A Depth-Optimal Canonical Form for Single-qubit Quantum Circuits

TL;DR

This paper introduces a depth-optimal canonical form for single-qubit quantum circuits in the H,T basis, enabling efficient exact and approximate decompositions with lower depth and resource requirements.

Contribution

It presents a novel depth-optimal canonical form for single-qubit circuits and extends it to approximate decompositions with reduced complexity and resource usage.

Findings

Lower-depth -approximate circuits than previous methods

Significantly fewer computational resources needed

Efficient extension to Solovay-Kitaev algorithms

Abstract

Given an arbitrary single-qubit operation, an important task is to efficiently decompose this operation into an (exact or approximate) sequence of fault-tolerant quantum operations. We derive a depth-optimal canonical form for single-qubit quantum circuits, and the corresponding rules for exactly reducing an arbitrary single-qubit circuit to this canonical form. We focus on the single-qubit universal H,T basis due to its role in fault-tolerant quantum computing, and show how our formalism might be extended to other universal bases. We then extend our canonical representation to the family of Solovay-Kitaev decomposition algorithms, in order to find an \epsilon-approximation to the single-qubit circuit in polylogarithmic time. For a given single-qubit operation, we find significantly lower-depth \epsilon-approximation circuits than previous state-of-the-art implementations. In addition, the implementation of our algorithm requires significantly fewer resources, in terms of computation memory, than previous approaches.