Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to performing robust computations. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, which can dramatically reduce error rates while introducing little or no experimental overhead. Moreover, we prove that our technique is robust to the inevitable variation in errors over the randomizing gates and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity---comparable to fidelities realized in current experiments---are sufficient to achieve fault-tolerant quantum computation. Furthermore, the worst-case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking protocols, enabling a rigorous certification of the performance of a quantum computer.
Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to performing robust computations. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, which can dramatically reduce error rates while introducing little or no experimental overhead. Moreover, we prove that our technique is robust to the inevitable variation in errors over the randomizing gates and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity---comparable to fidelities realized in current experiments---are sufficient to achieve fault-tolerant quantum computation. Furthermore, the worst-case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking protocols, enabling a rigorous certification of the performance of a quantum computer.
Current hardware for quantum computing suffers from high levels of noise, and so to achieve practical fault-tolerant quantum computing will require powerful and efficient methods to correct for errors in quantum circuits. Here, we explore the role and effectiveness of using noise tailoring techniques to improve the performance of error correcting codes. Noise tailoring methods such as randomized compiling (RC) convert complex coherent noise processes to effective stochastic noise. While it is known that this can be leveraged to design efficient diagnostic tools, we explore its impact on the performance of error correcting codes. Of particular interest is the important class of coherent errors, arising from control errors, where RC has the maximum effect---converting these into purely stochastic errors. For these errors, we show here that RC delivers an improvement in performance of the concatenated Steane code by several orders of magnitude. We also show that below a threshold rotation angle, the gains in logical fidelity can be arbitrarily magnified by increasing the size of the codes. These results suggest that using randomized compiling can lead to a significant reduction in the resource overhead required to achieve fault tolerance.
Abstract Contemporary methods for benchmarking noisy quantum processors typically measure average error rates or process infidelities. However, thresholds for fault-tolerant quantum error correction are given in terms of worst-case error rates—defined via the diamond norm—which can differ from average error rates by orders of magnitude. One method for resolving this discrepancy is to randomize the physical implementation of quantum gates, using techniques like randomized compiling (RC). In this work, we use gate set tomography to perform precision characterization of a set of two-qubit logic gates to study RC on a superconducting quantum processor. We find that, under RC, gate errors are accurately described by a stochastic Pauli noise model without coherent errors, and that spatially correlated coherent errors and non-Markovian errors are strongly suppressed. We further show that the average and worst-case error rates are equal for randomly compiled gates, and measure a maximum worst-case error of 0.0197(3) for our gate set. Our results show that randomized benchmarks are a viable route to both verifying that a quantum processor’s error rates are below a fault-tolerance threshold, and to bounding the failure rates of near-term algorithms, if—and only if—gates are implemented via randomization methods which tailor noise.
Randomized compiling reduces the effects of errors on quantum computers by tailoring arbitrary Markovian errors into stochastic Pauli noise. Here we prove that randomized compiling also tailors non-Markovian errors into local stochastic Pauli noise and investigate the technique's limitations. We show through analysis and numerical results that randomized compiling alters errors in three distinct helpful ways. First, it prevents the coherent accumulation of errors (including hard to remove crosstalk effects) across gate cycles by destroying intercycle coherent correlations. Second, it converts individual gate cycle errors into Pauli noise. Finally, randomized compiling reduces the variability inherent to noisy devices. We confirm these theoretical predictions with the IBM Quantum Experience platform and describe experimental data that illustrates a drastic performance improvement across public devices. These results cement the importance of randomized compiling in near- and long-term quantum information processing.
The successful implementation of algorithms on quantum processors relies on the accurate control of quantum bits (qubits) to perform logic gate operations. In this era of noisy intermediate-scale quantum (NISQ) computing, systematic miscalibrations, drift, and crosstalk in the control of qubits can lead to a coherent form of error that has no classical analog. Coherent errors severely limit the performance of quantum algorithms in an unpredictable manner, and mitigating their impact is necessary for realizing reliable quantum computations. Moreover, the average error rates measured by randomized benchmarking and related protocols are not sensitive to the full impact of coherent errors and therefore do not reliably predict the global performance of quantum algorithms, leaving us unprepared to validate the accuracy of future large-scale quantum computations. Randomized compiling is a protocol designed to overcome these performance limitations by converting coherent errors into stochastic noise, dramatically reducing unpredictable errors in quantum algorithms and enabling accurate predictions of algorithmic performance from error rates measured via cycle benchmarking. In this work, we demonstrate significant performance gains under randomized compiling for the four-qubit quantum Fourier transform algorithm and for random circuits of variable depth on a superconducting quantum processor. Additionally, we accurately predict algorithm performance using experimentally measured error rates. Our results demonstrate that randomized compiling can be utilized to leverage and predict the capabilities of modern-day noisy quantum processors, paving the way forward for scalable quantum computing.
The characterization of noisy quantum circuits is an important step in the development of large-scale quantum computers. As experimental quantum architectures approach the threshold for fault-tolerant quantum computing, obtaining a benchmark of process fidelities is needed to verify the integrity of the system and ensure functionality. Additionally, the ability to successfully implement a quantum code of choice may depend on the type of noise that is present in the system. It is therefore necessary to characterize not only the fidelity of the process, but also the structure of the noise. \n \nRandomized benchmarking is a characterization protocol which extracts partial information from a noisy quantum process. This thesis gives a detailed review of the randomized benchmarking protocol, and presents the results of three collaborative projects which use techniques similar to randomized benchmarking to obtain previously hidden information. The first protocol is designed to obtain a benchmark on loss errors. This protocol also provides information on the detector efficiency and an additional constraint on estimated parameters from randomized benchmarking. The second protocol is an extension of the first, under the special case of coherent leakage errors, for which we obtain an estimate of the leakage rate. The third protocol provides a method of testing for spatial correlations in noisy processes. This test may be used to determine the correlation structure of errors, and the error correcting codes needed for successful experiments in a particular architecture. All protocols presented in this thesis are scalable, platform-independent and insensitive to state preparation and measurement errors. We also provide results of numerical tests of these protocols for simulated noise, which show their accuracy and robustness.
The promise of quantum computing with imperfect qubits relies on the ability of a quantum computing system to scale cheaply through error correction and fault tolerance. While fault tolerance requires relatively mild assumptions about the nature of qubit errors, the overhead associated with coherent and non-Markovian errors can be orders of magnitude larger than the overhead associated with purely stochastic Markovian errors. One proposal to address this challenge is to randomize the circuits of interest, shaping the errors to be stochastic Pauli errors but leaving the aggregate computation unaffected. The randomization technique can also suppress couplings to slow degrees of freedom associated with non-Markovian evolution. Here, we demonstrate the implementation of Pauli-frame randomization in a superconducting circuit system, exploiting a flexible programming and control infrastructure to achieve this with low effort. We use high-accuracy gate-set tomography to characterize in detail the properties of the circuit error, with and without the randomization procedure, which allows us to make rigorous statements about Markovianity as well as the nature of the observed errors. We demonstrate that randomization suppresses signatures of non-Markovian evolution to statistically insignificant levels, from a Markovian model violation ranging from $43\ensuremath{\sigma}$ to $1987\ensuremath{\sigma}$, down to violations between $0.3\ensuremath{\sigma}$ and $2.7\ensuremath{\sigma}$ under randomization. Moreover, we demonstrate that, under randomization, the experimental errors are well described by a Pauli error model, with model violations that are similarly insignificant (between $0.8\ensuremath{\sigma}$ and $2.7\ensuremath{\sigma}$). Importantly, all these improvements in the model accuracy were obtained without degradation to fidelity, and with some improvements to error rates as quantified by the diamond norm. This demonstrates the ability of Pauli-frame randomization to shape noise into forms that are more benign for quantum error correction and fault tolerance.