Hamiltonian simulation error and resources

Let $V$ be a finite-dimensional complex inner-product space and let $H=H^{\ast }$ be a specified endomorphism. Its exact time-$t$ evolution is $U_H(t)=e^{-itH}$ as defined by Finite Hermitian matrix dynamics. The notation $\mathrm{\parallel }T\mathrm{\parallel }$ denotes the operator norm of The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||. The target register's Hilbert space is identified with $V$ as part of the input. A Hamiltonian simulator at encoded time $t$ and tolerance $0<ϵ<1$ is a declared procedure whose output is a unitary $W$ on the stated target register satisfying $\mathrm{\parallel }W-U_H(t)\mathrm{\parallel }\le ϵ$. If its circuit uses auxiliary registers, the exact clean or dirty contract of Clean and dirty ancillas must make the induced target unitary $W$ well-defined; the error is measured on that target unitary. The scalar time and tolerance have exact finite descriptions under the declared input convention; their description lengths are recorded.

The resource record lists, separately, the number of calls to each named oracle, controlled calls, inverse calls, elementary gate occurrences, clean and dirty ancillary wires, coefficient bits, full circuit-description bits, and classical generation time. A coordinate not supplied by the access contract is unavailable, not zero. No matrix description supplies a circuit, oracle, controlled call, inverse call, or term exponential for free; uniform-family and cost conventions are those of Full-list uniform quantum circuit family, Circuit-family and computation cost model, Controlled Unitary Access, and Inverse Unitary Access. No polynomial resource bound is asserted by this definition.