Efficient Decomposition of Single-Qubit Gates into $V$ Basis Circuits

TL;DR

This paper introduces efficient algorithms for decomposing single-qubit gates into circuits over the universal V basis, improving compilation depth and precision compared to traditional methods.

Contribution

The authors present the first constructive algorithms for V basis circuit compilation, with one polynomial-time algorithm and a more precise, albeit exponential-time, search-based method.

Findings

Polynomial-time algorithm improves depth/precision over state-of-the-art

Exponential-time search yields shorter circuits in practice

Algorithms are the first of their kind for the V basis

Abstract

We develop the first constructive algorithms for compiling single-qubit unitary gates into circuits over the universal $V$ basis. The $V$ basis is an alternative universal basis to the more commonly studied $\{H,T\}$ basis. We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time (in precision $\log(1/\epsilon)$) and offers a depth/precision guarantee that improves upon state-of-the-art methods for compiling into the $\{H,T\}$ basis by factors ranging from 1.86 to $\log_2(5)$. The second algorithm is analogous to direct search and yields circuits a factor of 3 to 4 times shorter than our first algorithm, and requires time exponential in $\log(1/\epsilon)$; however, we show that in practice the runtime is reasonable for an important range of target precisions.