Abstract We investigate randomized benchmarking (RB) in a general setting with quantum gates that form a representation, not necessarily an irreducible one, of a finite group. We derive an estimate for the average fidelity, to which experimental data may then be calibrated. Furthermore, we establish that RB can be achieved by the sole implementation of quantum gates that generate the group as well as one additional arbitrary group element. In this case, we need to assume that the noise is close to being covariant. This yields a more practical approach to RB. Moreover, we show that RB is stable with respect to approximate Haar sampling for the sequences of gates. This opens up the possibility of using Markov chain Monte Carlo methods to obtain the random sequences of gates more efficiently. We demonstrate these results numerically using the well-studied example of the Clifford group as well as the group of monomial unitary matrices. For the latter, we focus on the subgroup with nonzero entries consisting of n th roots of unity, which contains T gates.
Abstract We investigate randomized benchmarking (RB) in a general setting with quantum gates that form a representation, not necessarily an irreducible one, of a finite group. We derive an estimate for the average fidelity, to which experimental data may then be calibrated. Furthermore, we establish that RB can be achieved by the sole implementation of quantum gates that generate the group as well as one additional arbitrary group element. In this case, we need to assume that the noise is close to being covariant. This yields a more practical approach to RB. Moreover, we show that RB is stable with respect to approximate Haar sampling for the sequences of gates. This opens up the possibility of using Markov chain Monte Carlo methods to obtain the random sequences of gates more efficiently. We demonstrate these results numerically using the well-studied example of the Clifford group as well as the group of monomial unitary matrices. For the latter, we focus on the subgroup with nonzero entries consisting of n th roots of unity, which contains T gates.
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.
Standard randomized benchmarking protocols entail sampling from a unitary 2-design, which is not always practical. In this article we examine randomized benchmarking protocols based on subgroups of the Clifford group that are not unitary 2-designs. We introduce a general method for analyzing such protocols and subsequently apply it to two subgroups, the group generated by controlled-not, Hadamard, and Pauli gates and that generated by only controlled-not and Pauli gates. In both cases the error probability can be estimated to within a factor of 2 or less where the factor can be arranged to be conservative and to decay exponentially in the number of qubits. For randomized benchmarking of logical qubits even better accuracy will typically be obtained. Thus, we show that sampling a distribution which is close to a unitary 2-design, although sufficient, is not necessary for randomized benchmarking to high accuracy.
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.
Any technology requires precise benchmarking of its components, and the quantum technologies are no exception. Randomized benchmarking allows for the relatively resource economical estimation of the average gate fidelity of quantum gates from the Clifford group, assuming identical noise levels for all gates, making use of suitable sequences of randomly chosen gates. In this work, we report significant progress on randomized benchmarking, by showing that it can be applied individually on a broad class of quantum gates outside the Clifford group, even for varying noise levels per quantum gate. This is possible at little overhead of quantum resources, but at the expense of a significant classical computational cost. At the heart of our analysis is a representation-theoretic framework which we bring into contact with classical estimation techniques based on bootstrapping and matrix pencils. We demonstrate the functioning of the scheme at hand of benchmarking tensor powers of T gates. Apart from its practical relevance, we expect this insight to be relevant as it highlights the role of assumptions made on unknown noise processes when characterizing quantum gates at high precision.
Abstract Randomized benchmarking is a technique for estimating the average fidelity of a set of quantum gates. However, if this gateset is not the multi-qubit Clifford group, robustly extracting the average fidelity is difficult. Here, we propose a new method based on representation theory that has little experimental overhead and robustly extracts the average fidelity for a broad class of gatesets. We apply our method to a multi-qubit gateset that includes the T -gate, and propose a new interleaved benchmarking protocol that extracts the average fidelity of a two-qubit Clifford gate using only single-qubit Clifford gates as reference.