FHB-n is a constant-depth, connectivity-free, analytically exact benchmark for single-qubit fidelity on NISQ quantum processors, derived from the structural isomorphism between the Fuxi "Earlier Heaven" hexagram system (伏羲先天六十四卦) and the n-dimensional hypercube graph Q_n.
This repository contains the English manuscript, the complete mathematical model, and independent verification code for every quantitative claim in the paper.
The 64 hexagrams under single-line changes form exactly the hypercube graph Q₆. The continuous-time quantum walk on Q_n has Hamiltonian H = Σₖ Xₖ, whose commuting terms factorize the evolution into n parallel single-qubit R_X rotations. Consequences: the walk hits the exact uniform distribution at t = π/4, returns exactly to its initial state at t = π, and the whole benchmark circuit has depth 3 regardless of qubit count with zero two-qubit gates. Under standard noise channels the return fidelity has closed forms — F_dep = (1−p/2)^2n, F_AD = (1−γ)^2n, F_RO = (1−ε)^n — whose small-error expansion F ≈ 1 − n(p + 2γ + ε) turns the benchmark into a calibrated linear probe of per-qubit error rates.
paper/
fhb.pdf — the paper, typeset (A4, 8 pages)
fhb.md — the paper (readable on GitHub)
fhb.tex — LaTeX source of the same manuscript
code/
fhb/ — pure-stdlib Python package: analytic model (walk, noise, Grover)
core.py — all closed forms with proofs referenced to the paper
fhb2.py — FHB-2 entangling extension: exact free-fermion reference (see docs/FHB2-NOTES.md)
qiskit_impl.py— optional Qiskit circuits (only file needing dependencies)
verify_claims.py— reproduces every number in the paper's tables; exits non-zero on mismatch
tests/ — unit tests (stdlib unittest)
verify/
verify.mjs — independent Node.js verification (zero dependencies, second implementation)
docs/
ROADMAP.md — research-direction assessment: what to build next, and rejected directions with reasons
FHB2-NOTES.md — FHB-2 derivation: Jordan-Wigner mapping, covariance evolution, validation protocol
FHB-1 is blind to two-qubit-gate quality. FHB-2 adds a nearest-neighbor RZZ/RX brickwork on a line — a matchgate (free-fermion) circuit whose ⟨Xⱼ⟩ / ⟨XⱼXₖ⟩ references are exactly computable in O(n²·d) via Majorana covariance evolution, at any depth and any n (n = 40 evaluates instantly). The (FHB-1, FHB-2) pair separates the single-qubit + readout error budget from the entangling-gate budget, both against exact references. Derivation and design constraints: docs/FHB2-NOTES.md; validation: 9 cross-checks against dense statevector simulation in code/tests/test_fhb2.py.
Two independent implementations (Python and JavaScript) check the same claims:
# Python (no dependencies)
cd code
python verify_claims.py
python -m unittest discover tests
# Node.js (no dependencies)
node verify/verify.mjsBoth scripts verify:
| Claim | Result |
|---|---|
| Exact uniform mixing at t = π/4 (Theorem 3) | max deviation 1.4 × 10⁻¹⁷ |
| Exact return at t = π (Theorem 4) | P = 1.000000000 |
| TVD to uniform = 63/64 at t ∈ {0, π/2, π} | exact |
| Closed-form noise fidelities vs. paper's Qiskit tables | agree within sampling error (13/15 settings; 2 documented outliers at extreme depolarizing rates) |
| Grover n=6 ideal success = 99.66% | exact |
| Q₆ graph facts (64 vertices, 192 edges, 6-regular, diameter 6) | exact |
The analytic model needs no simulator. To actually run the benchmark:
# pip install qiskit qiskit-aer
import math, sys
sys.path.insert(0, "code")
from fhb.qiskit_impl import fhb_core_circuit
qc = fhb_core_circuit(n=6, t=math.pi) # return-fidelity experiment, depth 3Score hardware output directly against the closed forms in code/fhb/core.py — no classical simulation required at any n.
FHB ran on three IBM Heron R2 processors (156 qubits each), 8192 shots per circuit, one batched job per backend; raw counts in code/results/, table regenerated by code/compare_results.py:
| Backend | F_ret (t=π) | (1−F)/n | Mixing TVD | FHB-2 mean dev | FHB-2 p fit |
|---|---|---|---|---|---|
| ibm_marrakesh | 0.9495 | 0.84% | 0.0392 | 0.0462 | 1.05% |
| ibm_fez | 0.9177 | 1.37% | 0.0337 | 0.0766 | 1.30% |
| ibm_kingston | 0.8888 | 1.85% | 0.0547 | 0.0229 | 0.47% |
What the data shows:
- The two error budgets decouple across devices. ibm_kingston has the worst single-qubit/readout budget (1.85%) but the best entangling-gate fit (0.47%); ibm_marrakesh is the reverse. A single holistic score would hide this — the (FHB-1, FHB-2) pair is designed precisely to separate it.
- The mixing point is noise-robust on all three devices (TVD 0.034–0.055, at or near the 8192-shot sampling floor ≈ 0.035), confirming its role as an execution sanity check rather than an error probe.
- Full π-periodicity is visible on hardware at all five time points on every backend.
- FHB-2 localizes faults: on ibm_fez, qubit 0 shows a −0.32 deviation in ⟨X₀⟩ while its other five qubits sit within ±0.09 — a single bad qubit/bond immediately visible in the per-qubit profile, invisible in any aggregate metric.
- Edge qubits (1 bond) deviate less than bulk qubits (2 bonds) on marrakesh and kingston, matching the bond-count prediction of the string-damping theorem.
- The hexagram–hypercube isomorphism is a formal mathematical fact. No claim is made that the historical originators of the hexagram system anticipated quantum mechanics (paper §10.1).
- FHB-n is a diagnostic, not an application benchmark: it certifies single-qubit health and readout, and is blind to two-qubit-gate quality (the Grover variant partially compensates).
- The Qiskit-simulated noise tables come from the v6 study; the analytic closed forms here reproduce them within 10⁵-shot sampling error, except depolarizing at 5%/10% (deviations +0.0026 / +0.0085, discussed in paper §6.3).
- Real-hardware validation is future work; the protocol is specified in paper §9.
MIT — see LICENSE.