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.
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.
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.
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.
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.
Randomised benchmarking is a widely used experimental technique to characterise the average error of quantum operations. Benchmarking procedures that scale to enable the characterisation of n-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomised benchmarking procedure over n-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabiliser codes. We present efficient methods for representing and composing group elements, sampling them uniformly and synthesising corresponding poly(n)-sized circuits. The procedure provides experimental access to two independent parameters that together characterise the average gate fidelity of a group element. A scalable procedure to determine the error of quantum operations has been developed by researchers in the United States. Andrew Cross and colleagues at the IBM T.J. Watson Research Center have expanded previously used protocols to a broader range of quantum circuits, thereby enabling error benchmarking for all common logical operations used in leading quantum computing protocols. The standard technique for error characterization in these devices is randomized benchmarking, which provides benefits such as insensitivity to measurement errors, but on the other hand only functions in a limited number of circumstances. In their theoretical study, the researchers demonstrate how randomized benchmarking approaches can be expanded to a much broader range of logical operations. The procedure can be readily implemented experimentally, and provides access to important experimental noise parameters in quantum computing.
Randomized benchmarking (RB) is a powerful method for determining the error rate of experimental quantum gates.Traditional RB, however, is restricted to gatesets, such as the Clifford group, that form a unitary 2-design.The recently introduced character RB can benchmark more general gates using techniques from representation theory; up to now, however, this method has only been applied to "multiplicity-free" groups, a mathematical restriction on these groups.In this paper, we extend the original character RB derivation to explicitly treat non-multiplicity-free groups, and derive several applications.First, we derive a rigorous version of the recently introduced subspace RB, which seeks to characterize a set of one-and two-qubit gates that are symmetric under SWAP.Second, we develop a new leakage RB protocol that applies to more general groups of gates.Finally, we derive a scalable RB protocol for the matchgate group, a group that like the Clifford group is nonuniversal but becomes universal with the addition of one additional gate.This example provides one of the few examples of a scalable non-Clifford RB protocol.In all three cases, compared to existing theories, our method requires similar resources, but either provides a more accurate estimate of gate fidelity, or applies to a more general group of gates.In conclusion, we discuss the potential, and challenges, of using non-multiplicity-free character RB to develop new classes of scalable RB protocols and methods of characterizing specific gates.
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.