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  • Open Access

Bosonic Randomized Benchmarking with Passive Transformations

Mirko Arienzo1,*, Dmitry Grinko2,3, Martin Kliesch1, and Markus Heinrich4,5

  • *Contact author: mirko.arienzo@tuhh.de
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PRX Quantum 6, 020305 – Published 7 April, 2025

DOI: https://doi.org/10.1103/PRXQuantum.6.020305

Abstract

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 bosonic Hilbert space. 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 passive bosonic RB. The protocol is built on top of the recently developed filtered RB framework [J. Helsen et al., PRX Quantum 3, 020357 (2022), M. Heinrich et al., Randomized benchmarking with random quantum circuits, arxiv:2212.06181 [quant-ph]] 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 postprocessing of the experimental data. We also analyze the sampling complexity of passive bosonic 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 the first results for heterodyne measurements.

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Popular Summary

Intrinsic noise in quantum devices makes developing scalable quantum technologies challenging. Therefore, it is crucial to have a standardized method for practically determining their error behavior. This task is more challenging for bosonic systems, for which practical protocols have been lacking. For discrete systems, randomized benchmarking (RB) is the gold standard for quantifying the average quality of quantum gates. This work bridges this gap and introduces an RB protocol for particle-number-preserving transformations in bosonic systems.

Our analysis is based on the Clebsch-Gordan decomposition for general unitary groups. By leveraging the relevant representation theory, we classically postprocess experimental data in a measure of the average quality of the implementation of passive transformations. The protocol is not affected by errors affecting the preparation of the input state and the measurement, and we observe a very mild scaling of the number of samples necessary to obtain accurate estimators with respect to the number of modes. Notably, particle loss is one of the major sources of noise in bosonic systems. By postselecting on particle-number-preserving events, we can also specialize the protocol to the estimation of the average loss parameters affecting the gates. To reduce the experimental effort, bosonic RB protocols, including either Gaussian input states or Gaussian measurement, can be developed (partial results are discussed in this paper). Extensions to active transformations are also of great interest. However, this faces convergency issues because of the noncompact nature of the symplectic group.

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PRX Quantum 6, 020305– Published 7 April, 2025
  • Received 27 September 2024
  • Revised 14 January 2025
  • Accepted 12 February 2025
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