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arXiv:1510.02720 · Bulletin of the American Physical Society

Scalable randomized benchmarking of non-Clifford gates

Andrew W. Cross, Easwar Magesan, Lev S. Bishop, John A. Smolin, Jay Gambetta

Published Oct 9, 2015• 5 citations
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Abstract

Randomized benchmarking is a widely used experimental technique to characterize the average error of quantum operations. Benchmarking procedures that scale to enable characterization 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 randomized benchmarking procedure over $n$-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabilizer codes. We present efficient methods for representing and composing group elements, sampling them uniformly, and synthesizing corresponding $\mathrm{poly}(n)$-sized circuits. The procedure provides experimental access to two independent parameters that together characterize the average gate fidelity of a group element.

DOI
10.48550/arxiv.1510.02720

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Randomized benchmarking is a widely used experimental technique to characterize the average error of quantum operations. Benchmarking procedures that scale to enable characterization of nnn-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 randomized benchmarking procedure over nnn-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabilizer codes. We present efficient methods for representing and composing group elements, sampling them uniformly, and synthesizing corresponding poly(n)\mathrm{poly}(n)poly(n)-sized circuits. The procedure provides experimental access to two independent parameters that together characterize the average gate fidelity of a group element.
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) protocols are the most widely used methods for assessing the performance of quantum gates.However, the existing RB methods either do not scale to many qubits or cannot benchmark a universal gate set.Here, we introduce and demonstrate a technique for scalable RB of many universal and continuously parametrized gate sets, using a class of circuits called randomized mirror circuits.Our technique can be applied to a gate set containing an entangling Clifford gate and the set of arbitrary single-qubit gates, as well as gate sets containing controlled rotations about the Pauli axes.We use our technique to benchmark universal gate sets on four qubits of the Advanced Quantum Testbed, including a gate set containing a controlled-S gate and its inverse, and we investigate how the observed error rate is impacted by the inclusion of non-Clifford gates.Finally, we demonstrate that our technique scales to many qubits with experiments on a 27-qubit IBM Q processor.We use our technique to quantify the impact of crosstalk on this 27-qubit device, and we find that it contributes approximately 2=3 of the total error per gate in random many-qubit circuit layers.
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.
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 is a powerful technique to efficiently estimate the performance and reliability of quantum gates, circuits, and devices. Here, we propose to perform randomized benchmarking in a coherent way, where superpositions of different random sequences rather than independent samples are used. We show that this leads to a uniform and simple protocol having significant advantages with respect to gates that can be benchmarked, and in terms of efficiency and scalability. We show that, e.g., universal gate sets, the set of nnn-qudit Pauli operators or more general sets including arbitrary unitaries, as well as a particular nnn-qudit Clifford gate using only the Pauli set, can be efficiently benchmarked. The price to pay is an additional complexity to add control to the involved quantum operations. However, we demonstrate that this can be done by using auxiliary degrees of freedom that are naturally available in basically any physical realization, and are independent of the gates to be tested.
With the development of controllable quantum systems, fast and practical characterization of multi-qubit gates has become essential for building high-fidelity quantum computing devices. The usual way to fulfill this requirement via randomized benchmarking demands complicated implementation of numerous multi-qubit twirling gates. How to efficiently and reliably estimate the fidelity of a quantum process remains an open problem. This work thus proposes a character-cycle benchmarking protocol and a character-average benchmarking protocol using only local twirling gates to estimate the process fidelity of an individual multi-qubit operation. Our protocols were able to characterize a large class of quantum gates including and beyond the Clifford group via the local gauge transformation, which forms a universal gate set for quantum computing. We demonstrated numerically our protocols for a non-Clifford gate—controlled- ( T X ) and a Clifford gate—five-qubit quantum error-correcting encoding circuit. The numerical results show that our protocols can efficiently and reliably characterize the gate process fidelities. Compared with the cross-entropy benchmarking, the simulation results show that the character-average benchmarking achieves three orders of magnitude improvements in terms of sampling complexity.