A nonempty positive-budget regime

Nonzero mathematical preparation and decoder errors can also meet the budget. A known source-code rotation $e^{-\mathrm{i}\alpha D}$, $\Vert D\Vert \le 1$, changes preparation by at most $|\alpha |$. A decoder mixture

differs by at most $\beta$. Taking $|\alpha |,\beta \le \epsilon /8$ suffices. Complete dilations remain necessary for postoperation residual claims.

example, not the reported run. For $d=2$, $c=1$, ${\omega }_0=3$, and $\epsilon =1/100$, take $T=40$, $M=440$, and $N=439$. These choices satisfy the sufficient inequalities, using $e<11/4$; they are deliberately conservative. Then

The nonzero detuning $\delta =1/16000$ gives ${\Delta }_T=1/400$. With fixed allowances summing to at most $1/400$, the recovery bound is below $1/100$.

This example was not executed in the supplied record. The reported run uses $T=1$, $N=8$, and no Hamiltonian perturbation test.