arXiv subjects

Joel J. Wallman

Publications and source records attributed to Joel J. Wallman.

At least 37 records · Page 2Linked to original sources

arXiv2021

Efficient learning of quantum noise

Noise is the central obstacle to building large-scale quantum computers. Quantum systems with sufficiently uncorrelated and weak noise could be used to solve computational problems that are intractable with current digital computers. There has been substantial progress towards engineering such systems. However, continued progress depends on the ability to characterize quantum noise reliably and efficiently with high precision. Here we describe such a protocol and report its experimental implementation on a 14-qubit superconducting quantum architecture. The method returns an estimate of the effective noise and can detect correlations within arbitrary sets of qubits. We show how to construct a quantum noise correlation matrix allowing the easy visualization of correlations between all pairs of qubits, enabling the discovery of long-range two-qubit correlations in the 14 qubit device that had not previously been detected. Our results are the first implementation of a provably rigorous and comprehensive diagnostic protocol capable of being run on state of the art devices and beyond. These results pave the way for noise metrology in next-generation quantum devices, calibration in the presence of crosstalk, bespoke quantum error-correcting codes, and customized fault-tolerance protocols that can greatly reduce the overhead in a quantum computation.

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arXiv2020

Simulating and mitigating crosstalk

We describe an efficient and scalable framework for modeling crosstalk effects on quantum information processors. By applying optimal control techniques, we show how to tuneup arbitrary high-fidelity parallel operations on systems with substantial local and nonlocal crosstalk. Simulations show drastically lower error rates for a 2D square array of 100 superconducting transmon qubits. These results suggest that rather than striving to engineer away undesirable interactions during fabrication, we can largely mitigate their effects through careful characterization and control optimization.

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arXiv2020

On the freedom in representing quantum operations

We discuss the effects of a gauge freedom in representing quantum information processing devices, and its implications for characterizing these devices. We demonstrate with experimentally relevant examples that there exists equally valid descriptions of the same experiment which distribute errors differently among objects in a gate-set, leading to different error rates. Consequently, it can be misleading to attach a concrete operational meaning to figures of merit for individual gate-set elements. We propose an alternative operational figure of merit for a gate-set, the mean variation error, and a protocol for measuring this figure.

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arXiv2019

Bounding the average gate fidelity of composite channels using the unitarity

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. We then derive a tighter bound that applies under additional information about the coherence of the error, namely, the unitarity, which can also be efficiently estimated via a related experimental protocol. This improved bound smoothly interpolates between the worst-case quadratic and best-case linear scalings for composite error channels. As an application we show how our analysis substantially improves the achievable precision on estimates of the error rates of individual gates under interleaved randomized benchmarking, enabling greater precision for current experimental methods to assess and tune-up control over quantum gate operations.

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arXiv2019

Multi-qubit Randomized Benchmarking Using Few Samples

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 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 show that the number of samples required for long sequence lengths can be made substantially smaller than previous rigorous results (even for single qubits) as long as the noise process under investigation is not unitary. Our results bring rigorous randomized benchmarking on systems with many qubits into the realm of experimental feasibility.

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arXiv2019

Clifford recompilation for faster classical simulation of quantum circuits

Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al, utilizes a randomized sparsification technique to approximate the output state of a quantum circuit by a stabilizer sum with a reduced number of terms. In this paper, we describe an improved Monte Carlo algorithm for performing randomized sparsification. This algorithm reduces the runtime of computing the approximate state by the factor $\ell/m$, where $\ell$ and $m$ are respectively the total and non-Clifford gate counts. The main technique is a circuit recompilation routine based on manipulating exponentiated Pauli operators. The recompilation routine also facilitates numerical search for Clifford decompositions of products of gates, which can further reduce the runtime in certain cases. It may additionally lead to a framework for optimizing circuit implementations over a gate set, reducing the overhead for state-injection in fault-tolerant implementations. We provide a concise exposition of randomized sparsification, and describe how to use it to estimate circuit amplitudes in a way which can be generalized to a broader class of gates and states. This latter method can be used to obtain additive error estimates of circuit probabilities with a faster runtime than the full techniques of Bravyi et al. Such estimates are useful for validating near-term quantum devices provided that the target probability is not exponentially small.

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arXiv2019

Phenomenological measure of quantum non-Markovianity

Non-Markovian effects are ubiquitous in physical quantum systems and remain a significant challenge to achieving high-quality control and reliable quantum computation, but due to their inherent complexity, are rarely characterized. Past approaches to quantifying non-Markovianity have concentrated on small, simple systems, and we show that such measures can capture properties that are irrelevant to applications. With this insight, we propose a method for constructing relevant quantifiers of non-Markovian dynamics and illustrate the scheme's computability by characterizing a model quantum dot array.

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arXiv2019

Estimating the Coherence of Noise

Noise mechanisms in quantum systems can be broadly characterized as either coherent (i.e., unitary) or incoherent. For a given fixed average error rate, coherent noise mechanisms will generally lead to a larger worst-case error than incoherent noise. We show that the coherence of a noise source can be quantified by the unitarity, which we relate to the average change in purity averaged over input pure states. We then show that the unitarity can be efficiently estimated using a protocol based on randomized benchmarking that is efficient and robust to state-preparation and measurement errors. We also show that the unitarity provides a lower bound on the optimal achievable gate infidelity under a given noisy process.

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arXiv2019

Statistical analysis of randomized benchmarking

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 experimental choices, RB+ efficiently provides estimates of error rates with multiplicative precision. Finally, we provide a simplified rigorous method for obtaining credible regions for parameters of interest and a heuristic approximation for these intervals that performs well in currently relevant regimes.

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arXiv2018

Coherence in quantum error-correcting codes

Typical studies of quantum error correction assume probabilistic Pauli noise, largely because it is relatively easy to analyze and simulate. Consequently, the effective logical noise due to physically realistic coherent errors is relatively unknown. Here, we prove that encoding a system in a stabilizer code and measuring error syndromes decoheres errors, that is, causes coherent errors to converge toward probabilistic Pauli errors, even when no recovery operations are applied. Two practical consequences are that the error rate in a logical circuit is well quantified by the average gate fidelity at the logical level and that essentially optimal recovery operators can be determined by independently optimizing the logical fidelity of the effective noise per syndrome.

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arXiv2018

Randomized Benchmarking under Different Gatesets

We provide a comprehensive analysis of the differences between two important standards for randomized benchmarking (RB): the Clifford-group RB protocol proposed originally in Emerson et al (2005) and Dankert et al (2006), and a variant of that RB protocol proposed later by the NIST group in Knill et al, PRA (2008). While these two protocols are frequently conflated or presumed equivalent, we prove that they produce distinct exponential fidelity decays leading to differences of up to a factor of 3 in the estimated error rates under experimentally realistic conditions. These differences arise because the NIST RB protocol does not satisfy the unitary two-design condition for the twirl in the Clifford-group protocol and thus the decay rate depends on non-invariant features of the error model. Our analysis provides an important first step towards developing definitive standards for benchmarking quantum gates and a more rigorous theoretical underpinning for the NIST protocol and other RB protocols lacking a group-structure. We conclude by discussing the potential impact of these differences for estimating fault-tolerant overheads.

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arXiv2018

From randomized benchmarking experiments to gateset circuit fidelity: how to interpret randomized benchmarking decay parameters

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.

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arXiv2018

Representations of the multi-qubit Clifford group

The Clifford group is a fundamental structure in quantum information with a wide variety of applications. We discuss the tensor representations of the $q$-qubit Clifford group, which is defined as the normalizer of the $q$-qubit Pauli group in $U(2^q)$. In particular, we characterize all irreducible subrepresentations of the two-copy representation $φ^{\otimes2}$ of the Clifford group on the matrix space $\mathbb{C}^{d\times d}\otimes \mathbb{C}^{d\times d}$ with $d=2^q$. In an upcoming companion paper we applied this result to cut down the number of samples necessary to perform randomised benchmarking, a method for characterising quantum systems.

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arXiv2018

Contextuality and The Single-Qubit Stabilizer Formalism

Contextuality is a fundamental non-classical property of quantum theory, which has recently been proven to be a key resource for achieving quantum speed-ups in some leading models of quantum computation. However, which of the forms of contextuality, and how much thereof, are required to obtain a speed-up in an arbitrary model of quantum computation remains unclear. In this paper, we show that the relation between contextuality and a compuational advantage is more complicated than previously thought. We achieve this by proving that generalized contextuality is present even within the simplest subset of quantum operations, the so-called single-qubit stabilizer theory, which offers no computational advantage and was previously believed to be completely non-contextual. However, the contextuality of the single-qubit stabilizer theory can be confined to transformations. Therefore our result also demonstrates that the commonly considered prepare-and-measure scenarios (which ignore transformations) do not fully capture the contextuality of quantum theory.

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arXiv2018

Contextuality bounds the efficiency of classical simulation of quantum processes

Contextuality has been conjectured to be a super-classical resource for quantum computation, analogous to the role of non-locality as a super-classical resource for communication. We show that the presence of contextuality places a lower bound on the amount of classical memory required to simulate any quantum sub-theory, thereby establishing a quantitative connection between contextuality and classical simulability. We apply our result to the qubit stabilizer sub-theory, where the presence of state-independent contextuality has been an obstacle in establishing contextuality as a quantum computational resource. We find that the presence of contextuality in this sub-theory demands that the minimum number of classical bits of memory required to simulate a multi-qubit system must scale quadratically in the number of qubits; notably, this is the same scaling as the Gottesman-Knill algorithm. We contrast this result with the (non-contextual) qudit case, where linear scaling is possible.

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arXiv2018

Bayesian Inference for Randomized Benchmarking Protocols

Randomized benchmarking (RB) protocols are standard tools for characterizing quantum devices. Prior analyses of RB protocols have not provided a complete method for analyzing realistic data, resulting in a variety of ad-hoc methods. The main confounding factor in rigorously analyzing data from RB protocols is an unknown and noise-dependent distribution of survival probabilities over random sequences. We propose a hierarchical Bayesian method where these survival distributions are modeled as nonparametric Dirichlet process mixtures. Our method infers parameters of interest without additional assumptions about the underlying physical noise process. We show with numerical examples that our method works robustly for both standard and highly pathological error models. Our method also works reliably at low noise levels and with little data because we avoid the asymptotic assumptions of commonly used methods such as least-squares fitting. For example, our method produces a narrow and consistent posterior for the average gate fidelity from ten random sequences per sequence length in the standard RB protocol.

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arXiv2018

Randomized benchmarking with gate-dependent noise

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 that it quantifies the average fidelity of the noise between ideal gates. We numerically demonstrate that our analysis is valid for strongly gate-dependent noise models. We also show why alternative analyses do not provide a rigorous justification for the empirical success of randomized benchmarking with gate-dependent noise.

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arXiv2017

Hard decoding algorithm for optimizing thresholds under general Markovian noise

Quantum error correction is instrumental in protecting quantum systems from noise in quantum computing and communication settings. Pauli channels can be efficiently simulated and threshold values for Pauli error rates under a variety of error-correcting codes have been obtained. However, realistic quantum systems can undergo noise processes that differ significantly from Pauli noise. In this paper, we present an efficient hard decoding algorithm for optimizing thresholds and lowering failure rates of an error-correcting code under general completely positive and trace-preserving (i.e., Markovian) noise. We use our hard decoding algorithm to study the performance of several error-correcting codes under various non-Pauli noise models by computing threshold values and failure rates for these codes. We compare the performance of our hard decoding algorithm to decoders optimized for depolarizing noise and show improvements in thresholds and reductions in failure rates by several orders of magnitude. Our hard decoding algorithm can also be adapted to take advantage of a code's non-Pauli transversal gates to further suppress noise. For example, we show that using the transversal gates of the 5-qubit code allows arbitrary rotations around certain axes to be perfectly corrected. Furthermore, we show that Pauli twirling can increase or decrease the threshold depending upon the code properties. Lastly, we show that even if the physical noise model differs slightly from the hypothesized noise model used to determine an optimized decoder, failure rates can still be reduced by applying our hard decoding algorithm.

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