Continuous-time Hamiltonian access cost model

Let $V$ be a finite-dimensional complex register and let $H=H^{\ast }$ be its declared generator. The mathematical target at time $t\in \mathbb{R}$ is $U_H(t)$ from Finite Hermitian matrix dynamics. A Hamiltonian-access contract records how this target and its input data are supplied; it does not equate a matrix description with an implementation.

The contract declares one input representation:

1. Explicit terms: a finite ordered list of Hermitian matrices $H_1,\dots ,H_m$, real coefficients $c_1,\dots ,c_m$ with their exact or finite-precision binary encodings, and the identity $H={\sum }_{j=1}^m{c}_j{H}_j$. If the implementation calls the individual operations $U_{c_j{H}_j}(t_j)$, the contract lists every operation, its order, and its charge. The equality of their composition with $U_H(t)$ must be proved or supplied as a separate simulation certificate; it does not follow from the term list alone.
2. Row-sparse input: an integer sparsity bound $d$, a row-location oracle, and an entry-value oracle returning each selected matrix entry to a declared coefficient precision. Each oracle's full unitary action on invalid encodings, and every forward, inverse, or controlled call actually used, are explicit parts of the contract.

The contract separately names an implementation of the requested evolution:

• an ideal continuous-time operation $U_H(t)$ with its declared evolution time and/or evolution-query charge; or
• a finite unitary circuit/simulator $V_{H,t,ϵ}$ with a supplied certificate $\mathrm{\parallel }{V}_{H,t,ϵ}-U_H(t){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le ϵ$ and its gate, query, description-bit, coefficient-precision, and classical-construction costs.

Initial-state preparation and final measurement are separate supplied stages. For each stage the contract records a triple $(q,g,b)$: $q$ is the number of declared oracle or evolution calls, $g$ is the finite gate count in the chosen gate model, and $b$ is the declared input, operation-description, coefficient, and precision-bit charge. Circuit description length and gate count follow Circuit-family and computation cost model; they are distinct coordinates. Evolution time is reported separately unless the chosen model explicitly includes it in a query coordinate. Serial composition adds the declared stage coordinates. A controlled or inverse operation is charged only if the model explicitly grants that access or supplies a construction, as in Controlled Unitary Access and Inverse Unitary Access.

A complexity claim for a family of such inputs must also specify a uniform generator for the full circuit and access descriptions, as in Full-list uniform quantum circuit family, and state the costs as functions of the input size. The matrix alone supplies no efficient $U_H(t)$, controlled call, inverse call, or zero-cost operation. In particular, a Boolean marked-item oracle does not by itself grant the continuous-time operation $e^{-it\mathrm{\mid }{e}_w⟩⟨e_w\mathrm{\mid }}$; any conversion or simulation and its precision are separately specified and charged.