Bosonic randomized benchmarking with passive transformations

Randomized benchmarking (RB) is the most commonly employed protocol for the characterization of unitary operations in quantum circuits due to its reasonable experimental requirements and robustness against state preparation and measurement (SPAM) errors. So far, the protocol has been limited to discrete or fermionic systems, whereas extensions to bosonic systems have been unclear for a long time due to challenges arising from the underlying infinite-dimensional Hilbert spaces. In this work, we close the gap for bosonic systems and develop an RB protocol to benchmark passive Gaussian transformations on any particle number subspace, which we call bosonic passive RB. The protocol is based on the recently developed filtered RB framework and is designed to isolate the multitude of exponential decays arising for passive bosonic transformations. We give explicit formulas and a Julia implementation for the necessary post-processing of the experimental data. We also analyze the sampling complexity of bosonic passive RB by deriving analytical expressions for the variance. They show a mild scaling with the number of modes, suggesting that passive bosonic RB is experimentally feasible for a moderate number of modes. We focus on experimental settings involving Fock states and particle number resolving measurements, but also discuss Gaussian settings, deriving first results for heterodyne measurements.