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GCD computations and variants of the Euclidean algorithm enjoy broad uses inboth classical and quantum algorithms. In this paper, we propose quantumcircuits for GCD computation with $O(n \log n)$ depth with O(n) ancillae. Priorcircuit construction needs $O(n^2)$ running time with O(n) ancillae. Theproposed construction is based on the binary GCD algorithm and it benefits fromlog-depth circuits for 1-bit shift, comparison/subtraction, and managingancillae. The worst-case gate count remains $O(n^2)$, as in traditionalcircuits.

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