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arxiv: 2606.09988 · v2 · pith:5CKBRHEQnew · submitted 2026-06-08 · 🪐 quant-ph · cond-mat.dis-nn· cond-mat.str-el· physics.comp-ph

Absence of poor local minima in matrix product states

Hao-Kai Zhang , Chenghong Zhu , Shuo Liu , Shi-Xin Zhang , Tao Xiang This is my paper

Pith reviewed 2026-07-02 22:33 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nncond-mat.str-elphysics.comp-ph
keywords matrix product stateslocal minimaenergy landscapegauge freedomtrainabilityvariational quantum algorithmsdensity matrix renormalization groupquantum circuits
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0 comments X

The pith

Matrix product states lack poor local minima in their energy landscapes because gauge freedom creates local overparametrization.

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

This paper resolves the apparent paradox that matrix product states train successfully in practice while quantum circuits generally suffer from poor local minima. It proves that MPS energy landscapes contain no poor local minima under conditions where brickwork circuits do. The proof shows that the gauge freedom allows an effective local overparametrization, causing local minima to concentrate near the global minimum. The central step is establishing invariance of the local minimum distribution when the orthogonality center is moved. Sympathetic readers would care because this accounts for the effectiveness of DMRG and guides design of trainable variational methods.

Core claim

The energy landscapes of matrix product states are free from poor local minima, under the same setting where brickwork circuits are not. The gauge freedom of MPS creates an effective local overparametrization that causes local minima to concentrate near the global minimum. The local minimum distribution is invariant under moves of the orthogonality center of MPS representations. Numerical experiments confirm that optimization of sequential circuits converges to near-optimal solutions even for random Hamiltonians, unlike brickwork circuits.

What carries the argument

Gauge freedom in MPS, which permits shifting the orthogonality center while preserving the represented state and creates local overparametrization.

If this is right

  • Optimization using sequential circuits reaches near-optimal values for random Hamiltonians.
  • MPS avoid trainability problems that affect brickwork circuit representations.
  • This underpins the practical success of density matrix renormalization group methods.
  • Variational algorithms based on MPS can be expected to converge reliably without poor minima traps.

Where Pith is reading between the lines

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

  • Similar gauge freedoms in other tensor network formats could yield comparable trainability benefits.
  • Ansatz designs for variational quantum eigensolvers might incorporate movable centers or equivalent freedoms to improve performance.
  • Testing the invariance on small bond-dimension MPS with known spectra could provide direct verification.

Load-bearing premise

The local minimum distribution of the MPS energy landscape is invariant under moves of the orthogonality center.

What would settle it

An explicit MPS representation and Hamiltonian containing a local minimum whose energy remains substantially above the global minimum after any allowed shift of the orthogonality center.

Figures

Figures reproduced from arXiv: 2606.09988 by Chenghong Zhu, Hao-Kai Zhang, Shi-Xin Zhang, Shuo Liu, Tao Xiang.

Figure 1
Figure 1. Figure 1: FIG. 1. Energy landscape schematics in the absence (a) or [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Local minimum correspondence. (a) and (c) depict [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The numerical training curves of (a) sequential cir [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Quantum circuits suffer from severe trainability issues: even shallow circuits are swamped with poor local minima. Yet matrix product states (MPS), which can be prepared by sequential circuits, are remarkably trainable in practice -- as demonstrated by decades of successful density matrix renormalization group calculations. In this work, we resolve this apparent paradox by proving that the energy landscapes of MPS are free from poor local minima, under the same setting where brickwork circuits are not. The key insight is that the gauge freedom of MPS creates an effective local overparametrization that causes local minima to concentrate near the global minimum, analogous to overparametrized classical neural networks. We rigorously prove that the local minimum distribution is invariant under moves of the orthogonality center of MPS representations. Numerical experiments further confirm that the optimization of sequential circuits converges to near-optimal solutions even for random Hamiltonians, in stark contrast to brickwork circuits. Our findings establish a theoretical understanding of the trainability of MPS, providing a valuable guide for designing variational quantum circuits and algorithms with better trainability in the future.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to resolve the trainability paradox between matrix product states (MPS) and brickwork quantum circuits by rigorously proving that MPS energy landscapes contain no poor local minima. The central argument is that gauge freedom induces an effective local overparametrization, established via a proof that the distribution of local minima is invariant under shifts of the orthogonality center; this is contrasted with brickwork circuits and supported by numerical optimization experiments on random Hamiltonians showing convergence to near-optimal solutions.

Significance. If the invariance result holds, the work supplies a concrete theoretical explanation for the decades-long empirical success of DMRG/MPS methods while identifying a structural reason why certain variational quantum circuits fail to train. The combination of a direct mathematical proof with reproducible numerical contrasts constitutes a clear advance in understanding overparametrization effects in quantum variational landscapes.

minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from an explicit statement of the precise setting (bond dimension, Hamiltonian class, system size) under which the invariance theorem is proven, to allow immediate comparison with the brickwork case.
  2. [Numerical experiments] Figure captions for the numerical experiments should include the precise optimization algorithm, number of random instances, and error bars or convergence criteria used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the central invariance result, and recommendation to accept. We are pleased that the connection to the trainability of DMRG/MPS methods and the contrast with brickwork circuits has been recognized as a clear advance.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central claim is a rigorous proof that the local-minimum distribution of the MPS energy landscape is invariant under orthogonality-center moves, which converts gauge freedom into effective overparametrization. This invariance is presented as a direct mathematical result rather than a fitted quantity, self-citation chain, or definitional renaming. No load-bearing step reduces by construction to its own inputs, and the argument is self-contained against the algebraic structure of MPS representations without invoking external fitted parameters or author-overlapping uniqueness theorems as the sole justification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the mathematical definition of MPS gauge freedom, canonical forms, and the invariance property under orthogonality center relocation; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption MPS representations admit gauge freedom via orthogonality center moves without changing the represented quantum state.
    Invoked as the key mechanism creating effective overparametrization (abstract key insight paragraph).
  • domain assumption The optimization setting for MPS is identical to that for brickwork circuits where poor minima exist.
    Stated as the comparison baseline for the absence claim.

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