Variational Quantum Simulation of Antiferromagnetic Hamiltonians
Part of ARPA Quantum Logical Systems — QONDRA · qondra@arpacorp.net
In May 2026, the University of Tokyo (Nakatsuji Lab) demonstrated 40-picosecond spin-orbit torque switching in Mn₃Sn / tantalum heterostructures — a Kagome antiferromagnet whose switching is rooted in spin–orbit coupling (SOC).
We cannot fabricate this material in silico, but we can simulate its quantum many-body physics using Variational Quantum Eigensolvers (VQE) and Quantum Approximate Optimization Algorithms (QAOA) on a classical simulator — and compare results directly to exact diagonalization and spectroscopic benchmarks.
spinq-vqe is the open-source Python package that implements this pipeline: lattice construction, variational ansätze, VQE runners, entanglement analysis, TeNPy DMRG references, NetKet Neural Quantum State baselines, an MLP surrogate for spin Hall angle, and a QAOA material-selection optimizer. All seven research notebooks are executed with published figures and reference data.
We simulate the Heisenberg antiferromagnet on a 1D Kagome strip (the geometry used in the notebooks):
H = J Σ_{<i,j>} S_i · S_j + D Σ_i (S_i^z)² + B Σ_i S_i^z
The pipeline builds the lattice graph (NetworkX), maps spin operators to Pauli strings (PennyLane), and runs VQE with COBYLA (primary) or Adam (diagnostic). Results are benchmarked against sparse exact diagonalization.
Decisions made in this release: HEA depth = 3 for N = 9; COBYLA over Adam (zero gradients from the |0⟩⊗N initial state); system sizes N = 9, 12, 18, and 24 with DMRG as the primary classical reference beyond sparse ED.
The spin Hall angle (θ_SH) is the figure of merit for spin-orbit torque efficiency. Selecting the top-k materials from N candidates is a combinatorial optimization problem, solved here with QAOA using a classical MLP surrogate as the oracle (Phase-A corpus: 32 materials in data/mp_theta_sh.csv; optional refresh via Materials Project API).
Materials Project supplies the structure metadata, while the θ_SH targets are a fixed illustrative oracle (illustrative_oracle in data/mp_theta_sh.csv). NB04 reports train / CV / hold-out surrogate metrics on the Phase-A corpus and runs QAOA / greedy / SA on a fixed historical N=12 pool with oracle=in_sample (pool LOOCV RMSE reported for honesty) — a reproducibility and method-demonstration workflow, not a materials-discovery claim.
Decisions made in this release: classical surrogate oracle (not raw DFT per evaluation); QAOA depths p = 1, 2, 3 compared against greedy and simulated-annealing baselines; k = 3 selected from N = 12 pool materials (full CSV is larger for diversity); p=1 (γ, β) landscape and depth-sensitivity plot (figures/qaoa_landscape.png) show that shallow QAOA can stall in a suboptimal basin while classical baselines reach the global optimum. A λ / budget / depth sweep on the same frozen oracle (figures/qaoa_sweep.png, data/qaoa_sweep.csv) checks that this is not a single COBYLA setting. A screening split (figures/qaoa_screening.png, data/qaoa_screening.csv) trains on the #20 25-row complement and deploys on the same N=12 pool with 4 unseen members. Surrogate honesty is reported as train / 5-fold CV / hold-out RMSE (figures/surrogate_predictions.png, data/surrogate_metrics.csv).
| N | Seeds | Mean E₀ | Std E₀ | Best E₀ | Error (best) | Notes |
|---|---|---|---|---|---|---|
| 9 | 5 | −1.23572 | 0.02853 | −1.28456 | 9.66% | HEA d=3, 27 params |
| 12 | 3 | −1.21520 | 0.02026 | −1.23859 | 16.33% | HEA d=2, 24 params |
| 9 | — | — | — | −1.42190399 | — | Exact diag., gap Δ ≈ 0 |
| 18 | — | — | — | −1.49962859 | — | Exact diag., gap Δ = 0.037 |
Adam / HEA d=3 at N=9 stalls at +0.141 (barren plateau). Seed-level statistics live in
data/vqe_results.csv and figures/vqe_seed_distribution.png.
Same normalized strip Hamiltonian as ED/DMRG/VQE. Complex RBM recovers ED; VQE remains ansatz-limited.
| N | NQS RBM E₀ | err vs ED | NQS ModPhase err | VQE best err |
|---|---|---|---|---|
| 9 | −1.42183 | 0.005% | 0.006% | 9.66% |
| 12 | −1.48015 | 0.018% | 1.99% | 16.33% |
Full comparison: data/method_comparison.csv, figures/nqs_*.png, notebook 07.
| Metric | Value | Interpretation |
|---|---|---|
| Mean single-site entropy | 0.9066 bits | Near-maximal → strong quantum fluctuations |
| Max single-site entropy | 1.000 bits | 7 of 9 sites maximally entangled |
| Sublattice I(A:B) | 3.689 bits | Strong inter-sublattice correlations |
| Mean pairwise MI | 0.227 bits | Non-local correlations (spin liquid signature) |
| Method | Total θ_SH | Selected | Notes |
|---|---|---|---|
| QAOA p=1 | 3.049 | W, Ta, Bi₂Se₃ | Best QAOA depth — still sub-optimal |
| QAOA p=2 | 3.049 | W, Ta, Bi₂Se₃ | Same selection as p=1 |
| QAOA p=3 | −0.451 | W, Ta, Pd | Deeper circuit — worse on this oracle |
| Greedy (classical) | 4.259 | Bi₂Se₃, CrTe₂, Mn₃Sn | Optimal on surrogate oracle |
| Sim. annealing | 4.259 | Mn₃Sn, CrTe₂, Bi₂Se₃ | Matches greedy |
Hold-out / CV on the 32-row corpus (data/surrogate_metrics.csv; regenerate with
python scripts/evaluate_surrogate.py):
| Split | n | RMSE | R² | Notes |
|---|---|---|---|---|
| Train (in-sample) | 25 | 0.116 | 0.93 | Fit set after 20% hold-out |
| 5-fold CV | 25 | 0.458 | −0.03 | Leakage-free Pipeline |
| Hold-out | 7 | 1.200 | 0.08 | W, Pd, MnPt, Bi₂Se₃, Ag, Sb₂Te₃, Mn₃Ga |
| QAOA pool in-sample | 12 | 0.006 | — | Oracle used for published totals |
| QAOA pool LOOCV | 12 | 1.71 | — | Honesty check; not the QAOA weights |
Best QAOA in the λ / budget / depth sweep is 3.570 (p=1, λ=5; W / CrTe₂ / Bi₂Se₃), still 0.69 below greedy 4.259. Extra COBYLA budget at λ=6 does not close the gap. The published table above is unchanged. The sweep figure reports θ_SH of the best-cost seed, matching the table.
A screening split trains on the #20 25-row complement and deploys on
the same N=12 pool (data/qaoa_screening.csv). Four pool members
(W, Pd, MnPt, Bi₂Se₃) were never in the fit. On the screening oracle,
greedy/SA still win (2.900 vs QAOA 2.505 at p=3). The published
in-sample table is unchanged.
This work sits at the intersection of frustrated magnetism, variational quantum algorithms, and spintronic materials design:
- Sachdev (1992) — Kagome Heisenberg antiferromagnet and spin-liquid phases
- Yan, Huse & White (2011) — spin liquid in the Kagome Heisenberg model
- Carleo & Troyer (2017) — Neural Quantum States / RBM wavefunctions
- Sinova et al. (2015) — spin Hall effects in materials
- Peruzzo et al. (2014) — original VQE
- Farhi et al. (2014) — original QAOA
- Cerezo et al. (2021) — barren plateaus in variational quantum algorithms
Full bibliography: REFERENCES.md (50+ entries).
| Component | Location | Description |
|---|---|---|
| Python package | src/spinq_vqe/ |
kagome, ansatz, vqe, entanglement, surrogate, qaoa, dmrg, nqs, utils |
| Notebooks | notebooks/01–07 |
Lattice/ED, VQE, entanglement, SOC QAOA, scaling, DMRG, NQS |
| Test suite | tests/ |
Eight modules, core suite < 90 s on CPU |
| Documentation | docs/ |
Physics, ansätze, API, notebooks, testing |
| Data & figures | data/, figures/ |
ED/VQE/QAOA/DMRG/NQS CSVs, surrogate_metrics.csv, publication plots |
- 2D periodic Kagome tiling — extend beyond the 1D strip geometry
- Sourced θ_SH audits — Phase A (32 illustrative rows) landed; later phases may add
sourced_primaryvalues or further diversity viascripts/fetch_mp_theta_sh.py - Paper draft — LaTeX manuscript targeting Physical Review B or npj Quantum Materials
DMRG (v0.1.5) and NQS (v0.1.6) comparisons are in the repository. Surrogate hold-out / CV (#20), the QAOA λ / budget / depth sweep (#21), and the train/pool screening split (#23) are in NB04.
spinq-vqe (this repo)
│
├──► spintronic-qrc [Kagome Hamiltonian as QRC reservoir]
├──► mtj-quantum-noise [AFM spin dynamics as decoherence source]
└──► quantum-hopfield-mram [Ising-limit Hamiltonian for memory landscape]
The Kagome AFM Hamiltonian computed here is intended as the physical foundation for related repos in the ARPA Spintronics QML program.
If you use this software, please cite using the metadata in CITATION.cff:
Peilivanidis, V., & ARPA Quantum Logical Systems (QONDRA). (2026). spinq-vqe: Variational Quantum Simulation of Antiferromagnetic Hamiltonians (v0.1.6). https://doi.org/10.5281/zenodo.21628505
Zenodo concept DOI: 10.5281/zenodo.21628505 (resolves to the latest archived version). Cite the software DOI for the code; cite any paper DOI separately when the manuscript is published.
Last updated: 2026-09-18 · Part of the ARPA Spintronics QML Research Program
The SOC notebook is a workflow benchmark on a fixed illustrative oracle: a Phase-A corpus of 32 committed targets supports surrogate train / CV / hold-out diagnostics (data/surrogate_metrics.csv); QAOA selects k=3 from a fixed N=12 pool. The totals in the tables and figures are objective values on that pool oracle, not verified material constants. data/theta_sh_sources.md defines the status contract and data/theta_sh_provenance.csv binds every oracle row to either primary evidence or an explicit illustrative status. Replacing any target requires regenerating the surrogate, classical baselines, QAOA runs, tables, and figures together.