Effective Unitary Matrix Input

A finite gate alphabet is a finite list of distinct labels, each assigned one exact matrix in $SU(2):=\{M\in M_2(\mathbb{C}):M^{\ast }M=I_2,\det M=1\}$. A Cauchy name for a target $U\in SU(2)$ is a deterministic procedure which, on unary input $1^b$, returns a rational $2\times 2$ complex matrix $Q_b$ whose real and imaginary entries differ from the corresponding entries of $U$ by at most $2^{-b}$. The name has a stated time bound $T(b)$. A precision request is a positive rational $\epsilon \in (0,1\mathrm{/}2)$ in binary, using the effective encoding convention of Effective binary encodings and total decoders.

The Cauchy name specifies the exact target through all its approximants; an individual $Q_b$ need not be unitary. An exact-real target handle and a Cauchy-name input are distinct interfaces: the first permits ideal exact comparisons, while a finite Turing implementation must use the second.