DOI: 10.5281/zenodo.18674828
title: Electron-Sector Integrity in Variational Quantum Eigensolvers — Computational Data and Validation Tools

description: 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