Electron-Sector Integrity in Variational Quantum Eigensolvers — Computational Data and Validation Tools

Electron-Sector Integrity in Variational Quantum Eigensolvers — Computational Data and Validation Tools Project HELIOS — Quantum Clarity LLCPrincipal Investigator: Amit BrahmbhattDate: February 2026Companion Paper: "Electron-Sector Integrity in Variational Quantum Eigensolvers: A Detect→Fix→Verify Framework and a New Axis of VQE Reliability" (manuscript submitted)Companion Chemistry Dataset: "Dataset: Redox-Dependent Charge Redistribution in Molybdenum-Doped Cuprate Clusters" (Zenodo DOI: 10.5281/zenodo.18674751) Overview This dataset provides computational data, statevectors, and validation tools supporting a methods paper on electron-sector drift in VQE. Two contributions: Practical: VQE implementations using non-number-conserving ansätze on parity-adjusted Hamiltonians can converge to solutions in the wrong electron sector. Standard convergence diagnostics do not detect this. We provide a complete detect→fix→verify workflow with computationally negligible overhead. Conceptual: Electron-sector purity is an independent axis of VQE reliability — orthogonal to optimization landscape ruggedness and basin multiplicity. A VQE result can be smooth, well-converged, and reproducible, yet wrong because it inhabits the wrong electron sector. This orthogonality means sector purity cannot be inferred from any existing diagnostic and requires its own validation step. The Cu₅MoO₁₂ cation is the proof-of-concept testbed. The mechanism and fix are general. Abstract Background / Problem Variational quantum eigensolver (VQE) implementations that employ non-number-conserving ansätze on parity-adjusted Hamiltonians can converge to states outside the intended electron-number sector. Standard convergence diagnostics — energy tolerance, gradient norms, parameter update thresholds — do not detect this failure mode, enabling sector-mixed solutions to pass all quality filters while producing energetically biased and orbitally uninterpretable results. Method We introduce a Detect→Fix→Verify framework for electron-sector integrity based on: (i) Hamming-weight analysis of the converged statevector to detect sector mixing, (ii) a quadratic number penalty term H′ = H + λ(N̂ − N_t)² added to the Hamiltonian to enforce sector purity, and (iii) post-optimization validation via NOON diagnostics and purity metrics to confirm restoration. All penalty terms decompose into single-qubit Z and two-qubit ZZ Pauli operators, commuting with the electronic Hamiltonian's Z components and requiring no new measurement infrastructure. Results In 20-qubit VQE simulations of Cu₅MoO₁₂ redox states, unconstrained optimization converges to a sector-mixed cation solution with P(N=13) = 0.8669 despite satisfying standard energy convergence criteria. This sector mixing produces an energy lowering of 2.91 kcal/mol relative to the intended N=13 sector — exceeding chemical accuracy — and inflates fractional NOON counts from 2 to 4, generating false-positive multi-reference diagnoses. Penalty enforcement (λ = 0.5 Ha) restores P(N=13) = 0.9984 with 80× purity improvement and reduces fractional NOON count to 2. The penalty expectation value ⟨λ(N̂−13)²⟩ = 0.0008 Ha (≈ 0.49 kcal/mol) is well below chemical accuracy, confirming physical Hamiltonian energies are recovered. Discovery We establish electron-sector purity as an independent axis of VQE reliability, orthogonal to optimization landscape ruggedness (σ_tail) and basin multiplicity (inter-seed variance). A VQE solution can be smooth, well-converged, and reproducible while inhabiting the wrong electron sector. We propose a Sector Validation Checklist — P(N=target) > 99%, purity proxy < 0.01, ⟨N⟩(1-RDM) ≈ ⟨N⟩(Hamming) — as recommended diagnostic thresholds for non-number-conserving VQE prior to publication of energetics or orbital diagnostics. Core Results Failure Mode Demonstration Cu₅MoO₁₂ Cation (unconstrained VQE): Metric Value Threshold Status Energy convergence Satisfied Standard ✓ Passes P(N=13) 86.69% > 99% ✗ FAILS Purity proxy 0.240 < 0.01 ✗ FAILS ⟨N⟩(1-RDM) 12.984 Target: 13 ✗ FAILS Fractional NOONs 4/10 — Inflated The failure mode is invisible to standard diagnostics. Convergence is satisfied; the sector is wrong. Energy bias: Unconstrained VQE = −1935.8317 Ha. After sector enforcement: VQE = −1935.8270 Ha. Difference = 2.91 kcal/mol — exceeds chemical accuracy. Detect→Fix→Verify Results Sector metrics before/after penalty: Configuration λ (Ha) P(N=13) P(N=12) P(N=14) Purity Proxy Classification Unconstrained 0.0 86.69% 7.34% 5.67% 0.240 SECTOR-MIXED ✗ Penalty λ=0.5 0.5 99.84% 0.08% 0.08% 0.003 SECTOR-PURE ✓ Purity improvement: 80× NOON stabilization: Configuration Fractional Spatial NOONs Character Unconstrained 4/10 2 NOONs are sector-mixing artifacts Penalty (λ=0.5) 2/10 Clean open-shell singlet Key finding: Sector mixing inflates fractional NOON counts, producing false-positive multi-reference diagnoses. Sector validation must precede NOON-based orbital analysis. Three Axes of VQE Reliability (Conceptual Framework) Landscape Basins Sector Interpretation Smooth Single Pure Ideal — reliable result Smooth Single Mixed Dangerous — looks correct, wrong sector Rugged Multiple Pure Complex landscape, but valid sector Rugged Multiple Mixed Multiple problems; full validation needed The "smooth + single + mixed" case is most dangerous: passes all conventional quality filters while being physically invalid. Because the axes are orthogonal, sector purity cannot be inferred from landscape or basin diagnostics. It requires its own validation step. Proposed Sector Validation Checklist Recommended diagnostic thresholds for publication-quality VQE: Metric Threshold Classification P(N=target) > 99% SECTOR-PURE — publication-ready P(N=target) 90–99% N-DOMINANT — apply penalty; verify P(N=target) < 90% SECTOR-MIXED — penalty required Purity proxy (1−ΣP²) < 0.01 Essentially pure Purity proxy > 0.10 High mixing — penalty required ⟨N⟩(1-RDM) − ⟨N⟩(Hamming) < 0.01 Internally consistent Dataset Files See README.md in dataset for complete file manifest and usage instructions. Key components: Statevectors (unconstrained, penalty-constrained, sector-drifted neutral) Natural Orbital Occupation Numbers (NOONs) Validation tools: compute_sector_metrics.py, add_number_penalty.py, extract_noons_from_statevector.py, calculate_penalty_expectation.py, validate_sector.sh Requirements: requirements.txt (numpy, openfermion, pyscf) Figures: Convergence curves (PDF + PNG) Citation If you use this dataset or tools, please cite: This dataset:Brahmbhatt, A. (2026). Electron-Sector Integrity in Variational Quantum Eigensolvers — Computational Data and Validation Tools [Data set]. Zenodo. https://doi.org/10.5281/zenodo.18674828 Companion chemistry dataset:Brahmbhatt, A. (2026). Dataset: Redox-Dependent Charge Redistribution in Molybdenum-Doped Cuprate Clusters — VQE Computational Data [Data set]. Zenodo. https://doi.org/10.5281/zenodo.18674751变分量子本征求解器（Variational Quantum Eigensolver, VQE）中的电子扇区完整性——计算数据与验证工具 HELIOS项目——Quantum Clarity LLC 首席研究员：Amit Brahmbhatt 日期：2026年2月 配套论文：《变分量子本征求解器中的电子扇区完整性：一种Detect→Fix→Verify框架与VQE可靠性新维度》（已投稿手稿） 配套化学数据集：《钼掺杂铜酸盐团簇的氧化还原依赖电荷再分配数据集》（Zenodo DOI: 10.5281/zenodo.18674751） 概述 本数据集提供计算数据、态矢量（statevector）与验证工具，支撑一篇关于VQE中电子扇区漂移的方法学论文。 本工作包含两项贡献： 1. 实践层面：在经宇称调整的哈密顿量上使用非粒子数守恒的ansatz试探态进行VQE计算时，算法可能收敛至目标电子扇区之外的解。标准收敛诊断无法检测这一问题。我们提供一套计算开销可忽略的完整Detect→Fix→Verify工作流。 2. 概念层面：电子扇区纯度是VQE可靠性的独立维度——与优化地形崎岖度和优化盆地数量正交。VQE结果即便平滑收敛、收敛性良好且可复现，仍可能因处于错误的电子扇区而失效。这种正交性意味着无法通过现有诊断推断扇区纯度，需开展专属验证步骤。 Cu₅MoO₁₂阳离子作为概念验证测试平台，其相关机制与修正方案具有普适性。 摘要 背景与问题 在经宇称调整的哈密顿量上使用非粒子数守恒ansatz试探态的变分量子本征求解器（VQE）实现，可能收敛至目标电子数扇区之外的态。标准收敛诊断——能量容差、梯度范数、参数更新阈值——无法检测这一失效模式，导致扇区混合解通过所有质量过滤，却产生能量偏差且轨道无法解释的结果。 方法 我们提出一种面向电子扇区完整性的Detect→Fix→Verify框架，基于以下三点：(i) 对收敛态矢量开展汉明权重（Hamming-weight）分析以检测扇区混合；(ii) 向哈密顿量添加二次数惩罚项 $H' = H + lambda(hat{N} - N_t)^2$ 以强制扇区纯度；(iii) 经优化后通过自然轨道占据数（Natural Orbital Occupation Numbers, NOON）诊断与纯度指标开展验证，确认扇区纯度恢复。所有惩罚项均可分解为单量子比特泡利（Pauli）Z算子与双量子比特泡利ZZ算子，与电子哈密顿量的Z分量对易，无需新增测量基础设施。 结果 在针对Cu₅MoO₁₂氧化还原态的20量子比特VQE模拟中，无约束优化收敛至扇区混合的阳离子解，即便满足标准能量收敛标准，其 $P(N=13) = 0.8669$。该扇区混合相较于目标 $N=13$ 扇区产生2.91 kcal/mol的能量降低——超出化学精度阈值——并将分数型自然轨道占据数从2虚增为4，生成假阳性多参考诊断结果。施加惩罚项（$lambda = 0.5$ Hartree）后，$P(N=13)$ 恢复至0.9984，扇区纯度提升80倍，且分数型自然轨道占据数降至2。惩罚项期望值 $langle lambda(hat{N}-13)^2 angle = 0.0008$ Hartree（≈0.49 kcal/mol）远低于化学精度阈值，证实物理哈密顿量能量得以恢复。 发现 我们确立电子扇区纯度为VQE可靠性的独立维度，与优化地形崎岖度（$sigma_{ ext{tail}}$）和优化盆地数量（种间方差）正交。VQE解即便平滑收敛、收敛性良好且可复现，仍可能处于错误的电子扇区。我们提出一份扇区验证清单——$P(N= ext{target}) > 99\%$、纯度代理指标 $<0.01$、$langle N angle_{(1 ext{-RDM})} approx langle N angle_{ ext{(Hamming)}}$——作为非粒子数守恒VQE在发表能量或轨道诊断结果前的推荐诊断阈值。 核心结果 失效模式演示 Cu₅MoO₁₂阳离子（无约束VQE）： | 指标 | 数值 | 阈值 | 状态 | | ---- | ---- | ---- | ---- | | 能量收敛性 | 满足 | 标准 | ✓ 通过 | | $P(N=13)$ | 86.69% | >99% | ✗ 未通过 | | 纯度代理指标 | 0.240 | <0.01 | ✗ 未通过 | | $langle N angle_{(1 ext{-RDM})}$ | 12.984 | 目标值：13 | ✗ 未通过 | | 分数型自然轨道占据数 | 4/10 | — | 虚增 | 该失效模式无法通过标准诊断检测。收敛性满足，但扇区错误。 能量偏差：无约束VQE = −1935.8317 Hartree。施加扇区约束后：VQE = −1935.8270 Hartree。差值为2.91 kcal/mol——超出化学精度阈值。 Detect→Fix→Verify结果 施加惩罚前后的扇区指标： | 配置 | $lambda$ (Hartree) | $P(N=13)$ | $P(N=12)$ | $P(N=14)$ | 纯度代理指标 | 分类 | | ---- | ---- | ---- | ---- | ---- | ---- | ---- | | 无约束 | 0.0 | 86.69% | 7.34% | 5.67% | 0.240 | 扇区混合 ✗ | | 惩罚项 $lambda=0.5$ | 0.5 | 99.84% | 0.08% | 0.08% | 0.003 | 扇区纯净 ✓ | 纯度提升：80倍 自然轨道占据数稳定化： | 配置 | 分数型空间自然轨道占据数 | 特征 | | ---- | ---- | ---- | | 无约束 | 4/10 | 2个自然轨道占据数为扇区混合伪影 | | 惩罚项（$lambda=0.5$） | 2/10 | 纯净开壳层单重态 | 关键发现：扇区混合会虚增分数型自然轨道占据数，生成假阳性多参考诊断结果。扇区验证必须先于基于自然轨道占据数的轨道分析。 VQE可靠性的三大维度（概念框架） | 优化地形 | 优化盆地 | 扇区状态 | 解释 | | ---- | ---- | ---- | ---- | | 平滑 | 单一 | 纯净 | 理想——结果可靠 | | 平滑 | 单一 | 混合 | 危险——外观正确，扇区错误 | | 崎岖 | 多个 | 纯净 | 地形复杂，但扇区有效 | | 崎岖 | 多个 | 混合 | 存在多重问题；需全面验证 | “平滑+单一+混合”的情况最为危险：通过所有常规质量过滤，却在物理上无效。由于各维度相互正交，无法通过地形或盆地诊断推断扇区纯度，需开展专属验证步骤。 推荐扇区验证清单 面向可发表级VQE的推荐诊断阈值： | 指标 | 阈值 | 分类 | | ---- | ---- | ---- | | $P(N= ext{target})$ | >99% | 扇区纯净——可发表 | | $P(N= ext{target})$ | 90–99% | 扇区主导——施加惩罚项并验证 | | $P(N= ext{target})$ | <90% | 扇区混合——需施加惩罚项 | | 纯度代理指标（$1-Sigma P^2$） | <0.01 | 近乎纯净 | | 纯度代理指标 | >0.10 | 高混合度——需施加惩罚项 | | $langle N angle_{(1 ext{-RDM})} - langle N angle_{ ext{(Hamming)}}$ | <0.01 | 内部一致 | 数据集文件 详见数据集中的README.md文件，获取完整文件清单与使用说明。 核心组件： 1. 态矢量（无约束、惩罚项约束、扇区漂移中性态） 2. 自然轨道占据数（NOONs） 3. 验证工具：compute_sector_metrics.py、add_number_penalty.py、extract_noons_from_statevector.py、calculate_penalty_expectation.py、validate_sector.sh 4. 依赖要求：requirements.txt（numpy、openfermion、pyscf） 5. 图表：收敛曲线（PDF + PNG格式） 引用 若使用本数据集或工具，请引用： 本数据集：Brahmbhatt, A. (2026). 变分量子本征求解器中的电子扇区完整性——计算数据与验证工具 [数据集]. Zenodo. https://doi.org/10.5281/zenodo.18674828 配套化学数据集：Brahmbhatt, A. (2026). 钼掺杂铜酸盐团簇的氧化还原依赖电荷再分配数据集——VQE计算数据 [数据集]. Zenodo. https://doi.org/10.5281/zenodo.18674751