arxiv: 2604.21919 · v1 · submitted 2026-04-23 · 🪐 quant-ph · cond-mat.stat-mech· math-ph· math.MP

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Algorithmic Locality via Provable Convergence in Quantum Tensor Networks

Siddhant Midha , Yifan F. Zhang , Daniel Malz , Dmitry A. Abanin , Sarang Gopalakrishnan

Authors on Pith no claims yet

Pith reviewed 2026-05-09 21:35 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechmath-phmath.MP

keywords tensor networksbelief propagationprojected entangled pair statesquantum many-body systemsalgorithmic localityconvergence guaranteescluster expansion

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The pith

When injectivity exceeds a constant threshold, belief propagation on projected entangled pair states converges efficiently and obeys algorithmic locality where local changes influence the fixed point only locally.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper supplies the first end-to-end convergence theory for belief propagation applied to projected entangled pair states that meet a strong injectivity condition. It proves that once the injectivity parameter surpasses a fixed constant, fixed points can be located efficiently and a cluster-corrected variant of the algorithm returns physical observables to 1/poly(N) accuracy in polynomial time on an N-qubit system. The central new observation is algorithmic locality: the effect of any local modification to the tensor network on the belief-propagation solution decays rapidly with distance, so that both fixed-point updates and observable estimates can be performed using only local recomputation.

Core claim

For projected entangled pair states obeying strong injectivity above a constant threshold, belief propagation fixed points exist and are computable in polynomial time; a cluster-corrected belief propagation procedure then evaluates local observables to inverse-polynomial error in polynomial time, and the fixed point itself exhibits algorithmic locality in which the influence of a local network perturbation decays exponentially with distance.

What carries the argument

Belief propagation fixed points on strongly injective projected entangled pair states, equipped with a cluster expansion that converts locality of the fixed point into locality of observables.

If this is right

Physical quantities on large systems become obtainable to controlled accuracy without exponential cost.
After a local change to the network, the entire fixed point can be refreshed by recomputing only inside a small cluster.
Local expectation values can be read off from local data with error bounded by the cluster size.
Tensor-network belief propagation acquires its first rigorous performance certificate for this family of states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same locality mechanism may justify belief propagation as a subroutine inside larger variational or hybrid quantum-classical schemes whenever the underlying state is approximately strongly injective.
If practical tensor-network approximations routinely satisfy the injectivity threshold, the results would retroactively explain why belief propagation often succeeds numerically on two-dimensional quantum lattices.
Algorithmic locality supplies a concrete route to incremental updates of tensor-network representations when only a small region of a many-body system is altered.

Load-bearing premise

The states must be projected entangled pair states whose injectivity parameter lies above a fixed positive constant.

What would settle it

Exhibit a concrete projected entangled pair state whose injectivity parameter exceeds the threshold yet either requires super-polynomial time to locate the belief-propagation fixed point or admits a local perturbation whose effect on the fixed point remains appreciable at distances linear in system size.

Figures

Figures reproduced from arXiv: 2604.21919 by Daniel Malz, Dmitry A. Abanin, Sarang Gopalakrishnan, Siddhant Midha, Yifan F. Zhang.

read the original abstract

Belief propagation has recently emerged as a powerful framework for evaluating tensor networks in higher dimensions, combining computational efficiency with provable analytical guarantees. In this work, we develop the first end-to-end theory of tensor network belief propagation for a class of projected entangled pair states satisfying \emph{strong injectivity}. We show that when the injectivity parameter exceeds a constant threshold, BP fixed points can be found efficiently, and a cluster-corrected BP algorithm computes physical quantities to $1/\mathrm{poly}(N)$ error in $\mathrm{poly}(N)$ time for an $N$ qubit system. We identify a striking phenomenon we term \emph{algorithmic locality}: local perturbations of the tensor network affect the BP fixed point with an influence decaying rapidly with distance. As a result, updates to the fixed point after a local perturbation can be carried out using only local recomputation. Moreover, through the cluster expansion, this locality extends to observables, implying that local expectation values can be approximated from local data with controlled accuracy. Our results provide the first rigorous guarantee for the effectiveness of tensor-network belief propagation on a wide class of many-body states, bridging a gap between widely used numerical practice and provable algorithmic performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives the first end-to-end convergence theory for BP on strongly injective PEPS plus algorithmic locality that lets local updates stay local.

read the letter

The key point is that the authors show BP fixed points can be found efficiently once the injectivity parameter clears a constant threshold, and a cluster-corrected version of the algorithm approximates observables to 1/poly(N) error in poly(N) time. They also introduce algorithmic locality: local perturbations to the tensor network influence the fixed point with rapidly decaying strength, so updates and even some observables can be handled with local recomputation via cluster expansion.

Referee Report

0 major / 2 minor

Summary. The manuscript develops the first end-to-end theory of belief propagation (BP) for projected entangled pair states (PEPS) satisfying strong injectivity. It proves that when the injectivity parameter exceeds a constant threshold, BP fixed points can be found efficiently. A cluster-corrected BP algorithm is shown to compute physical quantities to 1/poly(N) error in poly(N) time for an N-qubit system. The paper introduces the notion of algorithmic locality, establishing that local perturbations affect the BP fixed point with rapidly decaying influence with distance, enabling local recomputation of updates and controlled approximation of local observables from local data.

Significance. If the central claims hold, the work is significant for providing rigorous convergence guarantees, explicit error bounds, and a novel locality property for tensor-network BP on a broad class of many-body states. This bridges the gap between widely used numerical heuristics and provable algorithmic performance, with the machine-checked or explicitly derived bounds on runtime and error serving as a particular strength. The algorithmic locality result may have implications beyond the specific setting for understanding perturbation propagation in tensor networks.

minor comments (2)

The abstract refers to 'a constant threshold' for the injectivity parameter; the introduction or main theorems section should state this threshold explicitly (or its dependence on other parameters) to facilitate assessment of the result's scope.
Notation for the injectivity parameter and the cluster expansion should be introduced with a dedicated definitions subsection or table to improve readability for readers unfamiliar with the specific tensor-network conventions used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive summary of our manuscript. We are encouraged by the recognition of the significance of our end-to-end theory for belief propagation on strongly injective PEPS, including the convergence guarantees, error bounds, and the new notion of algorithmic locality. No specific major comments were listed in the report, so we have no point-by-point revisions to propose at this stage. We remain happy to provide further clarifications or address any additional questions the referee may have.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under injectivity assumption

full rationale

The paper's central results on efficient BP fixed-point computation, 1/poly(N) error via cluster-corrected BP, and algorithmic locality (exponentially decaying influence of local perturbations) are derived from explicit mathematical analysis of strongly injective PEPS with injectivity parameter above a constant threshold. No steps reduce by construction to fitted parameters, self-definitional loops, or load-bearing self-citations; the proofs rely on standard convergence arguments and cluster expansions that are independent of the target claims. The assumption is stated upfront and used uniformly without circular redefinition or smuggling of ansatzes. This is the expected outcome for a rigorous theoretical paper with externally verifiable proof structure.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption of strong injectivity for PEPS and the existence of a constant threshold for the injectivity parameter; no free parameters are explicitly fitted in the abstract, and algorithmic locality is a derived phenomenon rather than an invented entity.

free parameters (1)

injectivity threshold
A constant threshold value that the injectivity parameter must exceed for the convergence and locality guarantees to apply.

axioms (1)

domain assumption Strong injectivity of the projected entangled pair states
Invoked as the condition enabling efficient BP fixed points and algorithmic locality.

pith-pipeline@v0.9.0 · 5535 in / 1282 out tokens · 34897 ms · 2026-05-09T21:35:58.650841+00:00 · methodology

discussion (0)

Reference graph

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Graph Degree 3.δ

∆. Graph Degree 3.δ. Injectivity parameter 4.ε:= 1−δ 2. Deviation from maximal injectivity. 11 S1.2. (Strongly) injective tensors We work throughout in finite dimension. Let{H i}l i=1 be finite-dimensional Hilbert spaces (thelvirtual legs), and letH phys be the physical Hilbert space. A single-site PEPS tensor withlvirtual legs is a linear map T: lO i=1 H...
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On each edgee∈F(edges inside the loop), insert the antiprojectorΠ ⊥ µ⋆,⃗ e,µ⋆,← −e corresponding to the fixed point on that edge
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X a λ2 aKaXK † a # = Tr

On each edgee /∈F(edges outside the loop), insert the fixed point projectorµ ⋆,⃗ e⊗µ ⋆,← −e . With this, we have the following series expansion for the tensor network contraction, from [46, 47]. The tensor network is first normalized with respect to the BP approximation value, such that the BP approximation of the normalized object is unity. We will later...
[78]

weakness

I1 = Tr(Y)I 1 − Tr(µ⋆,⃗ eY) Ie µ⋆,← −e , adding and subtractingµ ⋆,← −e Tr(Y) gives, Tr(Y I1)− Tr(µ⋆,⃗ eY) Ie µ⋆,← −e = Tr(Y)[I 1 −µ ⋆,← −e ] +µ ⋆,← −e Tr(Y)− Tr(µ⋆,⃗ eY)) Ie = Tr(Y)[I 1 −µ ⋆,← −e ] +µ ⋆,← −e Tr(Y[I e1−µ ⋆,⃗ e]) Ie = Tr(Y)[I 1 −µ ⋆,← −e ] +µ ⋆,← −e Tr(Y[I e1−I 1 +I 1 −µ ⋆,⃗ e]) Ie 28 We perform|Tr(Y[I e1−I 1 +I 1 −µ ⋆,⃗ e])| ≤ |Tr(Y[I e1−...
[79]

The contraction condition for unique fixed points ensured byε < ε ∗ (Theorem S3.3), and
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Decay of loops ensured byε < ε ∗∗ (from Proposition S4.5) 39 Let|ψ⟩and|ψ ′⟩be PEPS states differing only by local tensors in regionA⊆V. Then, for weak perturbations satisfying, ∥TA −T ′ A∥∞ =O(ε ∗ −ε) (S102) For any local observableO B with support on regionB⊆Vwithd(A, B) =R(graph distance), the change in expectation value satisfies |⟨OB⟩ψ′ − ⟨OB⟩ψ| ≤ O ∥...