Correctness of the small-time-aware simulator

If $\alpha \mathrm{\mid }t\mathrm{\mid }>ϵ$, the algorithm invokes OP4, and [F2] gives the clean-input isometry bound.

Suppose $\alpha \mathrm{\mid }t\mathrm{\mid }\le ϵ$. For a unit vector $\psi$, the differential identity [F4] and the fundamental theorem of calculus give $U_H(t)\psi -\psi ={\int }_0^t-iH{U}_H(s)\psi ds$. Each integrand has norm at most $\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$, because $U_H(s)$ preserves norm. The triangle inequality for Riemann sums and continuity yield $\mathrm{\parallel }(U_H(t)-I)\psi \mathrm{\parallel }\le \mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \alpha \mathrm{\mid }t\mathrm{\mid }\le ϵ$, using [F7]. Taking the maximum over unit $\psi$ proves the clean-input operator bound for the identity circuit. The work registers are exactly zero, so the same isometry bound holds.

In either branch, the implementation on clean inputs and the target $J_{\mathrm{o}\mathrm{u}\mathrm{t}}{e}^{-itH}$ are isometries whose operator-norm difference is at most $ϵ$, by steps 1.1 and 1.2. Apply [F6] with $\delta =ϵ$ to obtain the reference-system trace-distance conclusion after tracing work.