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arxiv: 2606.09995 · v1 · pith:YGIJNFTWnew · submitted 2026-06-08 · 🪐 quant-ph

Quantum resources in non-stoquastic quantum annealing

Chiara Capecci , Sebastian Nagies , Naga Dileep Varikuti , Philipp Hauke This is my paper

Pith reviewed 2026-06-27 16:05 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum annealingnon-stoquasticentanglement entropystabilizer Renyi entropyquantum resourcesphase transitionsquantum advantageclassical simulation hardness
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The pith

Non-stoquastic terms in quantum annealing maintain or increase the quantum resources that hinder classical simulation.

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

The paper measures entanglement entropy and stabilizer Rényi entropy along non-stoquastic annealing paths for the fully connected p-spin model and a geometrically local Ising model. It compares these quantities to the size of the spectral gap and finds that the resources either hold steady or grow larger in the deeply non-stoquastic regime. This pattern holds across the chosen paths even though the precise scaling depends on the model. A sympathetic reader cares because the same non-stoquastic catalysts that convert first-order phase transitions into second-order ones also keep the system hard for tensor-network and stabilizer-tableau simulators. The result ties measurable quantum features directly to the performance gain over stoquastic protocols.

Core claim

We address this question by computing quantum resources -- entanglement entropy and stabilizer Rényi entropy -- whose presence makes classical computations based on tensor networks and stabilizer-tableau methods exponentially hard. We compare these with the spectral gap along the annealing path for two paradigmatic benchmark models, the fully connected p-spin model and a geometrically local Ising model. While the exact behavior shows a subtle dependency on the underlying model and the annealing path, our numerics suggest consistently that the scaling of entanglement and non-stabilizerness is at least maintained in the deeply non-stoquastic regime and in some cases even significantly enhanced

What carries the argument

Entanglement entropy and stabilizer Rényi entropy, computed along the annealing path and compared to the spectral gap to quantify resources that render tensor-network and stabilizer methods hard.

If this is right

  • Tensor-network simulations of the annealing dynamics remain exponentially costly once non-stoquastic terms dominate.
  • Stabilizer-tableau methods encounter comparable exponential overhead from the elevated non-stabilizerness.
  • The performance advantage gained by converting first-order to second-order transitions is accompanied by sustained or larger quantum resources.
  • Classical methods that ignore these resources will continue to fail even when the gap is enlarged by the catalyst.

Where Pith is reading between the lines

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

  • The observed correlation could be used to screen candidate catalyst Hamiltonians by checking whether they also increase the chosen resource measures.
  • If the pattern holds for other geometrically local problems, non-stoquastic annealing would systematically separate quantum and classical simulability.
  • One could test whether deliberately suppressing the resource measures while retaining non-stoquasticity erases the gap improvement.
  • The same resource diagnostics might apply to other protocols that introduce sign-problem-inducing terms to obtain quantum advantage.

Load-bearing premise

The scaling behavior observed for the fully connected p-spin model and the geometrically local Ising model under the chosen annealing paths is representative of the broader class of problems where non-stoquastic catalysts convert first-order to second-order transitions.

What would settle it

A concrete counter-example would be any non-stoquastic path or model in which the gap improves markedly yet both entanglement entropy and stabilizer Rényi entropy fall below their stoquastic counterparts at corresponding points along the schedule.

Figures

Figures reproduced from arXiv: 2606.09995 by Chiara Capecci, Naga Dileep Varikuti, Philipp Hauke, Sebastian Nagies.

Figure 1
Figure 1. Figure 1: Illustration of two types of annealing schedules in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: First row: (a) energy gap, (b) bipartite entanglement entropy, and (c) stabilizer Rényi entropy as functions of the annealing parameters s and λ, for a p-spin model cost Hamiltonian with N = 30 and p = 5. All observables are evaluated on the instantaneous ground state of the Hamiltonian in Eq. (7). All quantities are shown in logarithmic scale to enhance the visibility of relevant features across the annea… view at source ↗
Figure 3
Figure 3. Figure 3: Top row: bipartite entanglement entropy of the instantaneous ground state along the annealing sweep as a function of s, for annealing paths defined by λ(s) = 1 − A sin(πs) with amplitudes A = 0, 0.5, 0.7, 0.9 (from left to right). For each amplitude, results are shown for system sizes N = 20, 30, 40, 50. Bottom row: corresponding stabilizer Rényi entropy density. Insets show the zoomed-in regions around th… view at source ↗
Figure 4
Figure 4. Figure 4: Graph structure of the cost Hamiltonian Hcost in the local Ising model defined by Eq. (11) for system sizes N = 8, 10, and 12 (introduced in Ref. [23]). Each graph con￾sists of two rings of N/2 qubits connected by two inter-ring couplings at diametrically opposite positions. Solid gray lines denote ferromagnetic couplings of strength Jij = −1, dashed black lines ferromagnetic couplings of strength Jij = −1… view at source ↗
Figure 5
Figure 5. Figure 5: Magnetization across Hamiltonian parameter space [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Energy gap and quantum resources evaluated on the instantaneous ground state of the Hamiltonian in Eq. (9), [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Finite-size scaling of (a) the minimum energy gap, (b) the maximum bipartite entanglement entropy, and (c) the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Representative spin configurations of the plateau [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Same as Fig. 8, but for [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Energy gap (top row), bipartite entanglement entropy (middle row), and stabilizer Rényi entropy density (bottom [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

Quantum annealing promises to solve combinatorial optimization problems by preparing the ground state of a target Hamiltonian. Standard annealing protocols are, however, stoquastic and can thus be simulated by sign-problem-free quantum Monte-Carlo methods. To obtain a true quantum advantage, it has been proposed to use non-stoquastic catalyst Hamiltonians. Active only at intermediate stages of the protocol, these can, for certain problems, convert first-order into second-order quantum phase transitions and thus permit an exponential speedup over the stoquastic protocol. At the same time, the non-stoquastic catalyst renders quantum Monte-Carlo methods inefficient. It remains, however, an open question how other classical computation methods are affected by the non-stoquastic terms. We address this question by computing quantum resources -- entanglement entropy and stabilizer R\'enyi entropy -- whose presence makes classical computations based on tensor networks and stabilizer-tableau methods exponentially hard. We compare these with the spectral gap along the annealing path for two paradigmatic benchmark models, the fully connected $p$-spin model and a geometrically local Ising model. While the exact behavior shows a subtle dependency on the underlying model and the annealing path, our numerics suggest consistently that the scaling of entanglement and non-stabilizerness is at least maintained in the deeply non-stoquastic regime and in some cases even significantly enhanced. Our results thus suggest that improvements of quantum performance in non-stoquastic annealing coincide with significant presence of quantum computational resources.

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 computes entanglement entropy and stabilizer Rényi entropy along non-stoquastic annealing paths for the fully connected p-spin model and a geometrically local Ising model, comparing their scaling to the spectral gap. The central claim is that these quantum resources are at least maintained (and sometimes enhanced) in the deeply non-stoquastic regime, suggesting that performance gains from converting first- to second-order transitions coincide with resources that render tensor-network and stabilizer-based classical methods exponentially costly. The abstract explicitly notes a model- and path-dependent subtlety in the observed behavior.

Significance. If the reported scaling holds beyond the two benchmarks, the work supplies concrete numerical evidence that non-stoquastic catalysis introduces quantum resources whose presence is directly tied to classical hardness, thereby supporting the prospect of genuine quantum advantage. The choice of entanglement entropy and stabilizer Rényi entropy is well-motivated, as both quantities have established links to the complexity of specific classical simulation techniques.

major comments (2)
  1. [Abstract / benchmark-models paragraph] Abstract and the paragraph introducing the benchmark models: the suggestion that resource scaling 'coincides with' performance improvement is load-bearing for the central claim, yet the numerics are confined to two specific models under chosen paths; the abstract itself flags a 'subtle dependency on model and annealing path,' so the representativeness for the broader class of first-to-second-order conversion problems is not established by the presented evidence.
  2. [Numerical results] Numerical-results section (comparison of scaling exponents): without reported system sizes, error bars, or explicit finite-size scaling analysis for the entropy quantities, it is difficult to assess whether the claimed 'maintenance or enhancement' of scaling is robust or an artifact of the accessible sizes.
minor comments (2)
  1. [Methods / definitions] Notation for the stabilizer Rényi entropy should be defined explicitly on first use, including the precise definition of the stabilizer group and the order of the Rényi entropy employed.
  2. [Figures] Figure captions should state the annealing schedule parameter values and the range of system sizes shown, to allow immediate assessment of the plotted data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below, providing the strongest honest defense of the manuscript's scope and claims while noting where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract / benchmark-models paragraph] Abstract and the paragraph introducing the benchmark models: the suggestion that resource scaling 'coincides with' performance improvement is load-bearing for the central claim, yet the numerics are confined to two specific models under chosen paths; the abstract itself flags a 'subtle dependency on model and annealing path,' so the representativeness for the broader class of first-to-second-order conversion problems is not established by the presented evidence.

    Authors: The manuscript explicitly frames its contribution around two paradigmatic benchmark models (the fully connected p-spin model and a geometrically local Ising model) chosen because they exhibit the known first-to-second-order transition conversion under non-stoquastic terms. This choice enables a direct, controlled comparison between the scaling of quantum resources and the spectral gap. The abstract already states that 'the exact behavior shows a subtle dependency on the underlying model and the annealing path' and qualifies the conclusion as what 'our numerics suggest.' The central claim is therefore limited to the observation that, within these benchmarks, resource maintenance or enhancement coincides with the performance gains; no broader universality is asserted. We view the acknowledged limitations as already sufficient to bound the scope. revision: no

  2. Referee: [Numerical results] Numerical-results section (comparison of scaling exponents): without reported system sizes, error bars, or explicit finite-size scaling analysis for the entropy quantities, it is difficult to assess whether the claimed 'maintenance or enhancement' of scaling is robust or an artifact of the accessible sizes.

    Authors: We agree that explicit reporting of these details will improve clarity. In the revised manuscript we will state the system sizes used for each model and path, include error bars (from sampling or ensemble averaging where relevant), and add a finite-size scaling discussion for both entanglement entropy and stabilizer Rényi entropy to demonstrate that the reported trends are not artifacts of the accessible sizes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; quantities computed directly from Hamiltonians

full rationale

The paper computes entanglement entropy and stabilizer Rényi entropy directly along the annealing paths of two explicit benchmark Hamiltonians and compares their scaling to the spectral gap. These resource measures are obtained from the eigenstates or density matrices of the given models without being defined in terms of the performance improvement or fitted to produce the claimed coincidence. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain appears in the derivation; the numerics constitute independent evidence for the suggested correlation. The representativeness concern raised by the skeptic is a question of external validity, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields minimal ledger. Relies on standard quantum-information definitions of the resource measures and on the assumption that the two benchmark models capture the relevant physics.

axioms (2)
  • standard math Standard definitions of entanglement entropy and stabilizer Rényi entropy in quantum many-body systems
    Invoked to quantify quantum resources that affect classical simulability.
  • domain assumption The two chosen models (p-spin and local Ising) are paradigmatic for studying non-stoquastic catalysis
    Stated in abstract as the basis for the numerical comparison.

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Forward citations

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