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

TL;DR

This paper introduces quantum circuits for GCD computation that achieve significantly reduced depth of O(n log n) while maintaining linear ancillae, improving over previous quadratic-depth circuits.

Contribution

The paper presents a novel quantum circuit design for GCD calculation with optimized depth using the binary GCD algorithm, reducing depth complexity from O(n^2) to O(n log n).

Findings

Achieves O(n log n) circuit depth for GCD computation.

Uses O(n) ancillae, matching prior resource bounds.

Maintains O(n^2) gate count, consistent with traditional circuits.

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.