Effective finite matrix encodings for local Hamiltonians

A finite-dimensional complex matrix input of dimension $d$ is effectively encoded if its complete binary description specifies $d$ and conforms to one fixed deterministic entry-approximation machine. Given the description, an entry index, and unary precision request $1^p$, that machine returns rational real and imaginary parts within $2^{-p}$ of the exact entry in time bounded by one fixed polynomial in the total description length plus $p$, uniformly over all valid encodings. Rational entries use their exact binary numerator and denominator; algebraic entries use a finite encoding compatible with this same evaluator and bound. A uniform matrix family has one fixed deterministic algorithm which, given the unary family index $1^n$ and any explicitly encoded family parameters, outputs the complete description of that member in time bounded by one fixed polynomial in $n$ plus the parameter length, uniformly over all valid indices and parameters. Hermiticity, dimensions, and norm bounds are promises on the exact matrices, not conclusions inferred from a finite approximation.

This interface includes finite output size and all approximation bit costs. It does not supply a matrix oracle, exact comparisons on arbitrary computable reals, or a free unitary implementation.