Nadarasa · v0.4.1 · Selene experiment
G16 — Ethylene QPDE on Selene
Quantum Phase Difference Estimation on the two-qubit π/π* active space of ethylene, compiled with Guppy and executed on the Selene Quest emulator. The protocol follows the chemistry section of the SoftBank / Quantinuum “Quantum Computing Frontiers” white paper (July 2026): estimate the phase difference between two eigenstates so the large, uninteresting global phase cancels and fewer ancilla bits buy chemical accuracy.
What the evolution-time trick buys
Restricted to the single-excitation subspace {|01⟩, |10⟩}, the active-space Hamiltonian is diag_avg·I + off·X, so the exact propagator is an XX+YY rotation by φ = off·t. Choosing the evolution time so that φ lands on a short binary fraction of π — here φ = kπ/16 — makes the measured interference land on exactly representable angles instead of forcing extra ancilla resolution. The closed form is
p(q1 = 1) = (1 − sin(2φ) · sin(β)) / 2
Guppy angles are halfturns, so φ = kπ/16 is written angle(k/16) — never angle(math.pi/16), which would be the S gate.
Sweep · measured vs predicted p(q1 = 1)
| k | t (a.u.) | β = 0 | β = π/4 | β = π/2 | β = 3π/4 | β = π |
|---|---|---|---|---|---|---|
| 1 | 0.49 | 0.4788 / 0.5000✓ | 0.3564 / 0.3647✓ | 0.3079 / 0.3087✓ | 0.3604 / 0.3647✓ | 0.4927 / 0.5000✓ |
| 2 | 0.98 | 0.4976 / 0.5000✓ | 0.2495 / 0.2500✓ | 0.1375 / 0.1464✓ | 0.2383 / 0.2500✓ | 0.5078 / 0.5000✓ |
| 4 | 1.96 | 0.5061 / 0.5000✓ | 0.1509 / 0.1464✓ | 0.0000 / 0.0000✓ | 0.1423 / 0.1464✓ | 0.5066 / 0.5000✓ |
| 8 | 3.93 | 0.4783 / 0.5000✓ | 0.4946 / 0.5000✓ | 0.5190 / 0.5000✓ | 0.5110 / 0.5000✓ | 0.4995 / 0.5000✓ |
Each cell shows measured / predicted. A cell passes when |measured − predicted| ≤ 0.0442, the 4σ binomial threshold 4·√(0.5/shots) at 4096 shots.
Recovering the eigenvalue gap
| k | 2φ nominal | 2φ estimated | gap (Ha) | abs. error |
|---|---|---|---|---|
| 1 | 0.3927 | 0.3944 | 0.8035 | 0.0035 |
| 2 | 0.7854 | 0.8112 | 0.8263 | 0.0263 |
| 4 | 1.5708 | 1.5708 | 0.8000 | 0.0000 |
Mean estimate 0.8099 Ha against a reference gap of 0.8000 Ha — an error of 0.0099 Ha.
k = 8 is held out of the fit: at k=8 the phase 2*phi reaches pi, where the interference signal is stationary and the arcsine branch is degenerate — the QPDE mod-1 wrap. It is kept in the sweep as an aliasing control — every β column collapses to 0.5, which is exactly what the wrap predicts.
Is the residual shot noise or fit bias?
The headline fit above is ~4% off the reference gap. That number has two very different possible causes, and only one of them is fixable by buying shots. Re-running the β = π/2 column across four shot budgets and a wider k set separates them.
| Shots | Gap (Ha) | Rel. error | Spread over k |
|---|---|---|---|
| 256 | 0.8499 | 6.24% | 0.1225 |
| 1024 | 0.7844 | 1.95% | 0.0298 |
| 4096 | 0.8021 | 0.27% | 0.0149 |
| 16384 | 0.8039 | 0.49% | 0.0266 |
- Feeding the inverter exact probabilities returns 0.8000 Ha — a bias of exactly 0.0000 Ha. The principal-branch inversion is unbiased over k ∈ 1, 2, 3, 4, so the entire residual is statistical.
- Accordingly the error falls to 0.27% at 4096 shots once k = 3 is added to the fit — the original 4% came as much from averaging only three k-points as from shot count.
- k = 6 is a second aliasing control, distinct from k = 8: the branch condition is 2φ ≤ π/2, so any k > 4 inverts onto the wrong arm of the arcsine and returns ≈0.27 Ha. Measured, reported, never fitted.
Driver quantum/qpde/fitquality.py, noiseless Quest, 4 budgets × 5 k-cells, all cells PASS the 4σ threshold.
Caveats
- The active-space constants are order-of-magnitude values quoted from the SoftBank/Quantinuum white paper, not a converged electronic-structure calculation.
- Quest is a noiseless state-vector emulator; deltas here are pure shot noise, not hardware error.
- The subspace propagator is compiled exactly as an XX+YY rotation rather than Trotterised.
Reproduce
pip install --target .pydeps -r quantum/requirements.txt PYTHONPATH=.pydeps python -m quantum.qpde.sweep PYTHONPATH=.pydeps python -m quantum.qpde.validate PYTHONPATH=.pydeps python -m quantum.qpde.fitquality
Pins: guppylang 0.21.16 · selene-sim 0.2.18 · Quest (ideal). The sweep caches each cell under quantum/qpde/_cache_qpde/ and is resumable; results ship as static JSON, nothing quantum runs at request time.