Quantum Circuits for GCD Computation with $O(n \log n)$ Depth and O(n) Ancillae

Abstract: GCD computations and variants of the Euclidean algorithm enjoy broad uses in both classical and quantum algorithms. In this paper, we propose quantum circuits for GCD computation with $O(n \log n)$ depth with O(n) ancillae. Prior circuit construction needs $O(n^2)$ running time with O(n) ancillae. The proposed construction is based on the binary GCD algorithm and it benefits from log-depth circuits for 1-bit shift, comparison/subtraction, and managing ancillae. The worst-case gate count remains $O(n^2)$, as in traditional circuits.