Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets

TL;DR

This paper extends an efficient exact synthesis algorithm for single-qubit unitaries to Clifford-cyclotomic gate sets, enabling minimal decomposition into specific z-rotations for various n, but with limitations for most values.

Contribution

It generalizes the exact synthesis algorithm to Clifford-cyclotomic sets, providing minimal decompositions for a broader class of gates beyond Clifford+T.

Findings

The algorithm is optimal in the number of z-rotations used.

For n=4, the group of exactly synthesizable unitaries matches a specific algebraic ring.

The characterization holds for some small n but fails for most positive even integers.

Abstract

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer $n$ we consider the "Clifford-cyclotomic" gate set consisting of the Clifford group plus a z-rotation by $\frac{\pi}{n}$. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of $\frac{\pi}{n}$ z-rotations. For the Clifford+T case $n=4$ the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring $\mathbb{Z}[e^{i\frac{\pi}{n}},1/2]$. We prove that this characterization holds for a handful of other small values of $n$ but the fraction of positive even integers for which it fails to hold is 100%.