We theoretically consider a cross-resonance (CR) gate implemented by pulse sequences proposed by Calderon-Vargas & Kestner, Phys. Rev. Lett. 118, 150502 (2017). These sequences mitigate systematic error to first order, but their effectiveness is limited by one-qubit gate imperfections. Using additional microwave control pulses, it is possible to tune the effective CR Hamiltonian into a regime where these sequences operate optimally. This improves the overall feasibility of these sequences by reducing the one-qubit operations required for error correction. We illustrate this by simulating randomized benchmarking for a system of weakly coupled transmons and show that while this novel pulse sequence does not offer an advantage with the current state of the art in transmons, it does improve the scaling of CR gate infidelity with one-qubit gate infidelity.
We implement a complete randomized benchmarking protocol on a system of two superconducting qubits. The protocol consists of randomizing over gates in the Clifford group, which experimentally are generated via an improved two-qubit cross-resonance gate implementation and single-qubit unitaries. From this we extract an optimal average error per Clifford of 0.0936. We also perform an interleaved experiment, alternating our optimal two-qubit gate with random two-qubit Clifford gates, to obtain a two-qubit gate error of 0.0653. We compare these values with a two-qubit gate error of ~0.12 obtained …
We study theoretically the responses of the dynamically corrected gates to time-dependent noises in the exchange-only spin qubit system. We consider $1/f$ noises having spectra proportional to $1/ω^α$, where the exponent $α$ indicates the strength of correlation within the noise. The quantum gate errors due to noises are extracted from a numerical simulation of Randomized Benchmarking, and are compared between the application of uncorrected operations and that of dynamically corrected gates robust against the hyperfine noise. We have found that for $α\gtrsim1.5$, the dynamically corrected gate…
We present measurements of single-qubit gate errors for a superconducting qubit. Results from quantum process tomography and randomized benchmarking are compared with gate errors obtained from a double pi pulse experiment. Randomized benchmarking reveals a minimum average gate error of 1.1+/-0.3% and a simple exponential dependence of fidelity on the number of gates. It shows that the limits on gate fidelity are primarily imposed by qubit decoherence, in agreement with theory.
We perform randomized benchmarking on neutral atomic quantum bits (qubits) confined in an optical lattice. Single qubit gates are implemented using microwaves, resulting in a measured error per randomized computational gate of 1.4(1) x 10^-4 that is dominated by the system T2 relaxation time. The results demonstrate the robustness of the system, and its viability for more advanced quantum information protocols.
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 two or less where the factor can be arranged to b…
We construct a gate and time-independent noise model that results in the output of a logical randomized benchmarking protocol oscillating rather than decaying exponentially. To illustrate our idea, we first construct an example in standard randomized benchmarking where we assume the existence of ``hidden'' qubits, permitting a choice of representation of the Clifford group that contains multiplicities. We use the multiplicities to, with each gate application, update a hidden memory of the gate history that we use to circumvent theorems which guarantee the output decays exponentially. In our fo…
Extrapolating physical error rates to logical error rates requires many assumptions and thus can radically under- or overestimate the performance of an error correction implementation. We introduce logical randomized benchmarking, a characterization procedure that directly assesses the performance of a quantum error correction implementation at the logical level, and is motivated by a reduction to the well-studied case of physical randomized benchmarking. We show that our method reliably reports logical performance and can estimate the average probability of correctable and uncorrectable error…
Randomized benchmarking (RB) protocols have become an essential tool for providing a meaningful partial characterization of experimental quantum operations. While the RB decay rate is known to enable estimates of the average fidelity of those operations under gate-independent Markovian noise, under gate-dependent noise this rate is more difficult to interpret rigorously. In this paper, we prove that single-qubit RB decay parameter $p$ coincides with the decay parameter of the gate-set circuit fidelity, a novel figure of merit which characterizes the expected average fidelity over arbitrary cir…
Randomized Benchmarking allows to efficiently and scalably characterize the average error of an unitary 2-design such as the Clifford group $\mathcal{C}$ on a physical candidate for quantum computation, as long as there are no non-computational 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 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 s…
We demonstrate a method to obtain the spectra of $1/f$ noises in spin-qubit devices from randomized benchmarking, assisted by supervised learning. The noise exponent, which indicates the correlation within the noise, is determined by training a double-layer neural network with the ratio between the randomized benchmarking results of pulse sequences that correct noise and not. After the training is completed, the neural network is able to predict the exponent within an absolute error of about 0.05, comparable with existing methods. The noise amplitude is then evaluated by training another neura…
Building upon the demonstration of coherent control and single-shot readout of the electron and nuclear spins of individual 31-P atoms in silicon, we present here a systematic experimental estimate of quantum gate fidelities using randomized benchmarking of 1-qubit gates in the Clifford group. We apply this analysis to the electron and the ionized 31-P nucleus of a single P donor in isotopically purified 28-Si. We find average gate fidelities of 99.95 % for the electron, and 99.99 % for the nuclear spin. These values are above certain error correction thresholds, and demonstrate the potential …
Randomized benchmarking is routinely used as an efficient method for characterizing the performance of sets of elementary logic gates in small quantum devices. In the measurement-based model of quantum computation, logic gates are implemented via single-site measurements on a fixed universal resource state. Here we adapt the randomized benchmarking protocol for a single qubit to a linear cluster state computation, which provides partial, yet efficient characterization of the noise associated with the target gate set. Applying randomized benchmarking to measurement-based quantum computation exh…
Randomized benchmarking is a promising tool for characterizing the noise in experimental implementations of quantum systems. In this paper, we prove that the estimates produced by randomized benchmarking (both standard and interleaved) for arbitrary Markovian noise sources are remarkably precise by showing that the variance due to sampling random gate sequences is small. We discuss how to choose experimental parameters, in particular the number and lengths of random sequences, in order to characterize average gate errors with rigorous confidence bounds. We also show that randomized benchmarkin…
To improve the performance of multi-qubit algorithms on quantum devices it is critical to have methods for characterizing non-local quantum errors such as crosstalk. To address this issue, we propose and test an extension to the analysis of simultaneous randomized benchmarking data -- correlated randomized benchmarking. We fit the decay of correlated polarizations to a composition of fixed-weight depolarizing maps to characterize the locality and weight of crosstalk errors. From these errors we introduce a crosstalk metric which indicates the distance to the closest map with only local errors.…
We investigate randomized benchmarking 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 randomized benchmarking 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 randomized benchmarking. Moreove…
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-two 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 new models for fidelity decay curves under certain types of non-Markovian noise models such as 1/f and lea…
Randomized benchmarking has emerged as a popular and easy-to-implement experimental technique for gauging the quality of gate operations in quantum computing devices. A typical randomized benchmarking procedure identifies the exponential decay in the fidelity as the benchmarking sequence of gates increases in length, and the decay rate is used to estimate the fidelity of the gate. That the fidelity decays exponentially, however, relies on the assumption of time-independent or static noise in the gates, with no correlations or significant drift in the noise over the gate sequence, a well-satisf…
In randomized benchmarking of quantum logical gates, partial twirling can be used for simpler implementation, better scaling, and higher accuracy and reliability. For instance, for two-qubit gates, single-qubit twirling is easier to realize than full averaging. We analyze such simplified, partial twirling and demonstrate that, unlike for the standard randomized benchmarking, the measured decay of fidelity is a linear combination of exponentials with different decay rates (3 for two qubits and single-bit twirling). The evolution with the sequence length is governed by an iteration matrix, whose…
We report the first complete characterization of single-qubit and two-qubit gate fidelities in silicon-based spin qubits, including cross-talk and error correlations between the two qubits. To do so, we use a combination of standard randomized benchmarking and a recently introduced method called character randomized benchmarking, which allows for more reliable estimates of the two-qubit fidelity in this system. Interestingly, with character randomized benchmarking, the two-qubit CPhase gate fidelity can be obtained by studying the additional decay induced by interleaving the CPhase gate in a r…
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…
The performance of quantum gates is often assessed using some form of randomized benchmarking. However, the existing methods become infeasible for more than approximately five qubits. Here we show how to use a simple and customizable class of circuits -- randomized mirror circuits -- to perform scalable, robust, and flexible randomized benchmarking of Clifford gates. We show that this technique approximately estimates the infidelity of an average many-qubit logic layer, and we use simulations of up to 225 qubits with physically realistic error rates in the range 0.1-1% to demonstrate its scala…
The control and handling of errors arising from cross-talk and unwanted interactions in multi-qubit systems is an important issue in quantum information processing architectures. We introduce a benchmarking protocol that provides information about the amount of addressability present in the system and implement it on coupled superconducting qubits. The protocol consists of randomized benchmarking each qubit individually and then simultaneously, and the amount of addressability is related to the difference of the average gate fidelities of those experiments. We present the results on two simila…
We show that non-exponential fidelity decays in randomized benchmarking experiments on quantum dot qubits are consistent with numerical simulations that incorporate low-frequency noise. By expanding standard randomized benchmarking analysis to this experimental regime, we find that such non-exponential decays are better modeled by multiple exponential decay rates, leading to an instantaneous control fidelity for isotopically-purified-silicon MOS quantum dot qubits which can be as high as 99.9% when low-frequency noise conditions and system calibrations are favorable. These advances in qubit ch…
Randomized benchmarking is a useful scheme for evaluation the average fidelity of a noisy quantum circuit. However, it is insensitive to the unitary error. Here, we propose a method of randomized benchmarking in which a unitary t-design is applied and by which the unitary error estimation can be converted to analysis of pseudo-randomness on a set of unitary operators. We give a bound on the number of randomized benchmarking sequences, when performing a unitary t-design on n-qubit d-dimensional system. By applying local random unitary operators, a decomposition of a unitary t-design, the bound …
We describe a simple randomized benchmarking protocol for quantum information processors and obtain a sequence of models for the observable fidelity decay as a function of a perturbative expansion of the errors. We are able to prove that the protocol provides an efficient and reliable estimate of an average error-rate for a set operations (gates) under a general noise model that allows for both time and gate-dependent errors. We determine the conditions under which this estimate remains valid and illustrate the protocol through numerical examples.
One of the main challenges in building a quantum processor is to characterize the environmental noise. Noise characterization can be achieved by exploiting different techniques, such as randomization where several sequences of random quantum gates are applied to the qubit under test to derive statistical characteristics about the affecting noises. A scalable and robust algorithm able to benchmark the full set of Clifford gates using randomization techniques is called randomized benchmarking. In this study, we simulated randomized benchmarking protocols in a semiconducting all-electrical three-…
We present a method for optimizing quantum control in experimental systems, using a subset of randomized benchmarking measurements to rapidly infer error. This is demonstrated to improve single- and two-qubit gates, minimize gate bleedthrough, where a gate mechanism can cause errors on subsequent gates, and identify control crosstalk in superconducting qubits. This method is able to correct parameters to where control errors no longer dominate, and is suitable for automated and closed-loop optimization of experimental systems.
We describe how randomized benchmarking can be used to reconstruct the unital part of any trace-preserving quantum map, which in turn is sufficient for the full characterization of any unitary evolution, or more generally, any unital trace-preserving evolution. This approach inherits randomized benchmarking's robustness to preparation, measurement, and gate imperfections, therefore avoiding systematic errors caused by these imperfections. We also extend these techniques to efficiently estimate the average fidelity of a quantum map to unitary maps outside of the Clifford group. The unitaries we…
Randomized benchmarking is a technique for estimating the average fidelity of a set of quantum gates. For general gatesets, however, it is difficult to draw robust conclusions from the resulting data. Here we propose a new method based on representation theory that has little experimental overhead and applies to a broad class of benchmarking problems. As an example, we apply our method to a gateset that includes the $T$-gate, and analyze a new interleaved benchmarking protocol that extracts the average fidelity of a 2-qubit Clifford gate using only single-qubit Clifford gates as reference.
We describe and expand upon the scalable randomized benchmarking protocol proposed in Phys. Rev. Lett. 106, 180504 (2011) which provides a method for benchmarking quantum gates and estimating the gate-dependence of the noise. The protocol allows the noise to have weak time and gate-dependence, and we provide a sufficient condition for the applicability of the protocol in terms of the average variation of the noise. We discuss how state preparation and measurement errors are taken into account and provide a complete proof of the scalability of the protocol. We establish a connection in special …
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 app…
We analyze randomized benchmarking for arbitrary gate-dependent noise and prove that the exact impact of gate-dependent noise can be described by a single perturbation term that decays exponentially with the sequence length. That is, the exact behavior of randomized benchmarking under general gate-dependent noise converges exponentially to a true exponential decay of exactly the same form as that predicted by previous analysis for gate-independent noise. Moreover, we show that the operational meaning of the decay parameter for gate-dependent noise is essentially unchanged, that is, we show tha…
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-des…
A precise measurement of dephasing over a range of timescales is critical for improving quantum gates beyond the error correction threshold. We present a metrological tool, based on randomized benchmarking, capable of greatly increasing the precision of Ramsey and spin echo sequences by the repeated but incoherent addition of phase noise. We find our SQUID-based qubit is not limited by $1/f$ flux noise at short timescales, but instead observe a telegraph noise mechanism that is not amenable to study with standard measurement techniques.
With improved gate calibrations reducing unitary errors, we achieve a benchmarked single-qubit gate fidelity of 99.95% with superconducting qubits in a circuit quantum electrodynamics system. We present a method for distinguishing between unitary and non-unitary errors in quantum gates by interleaving repetitions of a target gate within a randomized benchmarking sequence. The benchmarking fidelity decays quadratically with the number of interleaved gates for unitary errors but linearly for non-unitary, allowing us to separate systematic coherent errors from decoherent effects. With this protoc…
Typical quantum gate tomography protocols struggle with a self-consistency problem: the gate operation cannot be reconstructed without knowledge of the initial state and final measurement, but such knowledge cannot be obtained without well-characterized gates. A recently proposed technique, known as randomized benchmarking tomography (RBT), sidesteps this self-consistency problem by designing experiments to be insensitive to preparation and measurement imperfections. We implement this proposal in a superconducting qubit system, using a number of experimental improvements including implementing…
Randomized benchmarking is an experimental procedure intended to demonstrate control of quantum systems. The procedure extracts the average error introduced by a set of control operations. When the target set of operations is intended to be the set of Clifford operators, the randomized benchmarking algorithm is particularly easy to perform and its results have an important interpretation with respect to quantum computation. The aim of the benchmark is to provide a simple, useful parameter describing the quality of quantum control with an experiment that can be performed in a standard way on an…
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 Clifford gates. In this work, we report significant progress on randomized benchmarking, by showing that it can be done for individual quantum gates outside the Clifford group, even for varying noise levels per quantum gate. This is possi…
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 with 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 $n-$qudit Pauli operators or more general sets including arbitrary …
We compute the expected randomized benchmarking sequence fidelity for a system subject to Gaussian time-correlated noise. For single qubit benchmarking we show that the expected sequence fidelity is given by the partition function of a long-range coupled spin-one Ising model, with each site in the Ising model corresponding to a free evolution interval. For d-state systems, the expected sequence fidelity is given by an Ising-like model partition function whose site variables are given by the weights of the adjoint representation of SU(d). A high effective temperature expansion for the partition…
Randomized benchmarking (RB) is an efficient and robust method to characterize gate errors in quantum circuits. Averaging over random sequences of gates leads to estimates of gate errors in terms of the average fidelity. These estimates are isolated from the state preparation and measurement errors that plague other methods like channel tomography and direct fidelity estimation. A decisive factor in the feasibility of randomized benchmarking is the number of sampled sequences required to obtain rigorous confidence intervals. Previous bounds were either prohibitively loose or required the numbe…
We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami [Sbornik: Mathematics 208, 1784 (2017)], we provide an alternative proof of Wallman's [Quantum 2, 47 (2018)] and Proctor's [Phys. Rev. Lett. 119, 130502 (2017)] bounds on the effect of gate-dependent noise on randomized benchmarking. We show explicitly that as long as our faulty gate-set is close to the targeted representation of the Clifford…
Quantum information processing offers promising advances for a wide range of fields and applications, provided that we can efficiently assess the performance of the control applied in candidate systems. That is, we must be able to determine whether we have implemented a desired gate, and refine accordingly. Randomized benchmarking reduces the difficulty of this task by exploiting symmetries in quantum operations. Here, we bound the resources required for benchmarking and show that, with prior information, we can achieve several orders of magnitude better accuracy than in traditional approaches…
The presence of correlations in noisy quantum circuits will be an inevitable side effect as quantum devices continue to grow in size and depth. Randomized Benchmarking (RB) is arguably the simplest method to initially assess the overall performance of a quantum device, as well as to pinpoint the presence of temporal-correlations, so-called non-Markovianity; however, when such presence is detected, it hitherto remains a challenge to operationally quantify its features. Here, we demonstrate a method exploiting the power of machine learning with matrix product operators to deduce the minimal aver…
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 th…
Being able to quantify the level of coherent control in a proposed device implementing a quantum information processor (QIP) is an important task for both comparing different devices and assessing a device's prospects with regards to achieving fault-tolerant quantum control. We implement in a liquid-state nuclear magnetic resonance QIP the randomized benchmarking protocol presented by Knill et al (PRA 77: 012307 (2008)). We report an error per randomized $\fracπ{2}$ pulse of $1.3 \pm 0.1 \times 10^{-4}$ with a single qubit QIP and show an experimentally relevant error model where the randomize…
Randomized benchmarking with non-Markovian noise and realistic finite-time gates
Pritzker School of Molecular Engineering, University of Chicago（芝加哥大学普里茨克分子工程学院）；National Center for Computational Sciences, Oak Ridge National Laboratory（橡树岭国家实验室国家计算科学中心）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）；Department of Physics and Astronomy, Northwestern University（西北大学物理与天文学系）
Leakage errors arise when the quantum state leaks out of some subspace of interest, for example, the two-level subspace of a multi-level system defining a computational `qubit' or the logical code space defined by some quantum error-correcting code or decoherence-free subspace. Leakage errors pose a distinct challenge to quantum control relative to the more well-studied decoherence errors and can be a limiting factor to achieving fault-tolerant quantum computation. Here we present scalable and robust randomized benchmarking protocols for quickly estimating the rates of both coherent and incohe…
Among the most popular and well studied quantum characterization, verification and validation techniques is randomized benchmarking (RB), an important statistical tool used to characterize the performance of physical logic operations useful in quantum information processing. In this work we provide a detailed mathematical treatment of the effect of temporal noise correlations on the outcomes of RB protocols. We provide a fully analytic framework capturing the accumulation of error in RB expressed in terms of a three-dimensional random walk in "Pauli space." Using this framework we derive the p…
Randomized benchmarking and variants thereof, which we collectively call RB+, are widely used to characterize the performance of quantum computers because they are simple, scalable, and robust to state-preparation and measurement errors. However, experimental implementations of RB+ allocate resources suboptimally and make ad-hoc assumptions that undermine the reliability of the data analysis. In this paper, we propose a simple modification of RB+ which rigorously eliminates a nuisance parameter and simplifies the experimental design. We then show that, with this modification and specific exper…
A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computin…
We characterize control of a qutrit implemented in the lowest three energy levels of a capacitively-shunted flux-biased superconducting circuit. Randomized benchmarking over the qutrit Clifford group yields an average fidelity of 98.89 $\pm$ 0.05%. For a selected subset of the Clifford group, we perform quantum process tomography and observe the behaviour of repeated gate sequences. Each qutrit gate is generated using only two-state rotations via a method applicable to any unitary. We find that errors are due to decoherence primarily and have a significant contribution from level shifts. This …
