arxiv: 2603.13536 · v1 · submitted 2026-03-13 · 🪐 quant-ph · cs.LG

Recognition: 2 theorem links

· Lean Theorem

Active Sampling Sample-based Quantum Diagonalization from Finite-Shot Measurements

Rinka Miura

Authors on Pith no claims yet

Pith reviewed 2026-05-15 11:25 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG

keywords active samplingsample-based quantum diagonalizationEpstein-Nesbet correctionfinite-shot measurementsquantum spin chainsground-state energyactive learningquantum hardware

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The pith

AS-SQD uses an Epstein-Nesbet acquisition function to select the most important basis states from finite-shot samples and recover lower ground-state energies than random or standard SQD.

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

The paper frames sample-based quantum diagonalization as an active learning problem in which measured bitstrings are used to build a subspace that is then expanded intelligently. An acquisition function drawn from second-order perturbation theory ranks candidate basis states connected to the current subspace by their estimated energy-lowering effect. On disordered Heisenberg and transverse-field Ising chains up to 16 qubits, including data taken from an IBM processor, this selection yields substantially lower absolute energy errors than either naive use of sampled states or blind random expansion.

Core claim

AS-SQD iteratively diagonalizes the Hamiltonian restricted to a growing set of measured basis states, generates all states connected by single Hamiltonian terms, and adds the candidates that score highest on the Epstein-Nesbet second-order energy correction; across simulated and hardware runs the procedure produces substantially smaller absolute energy errors than standard SQD or random expansion while remaining robust to 20 percent excited-state contamination and real SPAM errors.

What carries the argument

Epstein-Nesbet second-order energy correction used as an acquisition function that ranks candidate basis states by the estimated second-order lowering of the variational energy when each is added to the current diagonalized subspace.

Load-bearing premise

The Epstein-Nesbet second-order correction still ranks candidate states correctly even when the prepared state contains excited-state contamination and only finite-shot measurements are available.

What would settle it

A controlled experiment in which AS-SQD and random expansion are run to the same subspace size on a model where the second-order perturbation approximation is known to be poor, such as a strongly disordered or highly frustrated spin chain, and the energy errors of AS-SQD are not lower.

Figures

Figures reproduced from arXiv: 2603.13536 by Rinka Miura.

**Figure 4.** Figure 4: FIG. 4. Ablation study of acquisition functions at [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Representative error trace ( [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

read the original abstract

Near-term quantum devices provide only finite-shot measurements and prepare imperfect, contaminated states. This motivates algorithms that convert samples into reliable low-energy estimates without full tomography or exhaustive measurements. We propose Active Sampling Sample-based Quantum Diagonalization (AS-SQD), framing SQD as an active learning problem: given measured bitstrings, which additional basis states should be included to efficiently recover the ground-state energy? SQD restricts the Hamiltonian to a selected set of basis states and classically diagonalizes the restricted matrix. However, naive SQD using only sampled states suffers from bias under finite-shot sampling and excited-state contamination, while blind random expansion is inefficient as system size grows. We introduce a perturbation-theoretic acquisition function based on Epstein--Nesbet second-order energy corrections to rank candidate basis states connected to the current subspace. At each iteration, AS-SQD diagonalizes the restricted Hamiltonian, generates connected candidates, and adds the most valuable ones according to this score. We evaluate AS-SQD on disordered Heisenberg and Transverse-Field Ising (TFIM) spin chains up to 16 qubits under a preparation model mixing 80\% ground state and 20\% first excited state. Furthermore, we validate its robustness against real-world state preparation and measurement (SPAM) errors using physical samples from an IBM Quantum processor. Across simulated and hardware evaluations, AS-SQD consistently achieves substantially lower absolute energy errors than standard SQD and random expansion. Detailed ablation studies demonstrate that physics-guided basis acquisition effectively concentrates computation on energetically relevant directions, bypassing exponential combinatorial bottlenecks.

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

AS-SQD turns basis selection into an active loop using Epstein-Nesbet scores and shows measurable gains over random expansion on both simulations and IBM hardware even with 20% excited-state contamination.

read the letter

The core new piece is framing SQD as active learning: after diagonalizing the current restricted Hamiltonian, the method scores connected candidate bitstrings by their Epstein-Nesbet second-order correction and adds the highest-scoring ones. This replaces the naive use of only sampled states or blind random growth, and the paper demonstrates lower absolute energy errors than both baselines on disordered Heisenberg and TFIM chains up to 16 qubits. The hardware validation with real SPAM errors and the 80/20 ground/excited preparation model is the part that matters most for near-term relevance; the ablations indicate the guided selection does concentrate effort on energetically useful directions rather than wasting it on the combinatorial explosion. The soft spot is the assumption that the perturbative ranking remains trustworthy once the initial subspace carries excited-state bias. The paper tests exactly the 20% contamination case and still reports consistent improvement, so the concern does not appear to break the method in these regimes, but the ranking could still degrade for smaller gaps or higher contamination levels not explored here. This work is aimed at people implementing variational diagonalization on noisy hardware who already have access to bitstring samples. It is worth sending to peer review because the algorithmic idea is simple to reproduce, the experiments include hardware data, and the gains are shown against clear baselines.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Active Sampling Sample-based Quantum Diagonalization (AS-SQD), which frames SQD as an active learning problem and introduces a perturbation-theoretic acquisition function based on Epstein-Nesbet second-order energy corrections to rank and add candidate basis states connected to the current restricted subspace. It evaluates the method on disordered Heisenberg and TFIM spin chains up to 16 qubits under an 80/20 ground/excited state preparation model, plus physical samples from an IBM Quantum processor, claiming substantially lower absolute energy errors than standard SQD and random expansion, with ablation studies showing that the physics-guided selection concentrates on relevant directions.

Significance. If the empirical improvements hold under the reported conditions, the approach offers a concrete way to mitigate bias from finite-shot sampling and state-preparation errors in near-term quantum diagonalization algorithms by using perturbative guidance to expand the basis efficiently, avoiding both the bias of naive SQD and the combinatorial cost of exhaustive or random expansion.

major comments (2)

[Acquisition function definition and § on initial subspace construction] The acquisition function relies on the Epstein-Nesbet second-order correction to rank candidates relative to the current diagonalized subspace. However, the initial samples are drawn from an 80/20 ground/excited mixture, so the reference state already contains substantial contamination; the perturbative ranking assumes the current variational state is sufficiently close to the true ground state that higher-order terms and off-diagonal couplings to excited states remain negligible. The manuscript does not provide a quantitative error analysis or bound demonstrating that the ranking remains reliable under this contamination level, which directly affects whether the active loop selects the energetically most important states.
[Results and evaluation sections] The central empirical claim is that AS-SQD achieves substantially lower absolute energy errors than baselines across simulated and hardware evaluations. The provided abstract and summary contain no numerical values, error bars, number of trials, or tables quantifying the improvement (e.g., mean absolute error reduction on 16-qubit instances), making it impossible to assess effect size or statistical significance from the given material.

minor comments (1)

[Abstract] The acronym AS-SQD is used in the title and abstract before being defined; expand on first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments below and indicate the revisions we will make.

read point-by-point responses

Referee: [Acquisition function definition and § on initial subspace construction] The acquisition function relies on the Epstein-Nesbet second-order correction to rank candidates relative to the current diagonalized subspace. However, the initial samples are drawn from an 80/20 ground/excited mixture, so the reference state already contains substantial contamination; the perturbative ranking assumes the current variational state is sufficiently close to the true ground state that higher-order terms and off-diagonal couplings to excited states remain negligible. The manuscript does not provide a quantitative error analysis or bound demonstrating that the ranking remains reliable under this contamination level, which directly affects whether the active loop selects the energetically most important states.

Authors: We agree that a quantitative error analysis for the acquisition function under the 20% excited-state contamination would strengthen the manuscript. While our empirical evaluations demonstrate robust performance, we did not include a dedicated bound or sensitivity analysis in the original submission. In the revised manuscript, we will add a new subsection in the Methods or Results section that provides numerical experiments quantifying the ranking stability across different contamination levels and derives approximate error bounds based on perturbation theory to address this concern. revision: yes
Referee: [Results and evaluation sections] The central empirical claim is that AS-SQD achieves substantially lower absolute energy errors than baselines across simulated and hardware evaluations. The provided abstract and summary contain no numerical values, error bars, number of trials, or tables quantifying the improvement (e.g., mean absolute error reduction on 16-qubit instances), making it impossible to assess effect size or statistical significance from the given material.

Authors: The full manuscript includes detailed results with numerical values, error bars, and statistics from multiple trials in the evaluation sections. However, we acknowledge that the abstract does not report specific numbers. We will revise the abstract to include key quantitative findings, such as the mean absolute energy errors and the number of trials performed, to allow readers to better assess the improvements. revision: yes

Circularity Check

0 steps flagged

Acquisition function derived from independent standard perturbation theory; no circularity

full rationale

The paper's central derivation introduces an acquisition function based on Epstein-Nesbet second-order energy corrections applied to the restricted Hamiltonian. This follows directly from established perturbation theory in quantum chemistry, without any reduction to fitted parameters from the final energy estimate, self-referential definitions, or load-bearing self-citations. The active sampling loop uses this external theoretical ranking to select basis states, and the overall claim of lower energy errors is evaluated empirically against baselines rather than forced by construction. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard quantum mechanical assumptions and the applicability of perturbation theory to the selected subspace under the given state preparation model.

axioms (1)

standard math The Epstein-Nesbet second-order perturbation correction can be used to estimate the energy lowering from adding a basis state to the subspace.
This is the core of the acquisition function and relies on standard quantum perturbation theory.

pith-pipeline@v0.9.0 · 5567 in / 1335 out tokens · 47352 ms · 2026-05-15T11:25:48.100582+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce a perturbation-theoretic acquisition function based on Epstein–Nesbet second-order energy corrections to rank candidate basis states... a(k)= |⟨k|H|ψ_S⟩|² / |E_S - H_kk|
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

AS-SQD consistently achieves substantially lower absolute energy errors than standard SQD and random expansion

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Critical Assessment of the Sample-Based Quantum Diagonalization for Heisenberg and Hubbard Models
quant-ph 2026-05 unverdicted novelty 5.0

SQD needs an exponentially increasing number of computational-basis configurations to approximate ground-state energies of Heisenberg and Hubbard models within fixed accuracy, even when configurations are chosen optim...

Reference graph

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