Randomized benchmarking allows one to efficiently and scalably characterize the average error of a unitary 2-design such as the Clifford group $\mathcal{C}$ on a physical candidate for quantum computation, as long as there are no noncomputational leakage levels in the system. We investigate the effect of leakage errors on randomized benchmarking induced from an additional level per physical qubit and provide a modified protocol that allows us to derive reliable estimates for the error per gate in their presence. We assess the variance of the sequence fidelity corresponding to the number of random sequences needed for valid fidelity estimation. Our protocol allows for gate-dependent error channels without being restricted to perturbations. We show that our protocol is compatible with interleaved randomized benchmarking and expand to benchmarking of arbitrary gates. This setting is relevant for superconducting transmon qubits, among other systems.
Randomized benchmarking allows one to efficiently and scalably characterize the average error of a unitary 2-design such as the Clifford group C on a physical candidate for quantum computation, as long as there are no noncomputational leakage levels in the system. We investigate the effect of leakage errors on randomized benchmarking induced from an additional level per physical qubit and provide a modified protocol that allows us to derive reliable estimates for the error per gate in their presence. We assess the variance of the sequence fidelity corresponding to the number of random sequences needed for valid fidelity estimation. Our protocol allows for gate-dependent error channels without being restricted to perturbations. We show that our protocol is compatible with interleaved randomized benchmarking and expand to benchmarking of arbitrary gates. This setting is relevant for superconducting transmon qubits, among other systems.
Accurate benchmarking of quantum gates is crucial for understanding and enhancing the performance of quantum hardware. A standard method for this is interleaved benchmarking, a technique which estimates the error on an interleaved target gate by comparing cumulative error rates of randomized sequences implemented with the interleaved gate and without it. In this work, we show both numerically and experimentally that the standard approach of interleaved randomized benchmarking (IRB), which uses the multi-qubit Clifford group for randomization, can produce highly inaccurate and even physically impossible estimates for the error on the interleaved gate in the presence of coherent errors. Fortunately we also show that interleaved benchmarking performed with cycle benchmarking, which randomizes with single qubit Pauli gates, provides dramatically reduced systematic uncertainty relative to standard IRB, and further provides as host of additional benefits including data reusability. We support our conclusions with a theoretical framework for bounding systematic errors, extensive numerical results comparing a range of interleaved protocols under fixed resource costs, and experimental demonstrations on three quantum computing platforms.
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.
Abstract Randomized benchmarking is an important protocol for robustly characterizing the error rates of quantum gates. The technique is typically applied to the Clifford gates since they form a group that satisfies a convenient technical condition of forming a unitary 2-design, in addition to having a tight connection to fault-tolerant quantum computing and an efficient classical simulation. In order to achieve universal quantum computing one must add at least one additional gate such as the T gate (also known as the π /8 gate). Here we propose and analyze a simple variation of the standard interleaved RB protocol that can accurately estimate the average fidelity of the T gate while retaining the many advantages of a unitary 2-design and the fidelity guarantees that such a design delivers, as well as the efficient classical simulation property of the Clifford group. Our work complements prior methods that have succeeded in estimating T gate fidelities, but only by relaxing the 2-design constraint and using a more complicated data analysis.
A common problem with physical qubits is that the state can leak outside the computational subspace. To characterize the problem, it is desirable to measure both the leakage probability and the fidelity of gates inside the computational subspace. Randomized benchmarking is a readily accessible method for estimating average gate fidelities and can be modified to account for leakage. However, in the presence of state preparation and measurement errors (SPAM) there are parameter degeneracies, and fidelities and leakage become underdetermined. We study the example of qudits with any number of leakage levels under the assumption of gate-independent noise. We show that randomized benchmarking sequences with leakage level decoherence can be analyzed as a Markov process with noisy classical preparations and measurements. This result generalizes to randomized benchmarking with random gates from an arbitrary group that contains a phase-decohering subgroup. We characterize the parameter degeneracy in inferring the Markov process, preparations and measurements. Based on this characterization we develop a measurement and inference method taking advantage of known conservative bounds on SPAM to constrain fidelity, leakage and other parameters of the Markov process, state preparations and measurements. As part of our study, we experimentally implemented the measurement method for a qubit defined on the one-excitation subspace of two transmons, for which there are two leakage levels. We determined confidence intervals accounting for parameter degeneracy for a number of performance parameters of the system. For leakage and fidelity, we compare these intervals to those obtained for several conventional methods for leakage benchmarking.
In this paper, we analyze the performance of randomized benchmarking protocols on gate sets under a variety of realistic error models that include systematic rotations, amplitude damping, leakage to higher levels, and 1/f noise. We find that, in almost all cases, benchmarking provides better than a factor-of-2 estimate of average error rate, suggesting that randomized benchmarking protocols are a valuable tool for verification and validation of quantum operations. In addition, we derive models for fidelity decay curves under certain types of non-Markovian noise models such as 1/f and leakage errors. We also show that, provided the standard error of the fidelity measurements is small, only a small number of trials are required for high-confidence estimation of gate errors.
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.