# Randomized benchmarking with gate-dependent noise

## TL;DR

This paper provides a rigorous analysis of randomized benchmarking under gate-dependent noise, showing it still exhibits exponential decay and accurately reflects average fidelity, even with strong gate dependence.

## Contribution

It proves that randomized benchmarking's exponential decay behavior persists under arbitrary gate-dependent noise, clarifying its operational meaning and validating its empirical effectiveness.

## Key findings

- Decay parameter quantifies average fidelity of gate-dependent noise
- Exponential decay converges to that of gate-independent noise
- Analysis validated with numerical simulations of strong gate-dependent noise

## Abstract

We analyze randomized benchmarking for arbitrary gate-dependent noise and prove that the exact impact of gate-dependent noise can be described by a single perturbation term that decays exponentially with the sequence length. That is, the exact behavior of randomized benchmarking under general gate-dependent noise converges exponentially to a true exponential decay of exactly the same form as that predicted by previous analysis for gate-independent noise. Moreover, we show that the operational meaning of the decay parameter for gate-dependent noise is essentially unchanged, that is, we show that it quantifies the average fidelity of the noise between ideal gates. We numerically demonstrate that our analysis is valid for strongly gate-dependent noise models. We also show why alternative analyses do not provide a rigorous justification for the empirical success of randomized benchmarking with gate-dependent noise.