Optimal ancilla-free Clifford+T approximation of z-rotations

TL;DR

This paper introduces a fast, near-optimal probabilistic algorithm for approximating single-qubit z-rotations with minimal T-count in Clifford+T circuits, requiring a factoring oracle or relying on a mild number-theoretic hypothesis.

Contribution

It presents a new algorithm that efficiently finds the shortest Clifford+T circuit approximations of z-rotations, improving over previous methods in both speed and optimality.

Findings

Algorithm achieves near-optimal T-count approximations.

Expected runtime is polynomial in the logarithm of the inverse error.

Works efficiently with or without a factoring oracle under certain hypotheses.

Abstract

We consider the problem of approximating arbitrary single-qubit z-rotations by ancilla-free Clifford+T circuits, up to given epsilon. We present a fast new probabilistic algorithm for solving this problem optimally, i.e., for finding the shortest possible circuit whatsoever for the given problem instance. The algorithm requires a factoring oracle (such as a quantum computer). Even in the absence of a factoring oracle, the algorithm is still near-optimal under a mild number-theoretic hypothesis. In this case, the algorithm finds a solution of T-count m + O(log(log(1/epsilon))), where m is the T-count of the second-to-optimal solution. In the typical case, this yields circuit approximations of T-count 3log_2(1/epsilon) + O(log(log(1/epsilon))). Our algorithm is efficient in practice, and provably efficient under the above-mentioned number-theoretic hypothesis, in the sense that its expected runtime is O(polylog(1/epsilon)).