Nine qubits are not the problem by themselves. The obstacle is the number and structure of the gates produced when a hardware compiler expands each multi-controlled F-move and routes the result over a sparse superconducting-qubit layout.
A statistically significant contrast is not enough. If most probability has leaked out of the encoded fusion subspace, the surviving signal is no longer an interpretable implementation of the intended experiment, and it is not evidence of quantum advantage.
From a compact diagram to thousands of entangling gates
The generic all-to-all decomposition used about 1,476 CX gates for each braid-order arm. On a heavy-hex proxy, the best order arm reached 2,926 CX gates at depth 6,854. The primary phase-sensitive filter was worse: about 4,300 CX gates at depth 10,009.
This is the classic difference between logical circuit size and compiled hardware cost. A four-controlled arbitrary single-qubit unitary is a compact box on paper, but a generic synthesis knows nothing about the Fibonacci matrix, the reachable control patterns, or clean work qubits elsewhere in the register.
The preregistered readiness gate
The noise study froze three simultaneous conditions:
- at least 80% probability of detecting the order contrast;
- at most 10% leakage from the encoded fusion subspace;
- at least 75% pass probability for the source-state filter.
Under mild one-qubit depolarizing noise, the contrast remained detectable with 128 shots, yet leakage was about 46%. Under mild two-qubit depolarizing noise, leakage rose to 61%. Mild thermal and combined proxies drove leakage to roughly 99%. More shots could estimate the damaged distribution more precisely, but could not restore the intended encoded state.
The verdict was therefore REVISE CIRCUIT/OBSERVABLE, not “run more shots.”
What mitigation did and did not do
Mitigation was audited rather than treated as a magic repair step.
| Method | Best observed effect | Why it did not change readiness |
|---|---|---|
| M3 readout mitigation | Restored readout-medium contrast from \(0.5562\) to \(0.6180\) | The raw readout-only case already passed |
| PEA/ZNE | Improved mild 1q and 2q contrast to \(0.5386\) and \(0.4715\) | Variance grew by \(5.86\times\) and \(4.44\times\); raw leakage still failed |
| Bounded PEC estimate | Conservative overhead \(11.77\times\) to \(19.14\times\) | Exceeded the frozen \(5\times\) cap |
| Additional shots | Increased detection power for attenuated contrasts | Did not repair state leakage or filter failure |
This does not mean the methods are useless. M3 correctly addresses measurement assignment error, and extrapolation can reduce bias in suitable observables. It means they cannot turn an excessively deep state-preparation circuit into faithful encoded dynamics merely by post-processing its output.
Fire Opal compatibility was a separate question
A later validate-only check submitted no shots and consumed no quantum time. Fire Opal accepted three ordered QASM payloads for ibm_fez, but the returned physical circuits were still deep: depth 1,665 with 598 two-qubit gates for the primary filter, and depth 1,219 with 441 two-qubit gates for each order arm.
That result established input and workflow compatibility, not hardware accuracy or mitigation benefit. No Fire Opal execution is part of the hardware result reported in this series.
The useful failure
The readiness failure localized the bottleneck. The Fibonacci algebra was not wrong; the generic synthesis was wasteful. The next design loop therefore targeted the internal identity of F, reachable control subspaces, adjacent cancellations, and finally the measured observable itself.
This is also why a qubit count cannot be used as a shortcut for complexity. A nine-qubit circuit may be too deep for current hardware, while a much larger but highly structured circuit could in principle be easier. Gate topology, entangling depth, idle time, error model, and the observable all matter.