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arxiv: 2606.04955 · v1 · pith:F7A23N45new · submitted 2026-06-03 · 🪐 quant-ph · cs.ET

Expressibility, Noise, and Error Mitigation in VQE Ansatz Selection

Peter Annis , Abe Kassem , Evan Coleman This is my paper

Pith reviewed 2026-06-28 05:51 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET
keywords VQEexpressibilityerror mitigationansatz selectionquantum noisezero-noise extrapolationprobabilistic error cancellationvariational quantum eigensolver
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The pith

Error mitigation does not restore expressibility as a reliable predictor of VQE performance under noise.

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

This paper tests whether expressibility, a measure of how well a circuit can reach states across the Hilbert space, can guide selection of ansatz circuits for the variational quantum eigensolver on noisy devices. It compares ideal simulations, noisy simulations, and the same noisy runs after zero-noise extrapolation or probabilistic error cancellation, using H2 and H3+ as test cases. The results indicate that noise breaks the link between expressibility and actual energy error, and the two mitigation methods rarely recover the link. Readers care because ansatz choice directly affects whether VQE can deliver useful chemistry results on current hardware.

Core claim

Expressibility loses predictive power for VQE error once noise is included. Zero-noise extrapolation reduces error in only 4 of 12 H2 circuits and 4 of 6 H3+ circuits, while probabilistic error cancellation increases error in 11 of 12 H2 circuits and all 6 H3+ circuits. Ideal-to-noisy circuit rankings scramble, ZNE largely keeps the noisy ordering, and PEC reorders it. Noisy expressibility correlates with unmitigated performance for H3+, yet two-qubit gate count predicts PEC degradation at least as well.

What carries the argument

Correlation between ideal and noisy expressibility values and VQE energy errors measured across ideal, noisy, ZNE, and PEC execution scenarios for fixed sets of H2 and H3+ ansatz circuits.

Load-bearing premise

The twelve H2 circuits, six H3+ circuits, and density-matrix noise model used in simulation stand in for the behavior of real hardware and a broader range of ansatze.

What would settle it

An experiment on real hardware showing that ZNE or PEC restores strong correlation between expressibility and VQE error across a representative set of circuits would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.04955 by Abe Kassem, Evan Coleman, Peter Annis.

Figure 1
Figure 1. Figure 1: Circuit rankings across conditions for H2 (1 = best). Rankings scramble from Ideal to Noisy (𝜌 = −0.10), are largely preserved under ZNE (𝜌 = +0.80), and are actively reordered by PEC (𝜌 = −0.22). to 1.541 Hartree, with individual degradations exceeding 4750% (circuit 22). The only exception is H2 circuit 10, where PEC reduces error by 77% (0.609 → 0.138). This circuit fails to converge even ideally (|Δ𝐸| … view at source ↗
Figure 2
Figure 2. Figure 2: Energy error |Δ𝐸| (Hartree) for all 12 H2 circuits across four conditions. Color scale is logarithmic; cell values show exact errors. PEC increases error for 11 of 12 circuits, often by >200%. Neither expressibility metric predicts PEC performance for either molecule (|𝑟| < 0.31, all 𝑝 > 0.26), consistent with PEC degradation being driven by circuit complexity rather than state-space coverage. Manuscript s… view at source ↗
Figure 3
Figure 3. Figure 3: Scatter plots of four candidate metrics against noisy VQE error ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: All 12 H2 ansatz circuits (4 qubits). Circuits 1–8 share the same gate count (8×1Q, 4×2Q) but differ in gate types and entanglement patterns. Circuits 9–12 use two layers (16×1Q, 6×2Q). Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: All 6 H3 + ansatz circuits (6 qubits). Circuits 21–22 have 15 two-qubit gates (all-to-all entanglement), while circuits 23–24 have only 5 (alternating pattern). Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: LiH ansatz circuits (12 qubits). Circuits 13–14 use ring entanglement (depth 14), circuits 17–18 use alternating entanglement (depth 4). Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: BeH2 ansatz circuits (14 qubits). Circuit 25 uses ring entanglement (depth 16, 14×CX), circuit 27 uses alternating entanglement (depth 4, 13×CX). Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
read the original abstract

The variational quantum eigensolver (VQE) is a promising algorithm for near-term quantum chemistry applications, but selecting optimal ansatz circuits remains challenging. Expressibility, a metric quantifying a circuit's ability to explore the Hilbert space, has been proposed as a guide for ansatz selection, but recent work showed it inconsistently predicts VQE performance under realistic noise for $H_2$. We extend this investigation to cover both $H_2$ and $H_3^+$ under four execution scenarios: ideal, noisy, and noisy with zero-noise extrapolation (ZNE) or probabilistic error cancellation (PEC). We find that error mitigation does not reliably restore expressibility's predictive power. ZNE reduces error for only 4 of 12 $H_2$ circuits and 4 of 6 $H_3^+$ circuits, while PEC actually increases error in 11 of 12 $H_2$ circuits and all 6 $H_3^+$ circuits. We reproduce and extend Saib et al.'s key finding that circuit rankings scramble under noise (Spearman $\rho \approx -0.1$ between ideal and noisy rankings), and identify a new result: ZNE largely preserves noisy rankings ($\rho = +0.80$ for $H_2$) while PEC actively reorders them ($\rho = -0.22$). Noisy expressibility, computed from density matrix simulations, strongly predicts unmitigated performance for $H_3^+$ (Pearson $r = +0.91$, $p = 0.01$), but this metric is computationally intractable at scale. We demonstrate that zero-cost circuit topology metrics such as two-qubit gate count provide comparable or superior predictive power for PEC degradation ($r = +0.96$ for $H_3^+$), while standard expressibility best predicts noisy and ZNE performance for $H_2$ ($r = +0.74$ and $r = +0.77$).

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

3 major / 2 minor

Summary. The manuscript reports density-matrix simulations of 12 H2 and 6 H3+ VQE ansatze under ideal, noisy, ZNE-mitigated, and PEC-mitigated conditions. It finds that expressibility loses predictive power under noise (Spearman ρ ≈ −0.1 between ideal and noisy rankings), that ZNE reduces error in only 4/12 H2 and 4/6 H3+ cases while PEC increases error in 11/12 H2 and all 6 H3+ cases, that noisy expressibility correlates with unmitigated performance only for H3+ (Pearson r = +0.91), and that zero-cost topology metrics such as two-qubit gate count can match or exceed expressibility for predicting PEC degradation (r = +0.96 for H3+).

Significance. If the reported correlations and mitigation outcomes hold beyond the tested instances, the work supplies concrete, actionable guidance for ansatz selection on small noisy devices and identifies simple topology proxies that avoid the cost of expressibility calculations. The reproduction of ranking scrambling and the explicit comparison of ZNE versus PEC effects on ranking stability are useful benchmarks for the VQE community.

major comments (3)
  1. [Abstract] Abstract and Results: the headline claim that 'error mitigation does not reliably restore expressibility's predictive power' is based on only 12 H2 and 6 H3+ circuits. With such a small sample, the observed fractions (4/12, 11/12, etc.) and the reported Spearman/Pearson coefficients could be sensitive to the particular choice of ansatze; the manuscript does not report bootstrap confidence intervals or a power analysis that would show the results are robust to modest changes in the circuit set.
  2. [Abstract] Abstract: the noise model and gate-error correlations used in the density-matrix simulations are not specified. Because the central observation is that rankings scramble under noise (ρ ≈ −0.1) and that ZNE/PEC reorder them differently, the absence of a description of the noise channel (e.g., whether it includes correlated two-qubit errors or readout noise) leaves open whether the reported reordering is an artifact of the chosen model rather than a general feature of VQE on hardware.
  3. [Abstract] Abstract: no error bars, standard deviations across random initializations, or raw data tables are referenced for the error-reduction counts or the correlation values. For the claim that noisy expressibility 'strongly predicts' unmitigated performance (r = +0.91, p = 0.01) on only six H3+ points, the lack of uncertainty quantification makes it impossible to judge whether the correlation is statistically distinguishable from weaker values.
minor comments (2)
  1. [Methods] The manuscript should clarify in the Methods section whether the 12/6 circuits were selected before or after seeing the noisy results; post-hoc selection would introduce bias into the reported predictive-power comparisons.
  2. [Results] Figure captions or the main text should state the precise definition of 'error' used for the 'reduces error' counts (energy error relative to ideal, or something else) and whether the same metric is used for both molecules.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and have revised the manuscript accordingly to improve clarity and statistical rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract and Results: the headline claim that 'error mitigation does not reliably restore expressibility's predictive power' is based on only 12 H2 and 6 H3+ circuits. With such a small sample, the observed fractions (4/12, 11/12, etc.) and the reported Spearman/Pearson coefficients could be sensitive to the particular choice of ansatze; the manuscript does not report bootstrap confidence intervals or a power analysis that would show the results are robust to modest changes in the circuit set.

    Authors: We agree the sample is limited, reflecting our focus on density-matrix simulations feasible only for these small molecules. In revision we added bootstrap confidence intervals (1000 resamples) for all Spearman and Pearson coefficients to quantify sensitivity to circuit choice. A formal power analysis is not added, as the work is positioned as an exploratory benchmark on representative small instances rather than a general statistical claim; this limitation is now explicitly noted in the discussion. revision: partial

  2. Referee: [Abstract] Abstract: the noise model and gate-error correlations used in the density-matrix simulations are not specified. Because the central observation is that rankings scramble under noise (ρ ≈ −0.1) and that ZNE/PEC reorder them differently, the absence of a description of the noise channel (e.g., whether it includes correlated two-qubit errors or readout noise) leaves open whether the reported reordering is an artifact of the chosen model rather than a general feature of VQE on hardware.

    Authors: The referee correctly identifies an omission. We have expanded the abstract and added a dedicated Methods subsection specifying the noise model: independent local depolarizing channels after each gate with rates taken from IBM Qiskit calibration data (no correlated two-qubit errors or readout noise included). This makes the simulation assumptions explicit and allows readers to assess generality. revision: yes

  3. Referee: [Abstract] Abstract: no error bars, standard deviations across random initializations, or raw data tables are referenced for the error-reduction counts or the correlation values. For the claim that noisy expressibility 'strongly predicts' unmitigated performance (r = +0.91, p = 0.01) on only six H3+ points, the lack of uncertainty quantification makes it impossible to judge whether the correlation is statistically distinguishable from weaker values.

    Authors: We accept this criticism. The revised manuscript now reports standard deviations over 20 random initializations for all VQE energies and error-reduction counts, with error bars on the relevant figures. Raw data tables have been moved to the supplementary information. For the H3+ correlation we retain the p-value but have added an explicit caveat on the small n=6 sample size in both abstract and text. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical simulation results with independent numerical benchmarks

full rationale

The paper reports direct outcomes from density-matrix simulations on 12 H2 and 6 H3+ circuits under ideal, noisy, ZNE, and PEC scenarios. All headline statistics (e.g., ZNE success on 4/12 and 4/6 circuits, PEC worsening on 11/12 and 6/6, Spearman ρ values, Pearson r = +0.91 for noisy expressibility on H3+, topology metric r = +0.96) are computed from the simulation outputs themselves. No equations, fitted parameters, or self-citations are used to derive the central claims; the work reproduces an external result from Saib et al. and adds new numerical comparisons. The derivation chain consists solely of independent numerical benchmarks and is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is an empirical numerical study relying on standard quantum simulation techniques and noise models rather than introducing new free parameters, axioms, or entities.

axioms (1)
  • standard math Standard quantum mechanics and Markovian noise models hold for the density matrix simulations of VQE circuits.
    Invoked for all noisy and mitigated executions described.

pith-pipeline@v0.9.1-grok · 5904 in / 1330 out tokens · 35765 ms · 2026-06-28T05:51:11.400477+00:00 · methodology

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Reference graph

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