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arXiv:1608.02943 · Quantum Science and Technology

Estimating the fidelity of T gates using standard interleaved randomized benchmarking

Robin Harper, Steven T. Flammia

Published Feb 9, 2017• 40 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2058-9565/aa5f8d

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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.
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.
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.
Randomized benchmarking allows one to efficiently and scalably characterize the average error of a unitary 2-design such as the Clifford group C\mathcal{C}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.
We describe a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits. We apply this protocol to a superconducting qubit system and find a bounded average error of 0.003 [0,0.016] for the single-qubit gates X(π/2) and Y(π/2). These bounded values provide better estimates of the average error than those extracted via quantum process tomography.
Abstract There is currently a significant need for robust and efficient methods for characterizing quantum devices. While there has been significant progress in this direction, there remains a crucial need to precisely determine the strength and type of errors on individual gate operations, in order to assess and improve control as well as reliably bound the total error in a quantum circuit given some partial information about the errors on the components. In this work, we first provide an optimal bound on the total fidelity of a circuit in terms of component fidelities, which can be efficiently experimentally estimated via randomized benchmarking (RB). We then derive a tighter bound that applies under additional information about the coherence of the error, namely, the unitarity, which can also be estimated via a related experimental protocol. This improved bound smoothly interpolates between the worst-case quadratic and best-case linear scaling for composite error channels. As an application we show how our analysis substantially improves the achievable precision on estimates of the infidelities of individual gates under interleaved RB, enabling greater precision for current experimental methods to assess and tune-up control over quantum gate operations.
Randomized benchmarking provides a tool for obtaining precise quantitative estimates of the average error rate of a physical quantum channel. Here we define real randomized benchmarking, which enables a separate determination of the average error rate in the real and complex parts of the channel. This provides more fine-grained information about average error rates with approximately the same cost as the standard protocol. The protocol requires only averaging over the real Clifford group, a subgroup of the full complex Clifford group, and makes use of the fact that it forms an orthogonal 2-design. It therefore allows benchmarking of fault-tolerant gates for an encoding which does not contain the full Clifford group transversally. Furthermore, our results are especially useful when considering quantum computations on rebits (or real encodings of complex computations), in which case the real Clifford group now plays the role of the complex Clifford group when studying stabilizer circuits.