Matrix Product Quantum Channels

• Matrix Product Quantum Channels are one-dimensional tensor network representations of completely positive, trace-preserving maps that extend the MPS/MPO paradigm to quantum processes.
• They employ local tensor factorizations via Kraus, Choi, and circuit constructions to systematically encode many-body quantum channels with finite bond dimensions.
• MPQCs bridge tensor-network theory with variational algorithms, facilitating advances in error mitigation, state preparation, and the asymptotic analysis of open quantum systems.

Matrix Product Quantum Channels (MPQCs) are one-dimensional tensor-network descriptions of completely positive, trace-preserving maps in which a many-body quantum channel is encoded by local tensors linked by finite-dimensional virtual bonds. Across the literature, the term covers two closely related constructions: channels represented directly as matrix-product objects—via Kraus, Choi, superoperator, or transfer-map factorizations—and shallow matrix-product-structured circuits that induce channels by Stinespring dilation. In both readings, MPQCs extend the matrix product state/operator paradigm from states and observables to quantum processes, and they connect tensor-network theory, variational simulation, quantum error mitigation, state preparation, and the asymptotic analysis of open-system and transfer-channel dynamics (Stucchi et al., 20 Mar 2026, Filippov et al., 2022, Rudolph et al., 2022).

1. Formal definition and equivalent descriptions

A quantum channel $\Lambda$ is a linear, completely positive trace-preserving (CPTP) map acting on density operators, with Kraus form

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$

Its Choi matrix is

$$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$

with complete positivity equivalent to $J(\Lambda)\succeq 0$ and trace preservation equivalent to $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$. In the matrix-product-channel formulation, the many-body Kraus data are factorized along a chain by local tensors with virtual bond dimension $r$, so that the global channel inherits the structure of an MPO or MPDO-like tensor network. In the formulation used for translation-invariant channels, an MPQC is specified by a single repeated rank-6 tensor $A^{ij,op}_{mn}$, where $i,j$ are input indices, $o,p$ are output indices, and $m,n$ are virtual indices; repeating this tensor on every site defines a homogeneous channel $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$0 on the full chain (Filippov et al., 2022, Stucchi et al., 20 Mar 2026).

A second standard realization appears through matrix product states. For an MPS with local tensors $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$1, the associated transfer completely positive map is

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$2

and in Liouville form

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$3

This transfer channel is a canonical MPQC. More generally, MPO-realized channels have Kraus operators of matrix-product form, while MPS transfer maps are special cases in which the channel acts on the auxiliary bond space rather than directly on physical qubits (Albert, 2018).

The circuit perspective is compatible with the channel perspective. Shallow matrix-product-structured circuits built from stacked nearest-neighbor two-qubit unitaries induce channels through Stinespring dilation: given a unitary $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$4 on system plus ancilla $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$5,

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$6

Accordingly, shallow matrix-product circuits obtained from MPS decompositions can be viewed as MPQCs, and with ancillas they become matrix product quantum channels in the strict CPTP sense (Rudolph et al., 2022).

2. Tensor-network structure, local purification, and transfer operators

In Pauli-transfer or superoperator form, an $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$7-qubit channel can be represented by a $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$8 matrix

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$9

where $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$0 and $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$1 are Pauli strings. The central matrix-product construction is to factorize $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$2 as an MPO with small bond dimension $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$3, so that long operators are replaced by products of local tensors carrying only bounded virtual correlations. The same channel can equally be represented by its Choi state as an MPDO or LPDO; in that setting, local purification enforces positivity by construction, and the superoperator MPO is obtained by local reshaping of the purified Choi tensors (Guo et al., 2022).

The locally purified translation-invariant subclass has a particularly rigid structure. A homogeneous MPQC tensor $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$4 admits a local purification if there exists a repeated single-site tensor $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$5 with one input physical leg, one output physical leg, one purification leg of dimension $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$6, and two virtual legs of bond dimension $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$7, such that

$$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$8

where $$J(\Lambda)=(\Lambda\otimes \mathrm{Id})(|\delta\rangle\langle\delta|),$$9 is the corresponding homogeneous matrix product isometry. In this subclass, complete positivity is automatic, while trace preservation is equivalent to the global condition

$J(\Lambda)\succeq 0$0

for all $J(\Lambda)\succeq 0$1. The associated transfer operator on bond space is

$J(\Lambda)\succeq 0$2

and, under the usual normal or canonical assumptions, its subleading spectrum controls correlation decay through

$J(\Lambda)\succeq 0$3

This reproduces the standard MPS/MPDO transfer-operator mechanism, but in the homogeneous locally purified setting the constraints are stronger than generic exponential clustering because finite blocking yields a strict finite light-cone (Stucchi et al., 20 Mar 2026).

A key structural theorem states that any homogeneous matrix product isometry can be written, after blocking at most $J(\Lambda)\succeq 0$4 sites, as a depth-two brickwork circuit of finite-range isometric gates

$J(\Lambda)\succeq 0$5

with

$J(\Lambda)\succeq 0$6

As a result, outputs of the channel from product inputs have a strict, finite light-cone, and for regions $J(\Lambda)\succeq 0$7 separated by more than four sites after the chosen blocking one has $J(\Lambda)\succeq 0$8 (Stucchi et al., 20 Mar 2026).

3. From matrix product states to shallow MPQCs

For a chain of $J(\Lambda)\succeq 0$9 qubits with local dimension $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$0, an MPS can be written as

$$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$1

with local tensors $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$2 and maximal bond dimension $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$3. Canonicalization converts MPS tensors into isometries. If a core yields an isometry $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$4 of shape $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$5 with $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$6, it can be extended to a unitary

$$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$7

Composing these unitaries along the chain produces a single linear layer that exactly prepares the MPS when gates can act on up to $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$8 qubits. Because real hardware typically supports only one- and two-qubit gates, the problem becomes one of decomposing these multi-qubit isometries into stacked shallow layers of nearest-neighbor two-qubit unitaries, which is precisely the matrix-product-circuit architecture targeted in the decomposition work (Rudolph et al., 2022).

The main algorithmic construction is a layer-by-layer disentangling scheme. At step $$\mathrm{Tr}_{\mathrm{out}}[J(\Lambda)]=I_{\mathrm{in}}$$9, a copy of the current MPS is truncated to $r$0 across every bond using SVD, converted into a nearest-neighbor two-qubit layer $r$1, and the inverse layer $r$2 is applied to the original MPS to reduce its entanglement. Iterating yields layers $r$3 with fidelity objective

$r$4

To improve fixed-depth performance, the construction is combined with constrained local optimization of the two-qubit gates. For a circuit $r$5, the paper optimizes

$r$6

through environment tensors $r$7, local SVD updates $r$8, and the unitary replacement $r$9, with a learning-rate-controlled update

$A^{ij,op}_{mn}$0

Among six benchmarked protocols, the sequential growth strategy $A^{ij,op}_{mn}$1—alternating analytical $A^{ij,op}_{mn}$2 disentangling with joint optimization of all accumulated layers—was reported to achieve the highest fidelities at lower depths and with fewer sweeps. In 12-qubit tests with $A^{ij,op}_{mn}$3 target states, including a $A^{ij,op}_{mn}$4 antiferromagnetic Heisenberg ground state, a Bars-and-Stripes superposition, and a random MPS, its advantage was especially pronounced for $A^{ij,op}_{mn}$5, where it often delivered orders-of-magnitude lower infidelity per added layer than the baselines (Rudolph et al., 2022).

A related line of work addresses state preparation as data encoding. The Matrix Product Disentangler (MPD) compiles target MPS into $A^{ij,op}_{mn}$6 layers of nearest-neighbor two-qubit gates without ancillas, with overall depth $A^{ij,op}_{mn}$7 and classical runtime $A^{ij,op}_{mn}$8, while MPD+TNO adds tensor-network optimization using L-BFGS-B. Reported results include efficient encoding of functions up to low-degree piecewise polynomials with accuracy exceeding $A^{ij,op}_{mn}$9, and a 128$i,j$0128 ChestMNIST image on 14 qubits with fidelity exceeding $i,j$1 at total depth approximately $i,j$2 single-qubit rotation and two-qubit CNOT gates. In the MPQC interpretation, these compiled encoders are matrix-product-structured state-preparation channels acting on $i,j$3 (Green et al., 23 Feb 2025).

4. Variational MPQCs for mitigation, simulation, and hybrid workflows

The variational Matrix Product Channel formulation used for VQE mitigation represents a many-qubit channel by local Kraus tensors $i,j$4 with finite bond dimension $i,j$5. Summing over the local Kraus index yields positive semidefinite local cores

$i,j$6

so complete positivity is automatic. The technical obstacle is trace preservation, which is global in naive form. The key contribution is the derivation of floor-local linear constraints

$i,j$7

together with

$i,j$8

which enforce global TP while allowing single-site semidefinite updates under sweeping optimization. The VQE objective is

$i,j$9

and the training loop uses informationally complete POVMs, training and validation quasistates, and down/up sweeps over local SDPs. For fixed $o,p$0 and Hamiltonian term count, the algorithm scales linearly with the number of qubits $o,p$1, while each local SDP variable has dimension $o,p$2 after grouping indices (Filippov et al., 2022).

On the stretched water benchmark, the method was instantiated on 12 qubits for CAS$o,p$3 in cc-pVDZ with a Hamiltonian containing 551 Pauli strings. The reported total linear-cut entanglement entropy of the target ground state was approximately $o,p$4 bits, the DMRG bond-dimension-1 reference gave $o,p$5 Ha, DMRG with bond dimension 2 gave $o,p$6 Ha, and the hardware-efficient VQE ansatz gave $o,p$7 Ha. With informationally complete measurements, training and validation sets of $o,p$8 shots each, and bond dimension $o,p$9, the best validation estimate was $m,n$0 Ha, below the classical DMRG(bond=2) reference. The channel was interpreted as adding correlations and denoising measured quasistates without assuming a noise model (Filippov et al., 2022).

An MPO-based error-mitigation method gives a parallel MPQC perspective. There, the noisy block channel is learned in PTM form as an MPO and inverted variationally by minimizing

$m,n$1

The method was demonstrated on fully parallel depth $m,n$2 circuits with up to $m,n$3 qubits under local and global noise, and the circuit error was reported to be reduced by several times with only a small bond dimension $m,n$4 for the noise channel. The inverse MPO of the circuit achieved Frobenius distances $m,n$5–$m,n$6 from identity with $m,n$7, and the approach explicitly models correlated noise that product-channel mitigators neglect (Guo et al., 2022).

MPQCs also appear as simulation primitives. The TensorMixedStates Julia library implements vectorized mixed-state MPS and applies quantum channels as MPO superoperators on these states. It supports discrete-time non-unitary gates through apply, continuous-time Lindblad evolution through mixed-state TDVP and approx_W, and W-I/W-II MPO approximations up to order 4. The representation does not automatically enforce positivity of $m,n$8 nor exact normalization under truncation, so accuracy is monitored through diagnostics such as $m,n$9, $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$00, and operator-space entanglement entropy (OSEE) (Houdayer et al., 16 May 2025).

5. Asymptotic theory, ergodicity, and infinite-volume limits

The long-run behavior of MPQCs is governed by their peripheral spectrum. If $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$01 are right and left peripheral eigenoperators with $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$02 and $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$03, then the asymptotic projection onto the peripheral subspace is

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$04

When the peripheral spectrum contains only fixed points, $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$05; with rotating points, the iterates exhibit quasiperiodic motion on the peripheral subspace, while the Cesàro limit projects onto the fixed-point sector. For faithful channels with a full-rank fixed point, the Noether-like theorem states that if $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$06, then each Kraus operator satisfies

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$07

This constrains admissible peripheral modes and, in the diagonalizable case, forces $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$08 to be rational, so the peripheral symmetry generated by conserved quantities is finite (Albert, 2018).

For ergodic sequences of channels $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$09, a quantum Perron–Frobenius theory yields random full-rank density matrices $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$10 and $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$11 and a rank-one replacement asymptotic. Writing $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$12, the channel segment normalized by $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$13 converges exponentially in induced $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$14 norm to

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$15

Applied to MPS transfer channels, this gives an explicit thermodynamic-limit formula for local observables and an exponential decay bound for two-point connected correlations in the bulk. In the translation-invariant primitive case, the correlation length reduces to the familiar spectral formula

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$16

with $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$17 the second-largest eigenvalue of the transfer channel (Movassagh et al., 2019).

A product-level trace-Dobrushin theory reformulates these phenomena directly in terms of channel products. For a positive trace-preserving map $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$18 on $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$19, the centered trace-Dobrushin coefficient is

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$20

It is submultiplicative and vanishes exactly for replacement channels. For deterministic products, decay of $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$21 is equivalent to trace-norm forgetting and to asymptotic replacement by a moving replacement channel. In the two-sided setting, pullback forgetting yields a unique boundary state $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$22 and canonical replacement family $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$23. For stationary random CPTP cocycles, submultiplicativity produces a trace-Dobrushin Lyapunov exponent $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$24; almost sure negativity of this exponent is equivalent to quenched trace-norm memory loss and gives exponential forward and pullback convergence to a unique dynamically stationary random replacement channel. When the $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$25-mixing profile of the channel environment tends to zero, annealed estimates are super-polynomial, while independence gives annealed exponential estimates. These channel statements transfer directly to deterministic and stationary random inhomogeneous MPS in left-canonical CPTP gauge, yielding infinite-volume limits, quantitative boundary stability, and correlation bounds controlled by the same auxiliary product coefficients (Pathirana, 30 Apr 2026).

6. Classification, expressivity, and limitations

A recurrent misconception is that MPQCs are exhausted by shallow unitary nearest-neighbor circuits. The recent structural literature draws a sharper distinction. In the translation-invariant locally purified class, all channels belong to a single phase: any two channels in the homogeneous locally purified subclass with matching input and output dimensions can be continuously deformed into one another. This contrasts with the unitary setting, where one-dimensional quantum cellular automata, equivalently homogeneous matrix product unitaries, carry a nontrivial GNVW index

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$26

The locally purified channel index is therefore trivial, even though the corresponding isometries admit constant-depth brickwork realizations after finite blocking. At the same time, a broader class of translation-invariant channels can generate long-range entanglement, and these remain deterministically implementable in constant depth using two rounds of local measurements and feedforward; the GHZ channel is the canonical example (Stucchi et al., 20 Mar 2026).

Expressivity results sharpen the distinction between MPQCs and simpler local-channel networks. For matrix product channels with bond dimension $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$27, the paper on VQE mitigation shows that expressivity depends primarily on $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$28, and that

$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$29

It also gives strict inclusions comparing MPQCs with brickwall and ladder networks of local channels, and notes that a distant controlled-unitary $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$30 lies in $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$31 but not in brickwall or ladder families unless the number of layers satisfies $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$32. Applied to states, an MPC$$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$33 acting on an MPS or MPDO increases effective bond dimension multiplicatively, $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$34, which is why hybrid “quantum + classical” postprocessing can outperform a purely classical strategy at the same bond dimension (Filippov et al., 2022).

The same literature also records the principal limitations. In MPS-to-circuit compilation, required depth still scales with $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$35, and in practice may exceed the lower bound $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$36 for exact two-qubit preparation; the analytical $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$37 disentangler is heuristic and can plateau or even reduce fidelity if pushed too far without optimization (Rudolph et al., 2022). In variational channel learning, small bond dimension $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$38 may be insufficient to capture the needed correlations, whereas larger $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$39 worsens trainability and the cost of local SDPs; moreover, because the inverse of a general noise process is not CP, MPQC mitigation cannot realize exact error cancellation for arbitrary strong noise (Filippov et al., 2022). In data encoding, high-entanglement or volume-law targets require larger bond dimension and many layers, so MPD alone stagnates and even MPD+TNO must rely on deeper circuits or more sophisticated strategies (Green et al., 23 Feb 2025). In mixed-state simulation, truncation in vectorized MPDO-like representations can slightly violate positivity or trace preservation unless larger $$\Lambda(\rho)=\sum_\alpha K_\alpha \rho K_\alpha^\dagger,\qquad \sum_\alpha K_\alpha^\dagger K_\alpha=I.$$40, smaller time steps, or tighter cutoffs are used (Houdayer et al., 16 May 2025).

Open directions are correspondingly diverse. Reported priorities include extensions from MPS to TTNs and PEPS, hybrid classical–quantum subroutines for higher-dimensional tensor networks, better optimizers and noise-resilience analyses for hardware implementations, formal approximation bounds relating depth and fidelity, learning MPQCs from experimental data efficiently, and classification questions beyond translation invariance, including symmetry-protected phases for channels (Rudolph et al., 2022, Stucchi et al., 20 Mar 2026).