Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates

TL;DR

This paper presents an efficient algorithm for the exact synthesis of single-qubit unitaries using Clifford and T gates, proving optimality and establishing a mathematical equivalence with a specific ring of complex numbers.

Contribution

It introduces a new synthesis algorithm with optimal gate count guarantees and proves an equivalence between circuit implementable unitaries and a mathematical ring in the single-qubit case.

Findings

Efficient synthesis algorithm with optimal T and Hadamard gate count.

Proved equivalence between Clifford+T unitaries and a specific algebraic ring for single qubits.

Conjecture on extending the equivalence to multi-qubit systems.

Abstract

In this paper, we show the equivalence of the set of unitaries computable by the circuits over the Clifford and T library and the set of unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$, in the single-qubit case. We report an efficient synthesis algorithm, with an exact optimality guarantee on the number of Hadamard and T gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over the Clifford and T library and unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$ holds in the $n$-qubit case.