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.
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.
Errors are common issues in quantum computing platforms, among which leakage is one of the most-challenging to address. This is because leakage, i.e., the loss of information stored in the computational subspace to undesired subspaces in a larger Hilbert space, is more difficult to detect and correct than errors that preserve the computational subspace. As a result, leakage presents a significant obstacle to the development of fault-tolerant quantum computation. In this paper, we propose an efficient and accurate benchmarking framework called leakage randomized benchmarking (LRB), for measuring leakage rates on multi-qubit quantum systems. Our approach is more insensitive to state preparation and measurement (SPAM) noise than existing leakage benchmarking protocols, requires fewer assumptions about the gate set itself, and can be used to benchmark multi-qubit leakages, which has not been achieved previously. We also extended the LRB protocol to an interleaved variant called interleaved LRB (iLRB), which can benchmark the average leakage rate of generic n-site quantum gates with reasonable noise assumptions. We demonstrate the iLRB protocol on benchmarking generic two-qubit gates realized using flux tuning and analyzed the behavior of iLRB under corresponding leakage models. Our numerical experiments showed good agreement with the theoretical estimations, indicating the feasibility of both the LRB and iLRB protocols.
Leakage errors are unwanted transfer of population outside of a defined computational subspace and they occur in almost every platform for quantum computing. While prevalent, leakage is often overlooked when measuring and reporting the fidelity of quantum gates with standard methods. In fact, when leakage is substantial it can cause a large overestimation of fidelity from the typical method used to measure fidelity, randomized benchmarking. We provide several methods for properly estimating fidelity in the presence of leakage errors that are applicable in different error regimes with carefully chosen sequence lengths. Then, we numerically demonstrate the methods for two-qubit randomized benchmarking, which often have the largest errors. Finally, we reanalyze previously shared data from Quantinuum systems with some of the methods provided.
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 such as 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 number of sampled sequences to scale exponentially with the number of qubits in order to obtain a fixed confidence interval at a fixed error rate. Here, we show that, with a small adaptation to the randomized benchmarking procedure, the number of sampled sequences required for a fixed confidence interval is dramatically smaller than could previously be justified. In particular, we show that the number of sampled sequences required is essentially independent of the number of qubits and scales favorably with the average error rate of the system under investigation. We also investigate the fitting procedure inherent to randomized benchmarking in light of our results and find that standard methods such as ordinary least squares optimization can give misleading results. We therefore recommend moving to more sophisticated fitting methods such as iteratively reweighted least squares optimization. Our results bring rigorous randomized benchmarking on systems with many qubits into the realm of experimental feasibility.
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.
We present a general framework for the quantification and characterization of leakage errors that result when a quantum system is encoded in the subspace of a larger system. To do this we introduce metrics for quantifying the coherent and incoherent properties of the resulting errors and we illustrate this framework with several examples relevant to superconducting qubits. In particular, we propose two quantities, the leakage and seepage rates, which together with average gate fidelity allow for characterizing the average performance of quantum gates in the presence of leakage and show how the randomized benchmarking protocol can be modified to enable the robust estimation of all three quantities for a Clifford gate set.
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 circuits of operations from the gate-set. We also prove that, in the limit of high-fidelity single-qubit experiments, the possible alarming disconnect between the average gate fidelity and RB experimental results is simply explained by a basis mismatch between the gates and the state-preparation and measurement procedures, that is, to a unitary degree of freedom in labeling the Pauli matrices. Based on numerical evidence and physically motivated arguments, we conjecture that these results also hold for higher dimensions.