Nadarasa · v0.4.2 · Selene experiment
G18 — Laplacian moments on a WL-blind graph pair
The quantum-TDA half of the SoftBank / Quantinuum white paper estimates spectral moments T_k = tr(Δ^k) of a graph Laplacian from a propagator trace rather than from a classical eigendecomposition. The sharpest small test case is a pair of graphs that 1-dimensional Weisfeiler–Leman colour refinement cannot distinguish: every vertex in both graphs has degree 2, so the refinement stabilises immediately and returns identical colourings. Their Laplacian spectra are different, and so are their moments from k = 3 upward.
The pair
| Graph | V / E | Degrees | Betti (b₀, b₁) | Laplacian spectrum |
|---|---|---|---|---|
| C6 | 6 / 6 | 2, 2, 2, 2, 2, 2 | 1, 1 | 0.0, 1.0, 1.0, 3.0, 3.0, 4.0 |
| 2C3 | 6 / 6 | 2, 2, 2, 2, 2, 2 | 2, 2 | 0.0, 0.0, 3.0, 3.0, 3.0, 3.0 |
Same vertex count, same edge count, same degree sequence — but C6 is connected with one independent cycle and 2C3 is two disjoint triangles. Topology sees it (b₀ = 1 vs 2); colour refinement does not.
What the circuit measures
The Laplacian is padded to 2³ = 8 dimensions, rescaled by 4 so the spectrum sits in [0, 1], and decomposed into 20–22 Pauli strings. A Hadamard test on one ancilla plus three system qubits reads off ⟨x|e^(−iHτ)|x⟩ for each computational basis state x; averaging the eight of them gives the normalised trace f(τ) = tr(e^(−iHτ)) / 8. Real and imaginary parts come from the same circuit with and without the ancilla S† rotation. The propagator is Trotterised at 3 steps, which costs up to 1.6e-2 in trace error at the longest τ — about two point-σ, so the measured curves track the exact ones to a couple of percent rather than to shot noise. That systematic is per-graph and largely common-mode: the measured differences below still track the exact differences closely.
Deterministically sweeping all eight basis states beats a random purification here: at three qubits it is only 8 circuits per (τ, part), and it removes the state-preparation variance entirely.
Model-free separation: the trace curves themselves
Before fitting anything, the two measured curves can simply be subtracted. Each averaged point carries binomial error 0.0078, so the difference is a clean χ² test with 24 degrees of freedom.
| τ | Re C6 | Re 2C3 | Δ measured | Δ exact | σ |
|---|---|---|---|---|---|
| 0.50 | 0.9666 | 0.9647 | 0.0019 | 0.0001 | 0.2 |
| 0.86 | 0.8995 | 0.9008 | -0.0013 | 0.0010 | 0.2 |
| 1.23 | 0.8127 | 0.7996 | 0.0131 | 0.0041 | 1.7 |
| 1.59 | 0.6962 | 0.6851 | 0.0111 | 0.0108 | 1.4 |
| 1.95 | 0.5836 | 0.5480 | 0.0356 | 0.0228 | 4.6 |
| 2.32 | 0.4625 | 0.4166 | 0.0459 | 0.0410 | 5.9 |
| 2.68 | 0.3464 | 0.2949 | 0.0515 | 0.0655 | 6.6 |
| 3.05 | 0.2651 | 0.1841 | 0.0810 | 0.0951 | 10.4 |
| 3.41 | 0.2123 | 0.0708 | 0.1415 | 0.1275 | 18.1 |
| 3.77 | 0.1783 | 0.0207 | 0.1576 | 0.1588 | 20.2 |
| 4.14 | 0.1768 | -0.0096 | 0.1864 | 0.1845 | 23.9 |
| 4.50 | 0.1917 | 0.0113 | 0.1804 | 0.1997 | 23.1 |
χ² = 2643 over 24 points = 110.1 per degree of freedom, with a single point reaching 23.9σ. The emulator distinguishes the pair decisively, using a quantity 1-WL cannot see.
Recovered moments
Fitting the truncated Taylor series f(τ) = Σ (−iτ)^k m_k / k! converts the curves into moments. Two truncations are shown: order 8 (near-unbiased, high variance) and order 6 (low variance, but truncation pushes weight from T₃ into T₄).
| Graph | k | Exact T_k | Fit (order 8) | Fit (order 6) |
|---|---|---|---|---|
| C6 | 1 | 12 | 12.0 ± 0.1 | 11.5 ± 0.1 |
| 2 | 36 | 35.0 ± 0.6 | 34.9 ± 0.3 | |
| 3 | 120 | 118.0 ± 4.0 | 98.3 ± 1.4 | |
| 4 | 420 | 366.9 ± 32.4 | 360.8 ± 9.6 | |
| 2C3 | 1 | 12 | 12.2 ± 0.1 | 11.8 ± 0.1 |
| 2 | 36 | 36.6 ± 0.6 | 35.3 ± 0.3 | |
| 3 | 108 | 112.7 ± 4.0 | 99.0 ± 1.4 | |
| 4 | 324 | 367.4 ± 32.4 | 291.2 ± 9.6 |
Moment-by-moment separation
| k | Exact Δ | Measured Δ (order 8) | σ | Measured Δ (order 6) | σ |
|---|---|---|---|---|---|
| 1 | 0 | 0.1 | 0.7σ | 0.3 | 2.4σ |
| 2 | 0 | 1.6 | 1.8σ | 0.4 | 0.8σ |
| 3 | 12 | 5.3 | 0.9σ | 0.8 | 0.4σ |
| 4 | 96 | 0.5 | 0.0σ | 69.6 | 5.1σ |
T₁ and T₂ agree between the two graphs, as they must — those moments are fixed by the vertex and edge counts, which the pair shares. The separation appears only at higher order. At order 8 the T₃ gap (12) sits inside its own error bar, so on this shot budget the per-moment test is not conclusive; at order 6 the truncation reassigns the discrepancy to T₄, where it shows up at 5.1σ. The honest summary: the moment vector separates the pair, but attributing the separation to one specific k needs more shots.
Caveats
- Noiseless Quest. G17 is the template for what an H2-class error ladder would do to these curves; that ladder has not been run here.
- Trotter error is not negligible here. At 3 steps it reaches 1.6e-2 in the trace, comparable to two point-σ, and it accounts for part of the gap between the order-8 moment fits and the exact values. More steps would fix it at proportionally more circuit depth.
- Three system qubits, six-vertex graphs. This is a mechanism check, not a scaling claim — the interesting regime is where the Laplacian no longer fits in a classical eigensolver.
- High moments are dominated by fit variance, not by the emulator. The Taylor-fit conditioning (fit order and τ grid) dominates the error far more than the shot count does; doubling the order roughly quadruples σ(T₃).
- Results are committed static JSON. Nothing quantum runs at request time.
Reproduce
pip install --target .pydeps -r quantum/requirements.txt PYTHONPATH=.pydeps python -m quantum.tda.sweep
Backend: selene-sim Quest (noiseless). Each (graph, τ, basis state, part) circuit caches under quantum/tda/_cache_tda/, so the 384-circuit sweep is resumable across sandbox sessions.