Quantum Circuit Overhead

TL;DR

This paper introduces the Quantum Circuit Overhead (QCO) as a measure to evaluate the efficiency of quantum gate sets, providing numerical bounds and insights into optimal gate choices.

Contribution

It defines the QCO and T-QCO measures, demonstrating their usefulness through extensive numerical calculations and identifying optimal gate set completions.

Findings

T gate is highly non-optimal for Clifford set completion.

Numerical bounds reveal efficiency differences among gate sets.

Optimal gate set completions vary depending on subgroup structures.

Abstract

We introduce a measure for evaluating the efficiency of finite universal quantum gate sets $\mathcal{S}$, called the Quantum Circuit Overhead (QCO), and the related notion of $T$-Quantum Circuit Overhead ($T$-QCO). QCO compares the circuit length required by $\mathcal S$ with the best possible length among gate sets of the same size. The $T$-QCO adapts this idea to cost models in which only selected costly gates are counted, while cheap operations are absorbed into an effective gate set. We demonstrate the usefulness of the ($T$-)QCO by extensive numerical calculations of its upper bounds, providing insight into the efficiency of various choices of single-qubit $\mathcal{S}$, including Haar-random gate sets and the gate sets derived from finite subgroups, such as Clifford and Hurwitz groups. In particular, our results suggest that, in terms of the upper bounds on the $T$-QCO, the famous T gate is a highly non-optimal choice for the completion of the Clifford gate set, even among the gates of order 8. We identify the optimal choices of such completions for both finite subgroups.