Fast exact synthesis of two-qubit unitaries using a near-minimum number of T gates

Abstract

This paper focuses on exact synthesis of two-qubit unitaries using Clifford and $T$ gates. We propose an ancilla-free synthesis algorithm (i) which uses $T$ gates no more than ten times the minimum possible number of $T$ gates, also known as the $T$ count, and (ii) whose time complexity is linear with the $T$ count and thus instance optimal. Our synthesis algorithm relies on a characterization of the $T$ count of two-qubit unitaries based on Lie group homomorphism, which may be interest of its own. Precisely, we show that for any two-qubit unitary generated by Clifford and $T$ gates, its $T$ count is equivalent to the least denominator exponent of its SO(6) representation up to a factor of at most 10.