Term-list Hamiltonian access

A term-list access input consists of an integer $m\ge 1$, finite-dimensional self-adjoint endomorphisms $H_1,\dots ,H_m$ on one stated register, and the exact target $H={\sum }_{j=1}^m{H}_j$. A supported time code is a finite expression whose leaves are rational numbers or symbols ${\rho }_q$ denoting the unique positive real root of $x^q=4$ for an odd integer $q\ge 1$; Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$ gives a unique nonnegative root; it is positive because $0^q=0\mathrm{\ne }4$. Internal operations are addition, subtraction, multiplication, or division by a nonzero expression. Its exact real value and bit length are recorded. These codes include rational times and every coefficient-scaled rational time in the finite Suzuki recursion.

For each $j$, the input declares which time codes are supported and supplies either the exact operation $e^{-is{H}_j}$ or a full-list uniform circuit generator, in the sense of Full-list uniform quantum circuit family, that returns a unitary $V_j(s)$ together with a certified error ${\eta }_j(s)\ge 0$ satisfying $\mathrm{\parallel }{V}_j(s)-e^{-is{H}_j}\mathrm{\parallel }\le {\eta }_j(s)$. If a circuit uses auxiliary registers, it supplies the exact clean or dirty contract of Clean and dirty ancillas so that its induced operation $V_j(s)$ on the target is well-defined: clean registers start and return to zero for every target input, and dirty registers are acted on by the identity. The operator-norm error is measured on the induced target operation. The provider specifies how each supported code is passed to the generator and records its actual description length and generation time; no polynomial bound is inferred. This is an explicit access promise, not a claim that every matrix has such a family.

For each supported call the input records its named-oracle query counts, gate count, ancillary wires, description bits, coefficient bits, and classical generation cost. Costs of composing the calls into a full product and generating that product are separately recorded as $Q_{\mathrm{w}\mathrm{r}\mathrm{a}\mathrm{p}}$ and $G_{\mathrm{w}\mathrm{r}\mathrm{a}\mathrm{p}}$. A controlled form is available only when separately declared as the block operation of Controlled Unitary Access. Inverse access to a call is available only when separately declared under Inverse Unitary Access, with its own cost and error record. Forward access implies neither capability. The term list supplies no promise that any operation or coefficient computation is free.