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 . The protocol is built on top of the recently developed filtered RB framework [J. Helsen , PRX Quantum 3, 020357 (2022), M. Heinrich , 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.
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 . The protocol is built on top of the recently developed filtered RB framework [J. Helsen , PRX Quantum 3, 020357 (2022), M. Heinrich , 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.
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.
Abstract Noise characterization methods such as randomized benchmarking (RB) are critical for the development of scalable quantum computers. Modern RB protocols for multiqubit systems extract physically relevant error rates by exploiting the structure of the group representation generated by the set of benchmarked operations. However, existing techniques become prohibitively inefficient for representations that are highly reducible yet decompose into irreducible subspaces of high dimension. These situations prevail when benchmarking high-dimensional systems such as qudits or bosonic modes, where experimental control is limited to implementing a small subset of all possible unitary operations. We introduce a broad framework for enhancing the sample efficiency of RB that is sufficiently powerful to extend the practical reach of RB beyond the multiqubit setting. Our strategy, which applies to any benchmarking group, uses ‘synthetic’ quantum circuits with classical post-processing of both input and output data to leverage the full structure of reducible superoperator representations. To demonstrate the efficacy of our approach, we develop a detailed theory of RB for systems with rotational symmetry. Such systems carry a natural action of the group SU ( 2 ) , and they form the basis for several novel quantum error-correcting codes. We show that, for experimentally accessible high-spin systems, synthetic RB protocols can reduce the complexity of measuring rotationally invariant error rates by two orders of magnitude relative to standard approaches such as character RB.
Characterization of experimental systems is an essential step in developing and improving quantum hardware. A collection of protocols known as Randomized Benchmarking (RB) was developed in the past decade, which provides an efficient way to measure error rates in quantum systems. In a recent paper (arxiv:2010.07974), a general framework for RB was proposed, which encompassed most of the known RB protocols and overcame the limitation on error models in previous works. However, even this general framework has a restriction: it can only be applied to a finite group of gates. This does not meet the need posed by experiments, in particular the demand for benchmarking non-Clifford gates and continuous gate sets on quantum devices. In this work we generalize the RB framework to continuous groups of gates and show that as long as the noise level is reasonably small, the output can be approximated as a linear combination of matrix exponential decays. As an application, we numerically study the fully randomized benchmarking protocol (i.e. RB with the entire unitary group as the gate set) enabled by our proof. This provides a unified way to estimate the gate fidelity for any quantum gate in an experiment.
Randomized benchmarking refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates.These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors.Over the years many versions have been developed, however a comprehensive theoretical treatment of randomized benchmarking has been missing.In this work, we develop a rigorous framework of randomized benchmarking general enough to encompass virtually all known protocols as well as novel, more flexible extensions.Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any randomized benchmarking experiment is well described by a linear combination of matrix exponential decays.We complement this with a detailed analysis of the fitting problem associated with randomized benchmarking data.We introduce modern signal processing techniques to randomized benchmarking, prove analytical sample complexity bounds, and numerically evaluate performance and limitations.In order to reduce the resource demands of this fitting problem, we introduce novel, scalable postprocessing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of randomized benchmarking protocols.These postprocessing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking.Finally, we discuss, in full generality, how and when randomized benchmarking decay rates can be used to infer quality measures like the average fidelity.On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on randomized benchmarking by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing.
Analog quantum simulation allows for assessing static and dynamical properties of strongly correlated quantum systems to high precision. To perform simulations outside the reach of classical computers, accurate and reliable implementations of the anticipated Hamiltonians are required. To achieve those, characterization and benchmarking tools are a necessity. For digital quantum devices, randomized benchmarking can provide a benchmark on the average quality of the implementation of a gate set. In this work, we introduce a versatile framework for randomized analog benchmarking of bosonic and fermionic quantum devices implementing particle number preserving dynamics. The scheme makes use of the restricted operations which are native to analog simulators and other continuous variable systems. Importantly, like randomized benchmarking, it is robust against state preparation and measurement errors. We discuss the scheme's efficiency, derive theoretical performance guarantees and showcase the protocol with numerical examples.
In its many variants, randomized benchmarking (RB) is a broadly used technique for assessing the quality of gate implementations on quantum computers. A detailed theoretical understanding and general guarantees exist for the functioning and interpretation of RB protocols if the gates under scrutiny are drawn uniformly at random from a compact group. In contrast, many practically attractive and scalable RB protocols implement random quantum circuits with local gates randomly drawn from some gate-set. Despite their abundance in practice, for those non-uniform RB protocols, general guarantees for gates from arbitrary compact groups under experimentally plausible assumptions are missing. In this work, we derive such guarantees for a large class of RB protocols for random circuits that we refer to as filtered RB. Prominent examples include linear cross-entropy benchmarking, character benchmarking, Pauli-noise tomography and variants of simultaneous RB. Building upon recent results for random circuits, we show that many relevant filtered RB schemes can be realized with random quantum circuits in linear depth, and we provide explicit small constants for common instances. We further derive general sample complexity bounds for filtered RB. We show filtered RB to be sample-efficient for several relevant groups, including protocols addressing higher-order cross-talk. Our theory for non-uniform filtered RB is, in principle, flexible enough to design new protocols for non-universal and analog quantum simulators.