Built something I’d love eyes on and ideally a hardware run.
**What it is:** A continuous-time quantum walk on the 7-cube (heptaract) that achieves perfect state transfer from |0000000⟩ to |1111111⟩ using a depth-2 circuit — 7 Rx(π) gates, zero entanglement, P=1.0.
The Hamiltonian factors cleanly because the 7 axis terms commute:
`H = Σᵢ Xᵢ → exp(-iHt) = ⊗ᵢ Rx(2t)`
At t = π/2 that’s just 7 X-rotations. Verified 1024/1024 shots on Qiskit Aer.
**The benchmark:** 7 gates vs ~448 classical expected steps. Scales O(n) quantum vs O(n·2^n) classical. 64x speedup at n=7, exponential beyond.
**The weird extra finding:** When you map the 7 axes to just-intonation pitch ratios (2/1, 3/2, 4/3, 5/4, 6/5, 7/4, 9/8), the 128 vertices collapse to 32 pitch classes, and the tonic (1/1) is invariant under all 5040 axis-label permutations. The spectral gap driving the transfer (Δ=2) maps to the perfect fifth. Not sure what to make of that yet but it feels like it’s pointing at something.
**What I’m asking:**
1. Anyone want to run this on real hardware? Depth 2 is well within NISQ fidelity range on any current 7-qubit device.
2. Is the pitch-invariance result known? Feels like it might connect to algebraic graph theory.
Hey Catalina — took your nudge and ran it myself on real hardware. Got onto IBM’s free Open plan and submitted to ibm_kingston (156-qubit Heron r2), my own account, job completed.
I actually tested something more fundamental than the walk first: measuring the relationships between qubits without collapsing the individual values — the non-demolition / stabilizer idea. Prepared a 3-qubit GHZ state and read the pairwise correlations. Real counts, 4096 shots:
Individual values: ⟨Z_a⟩=+0.02, ⟨Z_b⟩=+0.02, ⟨Z_c⟩=+0.00 — essentially zero. Each qubit alone is unpredictable.
Relationships: ⟨Z_aZ_b⟩=+0.94, ⟨Z_bZ_c⟩=+0.95, ⟨Z_aZ_c⟩=+0.93 — locked near +1.
Cross-check: (a·b)(b·c)=0.90 vs measured a·c=0.93 — the relationships close consistently.
So on real metal, the machine couldn’t tell you the values but knew the relationships cold. The ~5–6% shortfall from 1.0 is ordinary device noise, not a flaw. Nothing oversold here — this is textbook stabilizer measurement, just cleanly reproduced on hardware I ran myself.
One thing I’m chasing next: you can invert those pair-distortions to back out each qubit’s individual readout fidelity (got 96.0 / 98.3 / 96.8% from this run), and with more qubits the pairwise reads massively over-determine the noise — solve and cross-check it. Will report back.
Quick follow-up — the walk itself, plus an answer to the pitch-invariance question.
The heptaract walk on hardware. Since the thread’s actually named for the walk, I ran it directly (free Open plan, 156-qubit Heron, 8192 shots) — the t = π/2 transfer |0000000⟩ → |1111111⟩:
P(|1111111⟩) = 94.7%
misses were almost all single-bit: 5.1% one bit off, 0.2% two bits off
Honest read: it’s 7 independent Rx(π) rotations with no entanglement, so there’s nothing to decohere — this is essentially a readout-error benchmark, and 94.7% ≈ (per-qubit readout fidelity)⁷. Clean confirmation of the state transfer, not evidence of a speedup. And to be upfront: the “64×” I’d floated is against a classical random walk — a deterministic classical machine flips 7 bits in 7 steps too, so I’ve dropped the speedup framing.
The pitch-class / tonic invariance — turns out it is a known result. As you weren’t sure about it, I decided to chase it down. The invariance of the tonic under permuting the seven axes is exactly the vertex-stabilizer of the hypercube’s automorphism group. For the 7-cube:
Aut(Q₇) = the hyperoctahedral group C₂ ≀ S₇ (signed permutations), order 2⁷·7! = 645,120.
The subgroup fixing the tonic vertex is S₇ (order 5,040) — the axis-permutations. That’s the “pitch-class invariance.”
So it’s textbook algebraic graph theory rather than anything new — which I take as a feature: the feeling was pointing at something real and citable. (Sanity-checked the group order by brute-forcing Q₂–Q₅ against 2ⁿ·n!.)
Also followed through on the noise-fingerprint idea from my last post: scaled to 7 qubits, the pairwise reads over-determine the per-qubit noise cleanly, and separating readout from entangling-gate error puts most of the noise in the entangling chain (~2.6% on this device), not readout.
Thanks again for the nudge to run these — learned a ton.
Thanks for sharing your update! I’m excited to see that you ran your idea on hardware. And thanks even more so for sharing your findings on the Forum; I’m confident that the community can learn a lot by reading through your posts.
And, yes, please do share what happens when you invert those pair-distortions to back out each qubit’s individual readout fidelity.