Theory of mirror benchmarking and demonstration on a quantum computer
Karl Mayer, A G Hall, Thomas M. Gatterman, Si Khadir Halit, Lee, Kenny, Justin Bohnet, Dan Gresh, Aaron M. Hankin, Kevin Gilmore, Gerber, Justin, Gaebler, John
A new class of protocols called mirror benchmarking was recently proposed to measure the system-level performance of quantum computers. These protocols involve circuits with random sequences of gates followed by mirroring, that is, inverting each gate in the sequence. We give a simple proof that mirror benchmarking leads to an exponential decay of the survival probability with sequence length, under the uniform noise assumption, provided the twirling group forms a 2-design. The decay rate is determined by a quantity that is a quadratic function of the error channel, and for certain types of errors is equal to the unitarity. This result yields a new method for estimating the coherence of noise. We present data from mirror benchmarking experiments run on the Honeywell System Model H1. This data constitutes a set of performance curves, indicating the success probability for random circuits as a function of qubit number and circuit depth.
A new class of protocols called mirror benchmarking was recently proposed to measure the system-level performance of quantum computers. These protocols involve circuits with random sequences of gates followed by mirroring, that is, inverting each gate in the sequence. We give a simple proof that mirror benchmarking leads to an exponential decay of the survival probability with sequence length, under the uniform noise assumption, provided the twirling group forms a 2-design. The decay rate is determined by a quantity that is a quadratic function of the error channel, and for certain types of errors is equal to the unitarity. This result yields a new method for estimating the coherence of noise. We present data from mirror benchmarking experiments run on the Honeywell System Model H1. This data constitutes a set of performance curves, indicating the success probability for random circuits as a function of qubit number and circuit depth.
Randomized benchmarking (RB) is widely used to measure an error rate of a set of quantum gates, by performing random circuits that would do nothing if the gates were perfect. In the limit of no finite-sampling error, the exponential decay rate of the observable survival probabilities, versus circuit length, yields a single error metric r. For Clifford gates with arbitrary small errors described by process matrices, r was believed to reliably correspond to the mean, over all Clifford gates, of the average gate infidelity between the imperfect gates and their ideal counterparts. We show that this quantity is not a well-defined property of a physical gate set. It depends on the representations used for the imperfect and ideal gates, and the variant typically computed in the literature can differ from r by orders of magnitude. We present new theories of the RB decay that are accurate for all small errors describable by process matrices, and show that the RB decay curve is a simple exponential for all such errors. These theories allow explicit computation of the error rate that RB measures (r), but as far as we can tell it does not correspond to the infidelity of a physically allowed (completely positive) representation of the imperfect gates.
Quantum computers have the potential to bring about a new age of technology, but in order for them to become a reality, the elephant in the room must be addressed, quantum errors. The complexity of quantum errors grows exponentially as the system size is increased; scalable characterization protocols which work under physically realistic noise are therefore needed to assess and optimize control. \n \nRandomized Benchmarking (RB) is a family of characterization protocols with the requisite qualities. RB protocols are independent of state preparation and measurement (SPAM) errors, and because they extract partial information about the error profiles in the form of figures of merit, they are scalable in the number of qubits. In this thesis, we study these figures of merit under various error models, gate-sets, and sample distributions, to further our understanding of their physical meaning, and to fine-tune the best practice for diagnosing errors in basic gate-sets. \n \nThe first study presented in this thesis looks at the physical meaning of the unitarity under physically realistic gate-dependent noise. Numerically, we demonstrate that under these realistic noise models, the unitarity obtained via Extended RB (XRB) overlaps strongly with the average of the unitarity of the errors over the set of individual operations. \n \nThe second study numerically and analytically proves that the Standard RB (SRB) decay parameter p coincides with the decay parameter of the novel figure of merit gate-set circuit fidelity. We also prove that, in the limit of high-fidelity single-qubit experiments, the disconnect between the average gate fidelity and SRB experimental results is caused by a basis mismatch between the gates and the state-preparation and measurement procedures. \n \nIn the third part of this thesis, we compare the results of two variations of the SRB protocol, one that samples uniformly (SRB) and one that samples non-uniformly (NIST RB) from the Clifford group. We prove that NIST RB produces an exponential fidelity curve that gives different infidelity results than SRB. \n \nThe fourth and final study gives a standard for assessing basic gate-sets. Experimental results are given from the assessment run on super-conducting and ion-trap devices. We also experimentally show how the systematic error bounds can be tightened by accounting for the error's degree of coherence, as well as using weaker randomizing sets.
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 spectrum gives the decay rates. For generic two-qubit gates one slowest exponential dominates and characterizes gate errors in three channels. Its decay rate is close, but different from that in the standard randomized benchmarking, and we find the leading correction. Using relations to the local invariants of two-qubit gates we identify all exceptional gates with several slow exponentials and analyze possibilities to extract their decay rates from the measured curves.
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 scalability. We then use up to 16 physical qubits of a cloud quantum computing platform to demonstrate that our technique can reveal and quantify crosstalk errors in many-qubit circuits.
Randomized benchmarking (RB) returns a single decay parameter or gate error rate, but when the noise is gate-dependent that parameter is notoriously difficult to interpret. We study RB and related random-circuit benchmarks under small gate-local Markovian noise by factorizing each noisy gate into its ideal action followed by the exponential of a small error generator and then perturbing the RB L matrix, which models the RB decay. For any low-error gate set that satisfies the requirements of RB (it is a sequence-asymptotic unitary 2-design), we show that Pauli-stochastic error rates contribute linearly to the RB decay parameter, while Hamiltonian error rates contribute through a gauge-invariant quadratic form. Pauli-correlation and active generators are absent from the linear term and enter only at higher order. For the single-qubit native gate set {Xπ/2,Yπ/2} we derive this quadratic form explicitly in terms of three first-order gauge-invariant (FOGI) coordinates: the two intrinsic over-rotations and one relational axis-tilt error. The resulting formula explains the dependence of DRB on the sampling distribution and the compilation dependence of CRB. These results turn RB decay rates into interpretable functions of experimentally meaningful elementary error rates.
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.