Randomized Benchmarking with Restricted Gate Sets

TL;DR

This paper explores randomized benchmarking protocols using subgroups of the Clifford group that are not unitary 2 designs, providing a general analysis method and demonstrating high-accuracy error estimation with restricted gate sets.

Contribution

It introduces a new analysis framework for randomized benchmarking with non-2-design subgroups and shows these can achieve accurate error estimates.

Findings

Error probability can be estimated within a factor of two or less.

The error estimate factor can be made conservative and decay exponentially with qubits.

Better accuracy is achievable for logical qubits.

Abstract

Standard randomized benchmarking protocols entail sampling from a unitary 2 design, which is not always practical. In this article we examine randomized benchmarking protocols based on subgroups of the Clifford group that are not unitary 2 designs. We introduce a general method for analyzing such protocols and subsequently apply it to two subgroups, the group generated by controlled-NOT, Hadamard, and Pauli gates and that generated by only controlled-NOT and Pauli gates. In both cases the error probability can be estimated to within a factor of two or less where the factor can be arranged to be conservative and to decay exponentially in the number of qubits. For randomized benchmarking of logical qubits even better accuracy will typically be obtained. Thus, we show that sampling a distribution which is close to a unitary 2 design, although sufficient, is not necessary for randomized benchmarking to high accuracy.