Efficient Unitarity Randomized Benchmarking of Few-qubit Clifford Gates

Unitarity randomized benchmarking (URB) is an experimental procedure for estimating the unitarity of implemented quantum gates independently of state preparation and measurement (SPAM) errors. The unitarity is a measure of coherence of a quantum gate that provides information independent of average fidelity. A central problem in the URB experiment is relating the number of data points to rigorous confidence intervals around the unitarity.
In this work we analyze the statistics of Clifford URB. We provide a bound on the required number of data points as a function of confidence and experimental parameters. The main ingredient for this result is a sharp bound on the variance of the data points due to the random sampling. Our result shows that estimating the unitarity of gates requires fewer data points as the gates become closer to being unitary, up to constant contribution due to SPAM errors. The bound is sufficiently sharp to benchmark small-dimensional systems in realistic parameter regimes using a modest number of data points. For example, we show that the unitarity of single-qubit Clifford gates can be rigorously estimated using few hundred data points under the assumption of gate-independent noise. This is a reduction of orders of magnitude compared to previously known bounds.