Exact (zero-error) polynomial-time quantum factoring over a fixed finite gate set

Theorem 1.1: a polynomial-time uniform family of polynomial-size quantum circuits over NOT, CNOT, Toffoli, Hadamard, $\mathrm{diag}(1,i)$, their inverses and singly controlled versions outputs the complete prime factorization of every $N\ge2$ with probability one, with gate and qubit counts polynomial in the bit length. The construction makes the true-order mass of an order trial an exact dyadic rational, then uses one amplitude-amplification step, and handles auxiliary factorizations of $p-1$ recursively with a deterministic primality test. Not shown: practical resource bounds (the polynomials are large) or any classical algorithm.

Theorem 1.1: a polynomial-time uniform family of polynomial-size quantum circuits over NOT, CNOT, Toffoli, Hadamard, $\mathrm{diag}(1,i)$, their inverses and singly controlled versions outputs the complete prime factorization of every $N\ge2$ with probability one, with gate and qubit counts polynomial in the bit length. The construction makes the true-order mass of an order trial an exact dyadic rational, then uses one amplitude-amplification step, and handles auxiliary factorizations of $p-1$ recursively with a deterministic primality test. Not shown: practical resource bounds (the polynomials are large) or any classical algorithm.