+1\} \) and teleports the content of every \(a+2L\) back to \(a\) along its backward chain. The depth depends neither on the distances nor on the number of gates, and no outcome is post-selected.

The average channel of a computational-basis measurement round is a finite adaptive protocol whose quantum-depth bound is the length of the round’s circuit.

The average map of a measurement round is trace preserving and completely positive. On a pure input \(|v\rangle \langle v|\), its output is \(\sum _m|V_mP_mUv\rangle \langle V_mP_mUv|\). In particular, if the round deterministically prepares \(\psi \) from a product input \(v\), and both input and target pure operators have trace one, its average output is \(|\psi \rangle \langle \psi |\).

Native onsite channels on \(d^k\) levels have physical-port depth zero. For \(d{\gt}0\) and \(N\geq 2\), a complete native matching layer has depth at most \(2k\). Every pair shares the same forward and backward routing; the cost has no multiplier equal to the number of gates. All unused wires are restored exactly, including their correlations with the data.

If each \(\Psi _J\) is trace preserving and completely positive, then so is \(\mathcal F\). If \(\Psi _J=\Psi \) is independent of \(J\), then \(\mathcal F=\Psi \circ \Phi \), where \(\Phi \) is the channel of the onsite instrument with its outcomes forgotten. A further linear map \(\Theta \) may be composed into every continuation in (3).

Suppose \(q{\gt}0\). Every \(P_S(p)\) is unitary. The Weyl product \(W_K(p)\) with factors \(W(p_x)\) off \(K\) and the identity on \(K\) satisfies \(W_K(p)=P_{I\setminus K}(p)\), and \(P_{\{ y\} }(p)\in \mathcal A_{\{ y\} }\) for every \(y\in I\). If \(y\notin S\), then

The port code stores the input on its original physical port and sets both local memory registers to zero. The data code stores its local register encoding on data digits, with zero port and scratch digits. Products of completed onsite channels convert these explicit pure codes in both directions at zero intersite depth. The channels are CPTP even outside the code, and the exact code identities hold on all logical operators.

If \(U\) has a physical-port unitary circuit of intersite depth at most \(T\), its placement along a port-preserving wire embedding has a physical-port unitary circuit of depth at most \(T\) in the enlarged layout. It acts as the identity on all additional wires.

Fix \(d\geq 1\) and a layout on a nonempty set of ring sites. For every finite physical-port channel protocol \(\Phi \) of depth at most \(T\), there are finitely many fresh environment wires with assigned spatial owners and an actual unitary \(U\) on the port-preserving extension such that

All fresh wires start in a product-zero state and are discarded only at the end. The equality holds on all original-wire operators and remains exact after tensoring every map with an arbitrary reference identity. No reset of the discarded environment is asserted.

Select one memory wire at each site. Swap it with the site’s physical port, apply a physical layer, and undo the onsite swaps. This acts exactly as the layer on the selected wires and as identity on all other wires, at physical depth one. Repeating this for \(k\) disjoint register digits implements the same matching permutation on each digit at depth at most \(k\), independently of the number of selected bonds. All unselected wires are restored on the whole operator space, without assuming they were initialized or uncorrelated.

Suppose an actual wire routing of physical depth \(T\) moves every channel in a commuting family into a single spatial site, possibly a different site for each member. Execute all those conjugated channels locally, then reverse the routing. The result is exactly the original family product, with physical depth at most \(2T\). The number of family members does not multiply this depth bound. The identities hold for every input operator, including arbitrary correlations with unused registers.

Let \(g_1,\dots ,g_n\) be register gates with pairwise disjoint stretches, each \(X\) a product of at most \(K\) unitaries on neighbouring sites. Then measurement rounds of total depth \(4s+K+2\) implement the product of the gates on the vectors with \(|0\rangle \) at the interiors of all the gates: the first registers are teleported next to the second ones as in Lemma 33.5.10, every \(X\) is applied to the \(2s\) consecutive sites \(a+2L,\dots ,a+2L+2s-1\) by a local circuit of depth \(K\), and the first registers are teleported back. The depth depends neither on the distances nor on the number of gates, and no outcome is post-selected.

For an arbitrary matrix on the selected register and its fresh environment, placing the matrix and then reducing the environment gives the placement of its local reduced map. Every unselected original wire remains an identity reference. The initialized environment-column blocks of the placed matrix are the placements of the corresponding local blocks.

The set in (12) contains zero and is bounded above by \(2\| B\| \). In particular, its supremum is finite and \(0\le d_y(B)\le 2\| B\| \). Every unitary \(U\in \mathcal A_{\{ y\} }\) satisfies \(\| [\widehat U,B]\| =\| [B,\widehat U]\| \le d_y(B)\). If these commutator norms are all at most a real number \(r\), then \(d_y(B)\le r\).

If \([\widehat U,B]=0\) for every \(U\in \mathcal A_{\{ y\} }\), then \(d_y(B)=0\). In particular, \(d_y(0)=0\), and every operator \(S\) on the spectator satisfies \(d_y(\mathbb 1_{\mathcal H}\otimes S)=0\).

For every \(c\in \mathbb {C}\),

Consequently \(d_y(-B)=d_y(B)\) and \(d_y(B-C)\le d_y(B)+d_y(C)\). For any finite set \(F\) and any family \((B_k)_{k\in F}\), \(d_y(\sum _{k\in F}B_k)\le \sum _{k\in F}d_y(B_k)\).

Let \(S,T\subseteq I\). If \(A\) on \((\mathbb {C}^d)^{\otimes I}\) is supported on \(S\) and \(B\) on \((\mathbb {C}^e)^{\otimes I}\) is supported on \(T\), then the matrix \(A\otimes B\), expressed in the coordinates \(p\), is supported on \(S\cup T\) in \((\mathbb {C}^{de})^{\otimes I}\). This is the support identification for the doubled local interactions in [ 248 ] , 02-information.tex, lines 416–424 and 513–524.

Suppose \(q{\gt}0\). For \(p\in (\mathbb {Z}/q\mathbb {Z}\times \mathbb {Z}/q\mathbb {Z})^I\), let \(W_K(p)\) be the product of the on-site Weyl operators \(W(p_x)\) at \(x\notin K\), with the identity at \(x\in K\). Then

Labels at retained sites are included in the sum; each choice of Weyl operators outside \(K\) consequently occurs \(q^{2|K|}\) times.

For every \(U\in \mathcal A_{I\setminus K}\), \([E_K^{\mathcal K}(B),\widehat U]=0\).

If \(q{\gt}0\), then \(\| E_K^{\mathcal K}(B)\| \le \| B\| \).

Suppose \(q{\gt}0\), and let \(\varepsilon :I\to \mathbb {R}\) satisfy \(\| [B,\widehat U]\| \le \varepsilon _y\) for each \(y\notin K\) and every unitary \(U\in \mathcal A_{\{ y\} }\). Then

Suppose \(q{\gt}0\). For physical operators \(A\) and spectator operators \(S\),

Retaining all physical sites gives \(E_I^{\mathcal K}(B)=B\).

Let \(I\) be a finite set of sites, let \(q\geq 1\), and let \(K\subseteq I\). Let \(H'\) and \(H\) be Hermitian operators on \((\mathbb {C}^q)^{\otimes I}\), and put \(G_K(t)=E_K(e^{itH'}(H'-H)e^{-itH'})\). There is a unique differentiable path \(U\) satisfying

For every real \(t\), the operator \(U(t)\) is unitary and acts on \(K\). Moreover, for any real \(t,\varepsilon \), if

for all \(s\) between \(0\) and \(t\), then

Empty site sets and empty regions are included. This is the supported relative evolution used in [ 248 ] .

A list \(h_1,\dots ,h_H\) of hops is valid if every hop \(h_j\) meets the hops \(h_{j+1},\dots ,h_H\) after it in the list only through its site \(c\): its sites \(e\) and \(f\) are not sites of \(h_{j+1},\dots ,h_H\), and its site \(c\) is not a site \(c\) or \(e\) of one of them. Its site \(c\) may be the site \(f\) of one of them, so the hops form chains \(c_0\to f_0=c_1\to f_1=\cdots \), the hops after \(h_j\) in the list acting before it. For a valid list and a local circuit \(U\), the teleportation round applies \(U\), the first gates of all hops in one layer, the second gates of all hops in one layer, measures the sites \(c\) and \(e\) of all hops, and corrects by the inverse of the product of the single-site unitaries that the hops leave. Its depth is the depth of \(U\) plus \(2\). It implements \(P_{h_1}\cdots P_{h_H}U\), with \(P_h\) the transposition of the sites \(c\) and \(f\) of \(h\), on the vectors \(|v\rangle \) such that \(U|v\rangle \) has \(|0\rangle \) at the sites \(e\) and \(f\) of every hop, with the scalar \(d^{-H}\) for every outcome.

Let \(U\) act on the original system and first environment, and let \(V\) act on the original system and second environment. In the three-block coordinates, their placed operators are \(U\otimes I_B\) and \(\widetilde V=\Sigma ^\dagger (V\otimes I_A)\Sigma \), where \(\Sigma \) sends \((s,a,b)\) to \((s,b,a)\). Initialization is \((X\otimes Z_A)\otimes Z_B\), and discarding the combined environment equals \(\operatorname{tr}_A\operatorname{tr}_B\) in these coordinates.

Let \(R\) be a normed star ring satisfying the \(C^*\)-identity. For \(B\in R\) and unitaries \(U,V\in R\),