# Approximate Randomized Benchmarking for Finite Groups

## TL;DR

This paper extends randomized benchmarking to finite groups with non-irreducible representations, providing practical methods for fidelity estimation and demonstrating stability under approximate sampling, with numerical examples including Clifford and monomial unitary groups.

## Contribution

It introduces a generalized approach to randomized benchmarking for finite groups, including methods requiring only group generators and an arbitrary element, and analyzes stability with approximate Haar sampling.

## Key findings

- Derived an estimate for average fidelity in generalized settings
- Proposed a practical implementation using group generators and one additional element
- Showed stability of randomized benchmarking under approximate Haar sampling

## Abstract

We investigate randomized benchmarking in a general setting with quantum gates that form a representation, not necessarily an irreducible one, of a finite group. We derive an estimate for the average fidelity, to which experimental data may then be calibrated. Furthermore, we establish that randomized benchmarking can be achieved by the sole implementation of quantum gates that generate the group as well as one additional arbitrary group element. In this case, we need to assume that the noise is close to being covariant. This yields a more practical approach to randomized benchmarking. Moreover, we show that randomized benchmarking is stable with respect to approximate Haar sampling for the sequences of gates. This opens up the possibility of using Markov chain Monte Carlo methods to obtain the random sequences of gates more efficiently. We demonstrate these results numerically using the well-studied example of the Clifford group as well as the group of monomial unitary matrices. For the latter, we focus on the subgroup with nonzero entries consisting of n-th roots of unity, which contains T gates.