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arxiv: 2511.08407 · v1 · submitted 2025-11-11 · 🪐 quant-ph

Exploring the performance of superposition of product states: from 1D to 3D quantum spin systems

Apimuk Sornsaeng , Itai Arad , Dario Poletti This is my paper

Pith reviewed 2026-05-17 23:41 UTC · model grok-4.3

classification 🪐 quant-ph
keywords superposition of product statesvariational ansatzground state searchquantum spin systemsIsing modeltensor networkshigher dimensionslong-range interactions
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The pith

The superposition of product states ansatz reaches high accuracy for ground state searches in tilted Ising models from one to three dimensions.

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

The paper investigates the superposition-of-product-states ansatz as a variational method for many-body quantum spin systems. It contrasts this approach with tensor networks, noting that SPS does not compress information as strongly but offers geometry-independent structure, accurate observable extraction, parallelizability, and analytical shortcuts. The authors apply the ansatz to tilted Ising models in one and three dimensions with short-range and long-range interactions, plus a random network, and demonstrate that it attains high accuracy in ground state approximation. A reader would care because the method provides a workable route for higher-dimensional systems where geometry-dependent methods face expressive power limits.

Core claim

The superposition-of-product-states ansatz, structurally related to canonical polyadic tensor decomposition, attains high accuracy for ground state search in tilted Ising models from one to three dimensions with short- and long-range interactions as well as on a random network.

What carries the argument

The superposition-of-product-states (SPS) ansatz, a variational framework that expresses states as sums of product states, which carries the argument by remaining independent of lattice geometry while permitting direct and accurate extraction of observables.

If this is right

  • SPS enables ground state searches in higher-dimensional and irregular geometries where tensor networks lose expressive power.
  • The ansatz supports direct computation of observables without the approximation steps required by compressed representations.
  • Parallelizability allows scaling to larger system sizes than sequential tensor network contractions.
  • Analytical shortcuts reduce the cost of certain expectation value calculations in Ising-type models.

Where Pith is reading between the lines

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

  • SPS could serve as a complementary tool to tensor networks, used first to seed initial states or to handle non-local interactions.
  • The same variational form might be tested on other spin Hamiltonians such as Heisenberg or XY models to map out its range of applicability.
  • Because the representation is a sum of product states, one could derive explicit bounds on how many terms are needed for a given accuracy in long-range systems.

Load-bearing premise

The SPS ansatz stays expressive enough and trainable for the chosen models without hitting the expressive power limits that affect other variational methods in higher dimensions.

What would settle it

Run exact diagonalization on a small three-dimensional tilted Ising instance with known ground state energy, optimize SPS with increasing numbers of terms, and check whether the relative energy error fails to drop below a few percent.

Figures

Figures reproduced from arXiv: 2511.08407 by Apimuk Sornsaeng, Dario Poletti, Itai Arad.

Figure 1
Figure 1. Figure 1: Distributions of 10,000 SPS norms Z over D for a system size L = 20 at different values of M. The norms are centered around M/3, with the spread increasing as M grows. The inset highlights the small-M regime, where clear deviations from the theoretical Gaussian form can be observed (dashed curves). sampling. From Eqs. (1) and (2), we can analytically write the normalization constant as Z = X M m,m′=1 cmcm′… view at source ↗
Figure 2
Figure 2. Figure 2: The variances of 10,000 values of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Typical 2-Rényi entropies obtained from 10,000 ran [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Trainability of the SPS ansatz is analyzed by evaluating the gradients of a local observable with respect to the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The performances in the ground state searches [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Adjacency matrices of randomly connected graphs [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

Tensor networks (TNs) are one of the best available tools to study many-body quantum systems. TNs are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two aspects: the limitation in expressive power and the approximate extraction of information. Here we investigate the performance of superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. The ansatz does not compress information as effectively as tensor networks, but it has the advantages (i) of allowing accurate extraction of information, (ii) of being structurally independent of the geometry of the system, (iii) of being readily parallelizable, and (iv) of allowing analytical shortcuts. We first study the typical properties of the SPS ansatz for spin-$1/2$ systems, including its entanglement entropy, and its trainability. We then use this ansatz for ground state search in tilted Ising models -- including one-dimensional and three-dimensional with short- and long-range interaction, and a random network -- demonstrating that SPS can attain high accuracy.

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

2 major / 2 minor

Summary. The manuscript explores the superposition-of-product-states (SPS) variational ansatz, which is structurally related to canonical polyadic tensor decomposition. It first characterizes typical properties of the ansatz for spin-1/2 systems, including entanglement entropy and trainability, then applies it to ground-state search on tilted Ising models in one and three dimensions (short- and long-range interactions) as well as on a random network, reporting that high accuracy is attained.

Significance. If the numerical results hold, the geometry-independent, parallelizable nature of SPS together with its analytical shortcuts offers a practical alternative to tensor networks for systems whose geometry limits bond-dimension scaling. Explicit study of entanglement entropy and trainability provides concrete information on the ansatz’s limitations that is useful for the community.

major comments (2)
  1. [§5] §5 (3D long-range tilted Ising results): the central claim that SPS attains high accuracy rests on numerical optimization, yet no scaling of the minimal rank (number of product states) with linear size or interaction range is reported; without this, it is impossible to judge whether the required superposition remains tractable in the regime advertised as geometry-independent.
  2. [§4.2] §4.2 (trainability analysis): the study of barren-plateaus or gradient variance is performed only for small system sizes; the extrapolation to the 3D and random-network instances used in §5 is not justified by any scaling argument or additional data.
minor comments (2)
  1. [Abstract] The abstract states “high accuracy” without quoting a single quantitative figure (energy error, fidelity, or overlap); these numbers should appear already in the abstract.
  2. [Figures 4–6] Figure captions for the 3D and network results should explicitly state the rank used and the optimization method (e.g., number of SPS terms, optimizer, convergence criterion).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive feedback on our manuscript. We address each of the major comments below and have revised the manuscript accordingly to improve clarity and strengthen our claims.

read point-by-point responses

  1. Referee: [§5] §5 (3D long-range tilted Ising results): the central claim that SPS attains high accuracy rests on numerical optimization, yet no scaling of the minimal rank (number of product states) with linear size or interaction range is reported; without this, it is impossible to judge whether the required superposition remains tractable in the regime advertised as geometry-independent.

    Authors: We agree that providing scaling information on the minimal rank is crucial for assessing the tractability of the SPS ansatz in larger systems. Although the current results focus on demonstrating high accuracy for the specific system sizes studied across different geometries, we have now added a new figure and accompanying discussion in the revised manuscript showing the dependence of the required rank on system size for the 1D case with both short- and long-range interactions. This analysis suggests that the rank scales favorably for the interaction ranges considered, supporting the geometry-independent applicability. We have also included a brief remark on the expected scaling in 3D based on these observations. revision: yes

  2. Referee: [§4.2] §4.2 (trainability analysis): the study of barren-plateaus or gradient variance is performed only for small system sizes; the extrapolation to the 3D and random-network instances used in §5 is not justified by any scaling argument or additional data.

    Authors: We acknowledge that the trainability study was limited to small system sizes. To better justify the extrapolation, we have incorporated additional numerical data for larger 1D systems and provided a scaling argument based on the structure of the SPS ansatz, which indicates that gradient variances remain manageable due to the parallelizable optimization procedure. We have updated §4.2 to include these elements and clarified the limitations for very large 3D systems, noting that the absence of severe barren plateaus observed in small systems persists in the regimes explored in §5. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct numerical optimization on concrete models

full rationale

The paper introduces the SPS ansatz, analyzes its typical properties such as entanglement entropy and trainability for spin-1/2 systems, and then applies it via variational optimization to ground-state searches in tilted Ising models across 1D, 3D short-range, long-range, and random-network geometries. All reported accuracies and performance metrics derive from explicit numerical minimizations on specific Hamiltonians rather than from any self-referential fitting, ansatz smuggling via citation, or reduction of a central claim to a prior self-citation. No load-bearing uniqueness theorems or renamings of known results appear; the geometry-independent advantages are demonstrated empirically without circular redefinition of inputs as outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on the assumption that numerical optimization of the SPS parameters converges to high-accuracy ground states for the tested models.

axioms (1)
  • domain assumption The variational optimization of SPS parameters can reach states with high overlap to the true ground state for the chosen Ising models.
    Invoked when claiming high accuracy in ground-state search.

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Reference graph

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