Remarks on Matsumoto and Amano's normal form for single-qubit Clifford+T operators

TL;DR

This paper provides a clear, simplified presentation of Matsumoto and Amano's unique normal form for single-qubit Clifford+T operators, including corollaries and an efficient synthesis algorithm, with implications for quantum circuit synthesis.

Contribution

It offers a streamlined exposition of the Matsumoto-Amano normal form and derives new corollaries, including an intrinsic characterization and an efficient synthesis algorithm.

Findings

Simplified proofs of Matsumoto and Amano's results

Intrinsic characterization of Clifford+T subgroup of SO(3)

Efficient T-optimal exact single-qubit synthesis algorithm

Abstract

Matsumoto and Amano (2008) showed that every single-qubit Clifford+T operator can be uniquely written of a particular form, which we call the Matsumoto-Amano normal form. In this mostly expository paper, we give a detailed and streamlined presentation of Matsumoto and Amano's results, simplifying some proofs along the way. We also point out some corollaries to Matsumoto and Amano's work, including an intrinsic characterization of the Clifford+T subgroup of SO(3), which also yields an efficient T-optimal exact single-qubit synthesis algorithm. Interestingly, this also gives an alternative proof of Kliuchnikov, Maslov, and Mosca's exact synthesis result for the Clifford+T subgroup of U(2).