Positronium and Penning Trap Hamiltonian Characterization with Sector-Audited VQE: Stability Benchmarks Across LiH, H₂, and Exotic Encoded Systems Using the Prometheus ELSD Platform

Positronium and Penning Trap Hamiltonian Characterization with Sector-Audited VQE: Stability Benchmarks Across LiH, H₂, and Exotic Encoded Systems Using the Prometheus ELSD Platform Multi-Seed VQE Auditing with the Prometheus Platform Document type: Technical Report — Verification & Validation (V&V) Author: Amit Brahmbhatt, Quantum-Clarity LLC Platform: Prometheus VQE Engine (penalized variant) Hardware: Ganymede workstation — NVIDIA L40S GPU (44.988 GB VRAM) Date: April 2026 Abstract We report a 50-statevector ensemble audit of the Prometheus ELSD (Electronic Landscape Stability Diagnostics) platform applied to two canonical VQE benchmark systems: lithium hydride (LiH) and molecular hydrogen (H₂). Three geometries of LiH — compressed, equilibrium, and stretched — were each characterized with 10 independent VQE seeds at UCCSD depth-6 / LANL2DZ / 20-qubit standard operating point, yielding 30 sector-verified statevectors. An additional 20 seeds across two H₂ geometries complete the dataset. All 50 statevectors pass dual sector audit (N-sector and S_z-sector clean) with zero failures. Exact σ values (ddof=0) are reported for all five geometries; three geometries classify as Rigid Stability and two as Coherent Open-Shell, where both Coherent Open-Shell assignments are threshold-driven by single-seed optimizer stalls within otherwise sector-clean, tightly clustered ensembles rather than by multi-basin physical landscapes or open-shell electronic structure. The dataset serves as a calibration anchor for the ELSD platform prior to deployment on transition-metal active site campaigns, and establishes a noise floor against which σ values from more complex systems can be interpreted. Two supplementary frontier diagnostics are included as appendices. Appendix A characterizes the hydrogen anion (H⁻) as a confirmed platform boundary condition: correlation-bound, inaccessible to gradient-based UCCSD, correctly identified as Model Pathology (barren plateau subtype), with CASCI recovering the correct energy within expected basis set error. Appendix B reports, to our knowledge, the first ELSD characterization of Penning trap normal modes and positronium via external Hamiltonian injection bypassing PySCF: positronium achieves chemical accuracy (Rigid Stability); axial and cyclotron modes of the BASE antiproton trap classify as Rigid Stability; the magnetron mode — an encoded inverted harmonic oscillator with no bounded variational ground state within the finite Fock truncation — classifies as Model Pathology (unbounded landscape subtype), consistent with its known role as the primary particle loss mechanism in physical antiproton traps. 1. Introduction The Prometheus ELSD platform produces sector-audited stability benchmarks (σ values in kcal/mol) for metal-centered active sites relevant to drug discovery, catalysis, and propulsion. Before trusting σ values on complex transition-metal systems — where ground truth is unknown — it is necessary to demonstrate that the engine recovers correct behavior on systems where the answer is known precisely. LiH and H₂ are the most widely cited VQE benchmark molecules in the quantum computing literature. H₂ was the first molecule simulated on quantum hardware (O'Malley et al., 2016). LiH serves as the standard four-electron physical system benchmark, though as noted in Section 2.1, the platform simulation uses the standard 10e/10o active space rather than a minimal 4-electron space. Both molecules have well-characterized potential energy surfaces (PES) from full configuration interaction (FCI) and coupled-cluster theory. This report documents the results of running the full ELSD audit protocol — 10-seed ensemble, sector enforcement, σ characterization — on five PES geometries across these two molecules. The dataset serves three purposes: Platform V&V: Confirms the Prometheus engine recovers correct PES shape and energetics on benchmark systems. Sector enforcement validation: Demonstrates that N-sector and S_z-sector penalties hold under PES stress across all seeds. Calibration anchor: Provides a record establishing a σ noise floor for downstream transition-metal campaign interpretation. 2. Methodology 2.1 Standard Operating Point All LiH runs use the Prometheus platform standard operating point: Parameter Value Qubits 20 Active space 10 electrons / 10 orbitals Ansatz UCCSD Circuit depth 6 Basis set LANL2DZ ddof 0 (registry convention) Pauli terms (approx.) ~3,100 Seeds per geometry 10 (seeds 0–9) Active space note: LiH is a four-electron physical system (Li: 2 core + 1 valence; H: 1 valence). The platform standard operating point uses 10e/10o on 20 qubits — larger than the minimal 4-electron space — for consistency across all campaigns including transition-metal systems. The physical sector targets are N=4 electrons and S_z=0, which the sector penalty enforces. The additional virtual orbitals are included to maintain platform uniformity; absolute energies reflect the LANL2DZ/10e/10o operating point, not a minimal-space calculation. Basis set note: LANL2DZ is an ECP basis optimized for transition metals and is the platform standard. It is not the conventional choice for LiH benchmarking (where cc-pVDZ or 6-31G* are typical). Absolute energies should not be compared directly to FCI/CBS literature values without basis set correction. The V&V claim rests on reproducibility, sector purity, and correct PES shape — not absolute energy accuracy relative to the CBS limit. 2.2 ELSD Audit Protocol Each geometry is run with seeds 0–9. For each seed: VQE optimization is performed with GPU-accelerated statevector simulation on the L40S. The resulting statevector is passed to decompose_energy.py for sector decomposition. Sector audit checks: (1) N-sector dominant at target electron count; (2) S_z-sector dominant at target value. A seed passes if both checks are clean (dom_p > 0.99 in the correct sector). σ (kcal/mol) is computed as the standard deviation of VQE energies across all 10 seeds (ddof=0), converted from Hartree via 627.509 kcal/mol/Ha. 2.3 Geometries LiH — three PES points: Label Bond Length (Å) Physical Regime LiH_compressed 0.798 Repulsive wall (0.5× equilibrium) LiH_eq 1.595 Equilibrium geometry LiH_stretched 3.190 Dissociation onset (2× equilibrium) H₂ — two PES points: Label Bond Length (Å) Physical Regime H2_eq 0.740 Equilibrium geometry H2_stretched 1.480 Dissociation onset (2× equilibrium) H₂ runs used cc-pVDZ basis (engine auto-selection for non-TM systems, full valence active space auto-detected). 3. Results 3.1 LiH — Raw VQE Energies (10 seeds × 3 geometries) LiH Equilibrium (1.595 Å) Seed VQE Energy (Ha) 0 −7.9836525917 1 −7.9835128784 2 −7.9832663536 3 −7.9819273949 4 −7.9836125374 5 −7.9836544991 6 −7.9836421013 7 −7.9836363792 8 −7.9836301804 9 −7.9835548401 HF reference (seed 0): −7.9836158670 Ha | HF qubit deviation: −0.000903 kcal/mol LiH Stretched (3.190 Å) Seed VQE Energy (Ha) 0 −7.9032778740 1 −7.9037251472 2 −7.9036064148 3 −7.9037194252 4 −7.9036483765 5 −7.9025230408 6 −7.9037041664 7 −7.9036974907 8 −7.9032559395 9 −7.9037270546 LiH Compressed (0.798 Å) Seed VQE Energy (Ha) 0 −7.7323246002 1 −7.7322788239 2 −7.7322854996 3 −7.7323365211 4 −7.7320728302 5 −7.7321715355 6 −7.7319145203 7 −7.7322111130 8 −7.7323327065 9 −7.7323298454 3.2 H₂ — Raw VQE Energies (10 seeds × 2 geometries) H₂ Equilibrium (0.74 Å) Seed VQE Energy (Ha) 0 −1.1263631582 1 −1.1288101673 2 −1.1287646294 3 −1.1289353371 4 −1.1287529469 5 −1.1287987232 6 −1.1287772655 7 −1.1288111210 8 −1.1287544966 9 −1.1287279129 Note on seed 0: Seed 0 lands at −1.1264 Ha, approximately 1.5 kcal/mol above the cluster formed by seeds 1–9 (σ = 0.0358 kcal/mol for seeds 1–9 alone). This is a single-seed optimizer stall — the optimizer reached a shallow local basin rather than the ground state. The statevector is sector-pure (N=2, S_z=0 clean). The distribution is not bimodal; this is not Multi-Basin behavior. H₂ Stretched (1.48 Å) Seed VQE Energy (Ha) 0 −1.0055429935 1 −1.0060112476 2 −1.0060659647 3 −1.0051498413 4 −1.0060530901 5 −1.0061186552 6 −1.0058400631 7 −1.0059812069 8 −1.0060089827 9 −1.0060392618 Note on seed 3: Seed 3 (−1.0051 Ha) is a mild straggler, ~0.6 kcal/mol above the main cluster. Same pattern as H₂ equilibrium seed 0 — shallow basin trap, not a second physical minimum. 3.3 Summary Statistics — All Five Geometries All σ values are exact (ddof=0, 10 seeds each): System Geometry Bond (Å) Basis Mean E (Ha) σ (Ha) σ (kcal/mol) ELSD Class LiH Compressed 0.798 LANL2DZ −7.7322258 0.00013195 0.0828 Rigid Stability LiH Equilibrium 1.595 LANL2DZ −7.9834090 0.00050632 0.3177 Coherent Open-Shell† LiH Stretched 3.190 LANL2DZ −7.9034885 0.00036420 0.2285 Rigid Stability H₂ Equilibrium 0.740 cc-pVDZ −1.1285496 0.00073081 0.4586 Coherent Open-Shell‡ H₂ Stretched 1.480 cc-pVDZ −1.0058811 0.00028970 0.1818 Rigid Stability †LiH equilibrium note: σ = 0.3177 kcal/mol places this marginally above the 0.3 kcal/mol Rigid Stability threshold. The spread is driven by seed 3 (−7.9819 Ha), which lands 1.07 kcal/mol below the 9-seed cluster (seeds 0,1,2,4–9: σ = 0.0494 kcal/mol, Rigid Stability). This is a single-seed optimizer stall on the deepest geometry — physically the most tightly bound state and therefore the one where a suboptimal basin is hardest to escape. The classification is Coherent Open-Shell by strict σ threshold; the underlying 9-seed behavior is Rigid Stability. ‡H₂ equilibrium note: σ = 0.4586 kcal/mol. Driven by seed 0 stall (~1.5 kcal/mol above cluster). Seeds 1–9: σ = 0.0358 kcal/mol (Rigid Stability). H₂ at equilibrium is a closed-shell singlet; the Coherent Open-Shell label reflects the σ threshold classification, not the electronic structure. The cc-pVDZ full-valence active space (larger than necessary for 2 electrons) increases landscape dimensionality and the probability of single-seed stalls. 3.4 PES Shape Validation The three-point LiH PES is physically correct: E (Ha) −7.73 ● Compressed (repulsive wall — highest energy) | −7.90 ● Stretched (dissociation onset — intermediate) | −7.98 ● Equilibrium (global minimum — lowest energy) The ordering compressed > stretched > equilibrium is the defining signature of a bound diatomic PES. Recovery of this ordering across 30 independent seeds with no ordering violations confirms correct landscape navigation across all three LiH PES points regardless of ELSD classification. 3.5 Sector Audit — Complete Summary System Seeds passing N-sector S_z-sector Result LiH compressed 10/10 CLEAN ✓ CLEAN ✓ All pass LiH equilibrium 10/10 CLEAN ✓ CLEAN ✓ All pass LiH stretched 10/10 CLEAN ✓ CLEAN ✓ All pass H₂ equilibrium 10/10 CLEAN ✓ CLEAN ✓ All pass H₂ stretched 10/10 CLEAN ✓ CLEAN ✓ All pass 50/50 statevectors sector-pure. Zero sector failures across the entire dataset. 4. Discussion 4.1 Platform Behavior PES shape recovery. The engine correctly resolves all three regimes of the LiH PES without geometry-specific tuning. The same penalty parameters, active space, and circuit depth that handle transition-metal active sites also recover the correct LiH PES shape and sector purity across all three geometries. Sector enforcement under stress. The compressed geometry (0.798 Å, repulsive wall) presents the steepest gradient and highest nuclear repulsion — the hardest convergence case. All 10 seeds pass dual sector audit at this geometry, validating that the penalty scheme holds under energetic stress. Single-seed optimizer stalls. Two geometries (LiH equilibrium and H₂ equilibrium) show one outlier seed each, driving their σ above the Rigid Stability threshold. Both outlier statevectors are sector-pure — the optimizer stalled at a wrong energy but remained in the correct physical sector. This distinguishes optimizer stalls from sector failures: the sector penalty is functioning correctly even when the energy optimizer is not. HF anchor fidelity. The HF qubit expectation deviates from PySCF HF by −0.000903 kcal/mol at LiH equilibrium, confirming faithful Hamiltonian construction via Jordan-Wigner transformation. Cross-basis consistency. LiH used LANL2DZ (platform standard); H₂ used cc-pVDZ (engine auto-selection for non-TM systems). Both produce physically correct results, confirming correct basis routing behavior. 4.2 σ Noise Floor and Transition-Metal Interpretation The LiH σ values (0.083–0.318 kcal/mol) establish a calibration baseline. When σ values from transition-metal campaigns are interpreted as Rigid Stability — Fe porphyrin (0.30–0.36 kcal/mol), NEP Zn²⁺ (0.10–0.28 kcal/mol), Ni methalox (0.16–0.41 kcal/mol) — this dataset confirms those values reflect genuine electronic landscape character rather than engine noise, since the same platform produces σ ≤ 0.32 kcal/mol on simple light molecules where the electronic structure is well-understood. 5. Conclusion The Prometheus ELSD platform recovers correct PES shape and sector-pure statevectors for LiH and H₂ across five geometries. All 50 statevectors pass dual sector audit with zero failures. Exact σ values (ddof=0) range from 0.0828 to 0.4586 kcal/mol; three geometries classify as Rigid Stability and two as Coherent Open-Shell, with both Coherent Open-Shell classifications driven by single-seed optimizer stalls rather than multi-basin physical landscapes. The dataset provides a σ noise floor calibration for interpreting downstream transition-metal campaign results. 6. Data Provenance Item Value Compute host Ganymede (manager@ganymede) Campaign directory /mnt/nvme2n1/compute_bench/antimatter_cpt/ (historical internal path name reflecting campaign origin; does not describe the LiH/H₂ benchmark content) Engine prometheus_vqe_engine_penalized.py Conda environment /mnt/nvme2n1/conda-envs/prometheus_platform GPU NVIDIA L40S, 44.988 GB VRAM LiH log files run_AntiLiH_{eq,stretched,compressed}_s{0-9}.log H₂ log files run_AntiH_H2_{eq,stretched}_s{0-9}.log Sector audit tool /mnt/nvme2n1/compute_bench/p450/scripts/decompose_energy.py Appendix A: ELSD Classification Thresholds Class σ (kcal/mol) Description Rigid Stability < 0.3 Tight single basin, reproducible across seeds Coherent Open-Shell 0.3 – 1.0 Stable but spread across seeds; may reflect open-shell electronics or single-seed optimizer stalls Multi-Basin > 1.0, multimodal Competing minima, landscape fragmented Model Pathology Non-convergent Sector leakage, optimizer failure, or unbounded landscape Model Pathology subtypes identified in supplementary diagnostics: Subtype Example Mechanism Path Forward Barren plateau H⁻ ion Non-perturbative correlation from HF; gradient absent CASCI-first hybrid protocol Energy scale mismatch Penning trap (pre-rescaling) System energy << convergence threshold RESCALE_FACTOR normalization Unbounded landscape Magnetron mode Inverted/truncated Hamiltonian, no bounded ground state Physically correct — report truncation depth Appendix B: Supplementary Frontier Diagnostics These sections document exploratory campaigns conducted after the primary V&V dataset was complete. They extend the platform boundary map but are not part of the core V&V claim. For Zenodo deposit purposes, this record is primarily a LiH/H₂ multi-seed VQE benchmark; the frontier diagnostics are supplementary and should be weighted accordingly in metadata and search indexing. B.1 H⁻ Ion — Boundary Condition: Barren Plateau Motivation: H⁻ is the matter analog of the anti-hydrogen ion produced in CERN's GBAR experiment. Under CPT symmetry, binding energies are identical. System: Two-electron anion (charge = −1), aug-cc-pVDZ basis, 2e/4o active space, 8 qubits. H⁻ is correlation-bound: HF predicts it as unbound; the anion exists solely through electron correlation. Results across three systematic attempts: Attempt Basis Active Space Qubits Correlation Captured v1 cc-pVDZ 2e/5o 10 0.0% v2 aug-cc-pVDZ 2e/9o 18 0.4% v3 aug-cc-pVDZ 2e/4o 8 0.1% Best attempt energetics (aug-cc-pVDZ, 2e/4o): Energy Value (Ha) HF −0.4868 VQE −0.4868 CASCI −0.5043 Correlation gap 11.0 kcal/mol VQE captured 0.009 kcal/mol (0.1%) Diagnosis: The failure is structural, not numerical. The HF reference (−0.4868 Ha) is correct (Mulliken charge = −1.000 confirmed). CASCI solves the problem exactly. VQE lands at HF regardless of active space size — the signature of a barren landscape: the correlation is non-perturbative from the HF reference, meaning gradient descent from HF produces no useful signal toward the correlated region of Hilbert space. ELSD Classification: Model Pathology (barren plateau) Path forward: CASCI/aug-cc-pVDZ recovers H⁻ energy within 0.24 kcal/mol of the exact non-relativistic value — within expected basis set error. The CASCI-first hybrid protocol (treat CASCI as primary result; flag VQE component as Model Pathology) extends platform reach to correlation-bound systems without hardware changes. B.2 Penning Trap Normal Modes and Positronium Motivation: These systems bypass PySCF entirely, using external Hamiltonian injection via vqe_antimatter_penning.py. All Hamiltonians are constructed from first principles using BASE experiment parameters (B = 1.5 T, V₀/d₀² = 10⁶ V/m²). Energy rescaling: Penning trap frequencies produce energies at 10⁻¹² to 10⁻⁹ Ha in SI-derived Hartree units — below the engine convergence threshold. A dimensionless rescaling factor anchors the axial zero-point energy to 0.5 Ha: RESCALE_FACTOR = TARGET_SCALE / E_z_physical = 0.5 Ha / 2.3674×10⁻¹⁰ Ha = 2.11×10⁹ All reported energies are rescaled values. Physical energies = VQE energy / RESCALE_FACTOR. The wavefunction topology is preserved exactly under this linear rescaling. Results: System Type Qubits VQE Error ELSD Class Notes Positronium (e⁺e⁻) Diagonal Coulomb 5 0.013 kcal/mol Rigid Stability E₁ = −0.25 Ha; μ = mₑ/2; no PySCF Axial mode (ωz) Normal oscillator 5 0.010 kcal/mol Rigid Stability BASE sideband cooling axis Cyclotron mode (ω₊) Normal oscillator 5 0.019 kcal/mol Rigid Stability BASE CPT measurement axis Magnetron mode (ω₋) Inverted oscillator 5 −331.97 kcal/mol Model Pathology See note below Magnetron classification note: The magnetron Hamiltonian is an inverted harmonic oscillator (H = −ℏω₋(n̂ + ½)) encoded in a finite 5-qubit Fock space (32 levels). Within this truncated space, higher Fock states have lower energy; there is no bounded variational ground state. The VQE optimizer correctly descends toward the lowest-energy state available — the Fock truncation boundary — rather than the zero-point state. This behavior reflects the mathematical structure of the encoded Hamiltonian (inverted + truncated), not a claim about physical antiproton dynamics beyond what the model encodes. The Model Pathology classification is appropriate: a gradient-based optimizer cannot be expected to find a bounded minimum in an unbounded landscape. The physical correspondence is noted for context: the mathematical instability of the inverted oscillator is consistent with the magnetron mode's known role as the primary particle loss channel in Penning traps, where resistive heating drives orbit growth toward electrode annihilation. This correspondence is illustrative, not a derived physical result. Three-mode stability portrait: Mode Rescaled ZPE ELSD Class Physical role Axial +0.500 Ha Rigid Stability Precision measurement axis Magnetron −0.0085 Ha Model Pathology Particle loss channel Cyclotron +3.662 Ha Rigid Stability CPT measurement axis To our knowledge, this is the first ELSD characterization of Penning trap motional modes and of positronium via external Hamiltonian injection. Quantum-Clarity LLC — April 2026 Provisional Patent Application 64/012,885 — Filed March 21, 2026