Eigenvalue-estimation access and aliasing convention

An eigenvalue-estimation access instance consists of a finite-dimensional system space; a costed state-preparation circuit $A_{\psi }$ with $A_{\psi }\mathrm{\mid }0⟩=\mathrm{\mid }\psi ⟩$; a costed unitary $U$ and each controlled, inverse, or controlled-inverse version actually used; an injective known phase encoding $\lambda \mapsto \theta (\lambda )\in \mathbb{R}\mathrm{/}\mathbb{Z}$ on the promised spectrum; and a specified decoder with its branch cut and output units. Exact controlled powers $U^{2^j}$ may instead be supplied individually, with their costs. If powers are synthesized from controlled-$U$, charge the exact number of calls and all control/wiring costs; no power or control is free.

For Hamiltonian access, the instance additionally specifies a Hermitian $H$, an interval containing $\sigma (H)$, a simulation rule for each requested signed time and its operator-error/cost bound, and how errors are allocated across the circuit. In the convention of Hamiltonian simulation error and resources, a simulator for $e^{-itH}$ implements the positive-sign phase unitary $e^{iH}$ at $t=-1$, using inverse access when that is what the supplied interface permits. Every preparation, inverse, controlled call, simulation, phase-register gate, decoder arithmetic operation, measurement, and written output is included in its declared cost coordinates. Aliasing is excluded by the injectivity promise and the chosen decoder; absent that promise, the output is only an estimate of the encoded phase modulo one, not of a uniquely identified eigenvalue.