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arXiv:1706.03787 · npj Quantum Information

Experimental quantum verification in the presence of temporally correlated noise

Sandeep Mavadia, Claire Edmunds, Cornelius Hempel, Harrison Ball, Federico Roy, Thomas M. Stace, Michael J. Biercuk

Published Jan 23, 2018• 57 citations
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Abstract

Abstract Growth in the capabilities of quantum information hardware mandates access to techniques for performance verification that function under realistic laboratory conditions. Here we experimentally characterise the impact of common temporally correlated noise processes on both randomised benchmarking (RB) and gate-set tomography (GST). Our analysis highlights the role of sequence structure in enhancing or suppressing the sensitivity of quantum verification protocols to either slowly or rapidly varying noise, which we treat in the limiting cases of quasi-DC miscalibration and white noise power spectra. We perform experiments with a single trapped 171Yb+ ion-qubit and inject engineered noise $$\left( { \propto \hat \sigma _z} \right)$$ ∝ σ ^ z to probe protocol performance. Experiments on RB validate predictions that measured fidelities over sequences are described by a gamma distribution varying between approximately Gaussian, and a broad, highly skewed distribution for rapidly and slowly varying noise, respectively. Similarly we find a strong gate set dependence of default experimental GST procedures in the presence of correlated errors, leading to significant deviations between estimated and calculated diamond distances in the presence of correlated $$\hat \sigma _z$$ σ ^ z errors. Numerical simulations demonstrate that expansion of the gate set to include negative rotations can suppress these discrepancies and increase reported diamond distances by orders of magnitude for the same error processes. Similar effects do not occur for correlated $$\hat \sigma _x$$ σ ^ x or $$\hat \sigma _y$$ σ ^ y errors or depolarising noise processes, highlighting the impact of the critical interplay of selected gate set and the gauge optimisation process on the meaning of the reported diamond norm in correlated noise environments.

DOI
10.1038/s41534-017-0052-0

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Abstract Growth in the capabilities of quantum information hardware mandates access to techniques for performance verification that function under realistic laboratory conditions. Here we experimentally characterise the impact of common temporally correlated noise processes on both randomised benchmarking (RB) and gate-set tomography (GST). Our analysis highlights the role of sequence structure in enhancing or suppressing the sensitivity of quantum verification protocols to either slowly or rapidly varying noise, which we treat in the limiting cases of quasi-DC miscalibration and white noise power spectra. We perform experiments with a single trapped 171Yb+ ion-qubit and inject engineered noise (∝σ^z)\left( { \propto \hat \sigma _z} \right)(∝σ^z​) ∝ σ ^ z to probe protocol performance. Experiments on RB validate predictions that measured fidelities over sequences are described by a gamma distribution varying between approximately Gaussian, and a broad, highly skewed distribution for rapidly and slowly varying noise, respectively. Similarly we find a strong gate set dependence of default experimental GST procedures in the presence of correlated errors, leading to significant deviations between estimated and calculated diamond distances in the presence of correlated σ^z\hat \sigma _zσ^z​ σ ^ z errors. Numerical simulations demonstrate that expansion of the gate set to include negative rotations can suppress these discrepancies and increase reported diamond distances by orders of magnitude for the same error processes. Similar effects do not occur for correlated σ^x\hat \sigma _xσ^x​ σ ^ x or σ^y\hat \sigma _yσ^y​ σ ^ y errors or depolarising noise processes, highlighting the impact of the critical interplay of selected gate set and the gauge optimisation process on the meaning of the reported diamond norm in correlated noise environments.
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 probability density function describing RB outcomes (averaged over noise) for both Markovian and correlated errors, which we show is generally described by a $\mathrm{\ensuremath{\Gamma}}$ distribution with shape and scale parameters depending on the correlation structure. Long temporal correlations impart large nonvanishing variance and skew in the distribution towards high-fidelity outcomes---consistent with existing experimental data---highlighting potential finite-sampling pitfalls and the divergence of the mean RB outcome from worst-case errors in the presence of noise correlations. We use the filter-transfer function formalism to reveal the underlying reason for these differences in terms of effective coherent averaging of correlated errors in certain random sequences. We conclude by commenting on the impact of these calculations on the utility of single-metric approaches to quantum characterization, verification, and validation.
Randomized benchmarking (RB) is a widely adopted protocol for estimating the average gate fidelity in quantum hardware. However, its standard formulation relies on the assumption of temporally uncorrelated noise, an assumption often violated in current devices. In this work, we derive analytic expressions for the average sequence fidelity (ASF) in the presence of temporally correlated (non-Markovian) noise with classical memory, including cases where such correlations originate from interactions with a quantum environment. We show how the ASF can be interpreted to extract meaningful benchmarking parameters under such noise and identify classes of interaction Hamiltonians that render temporal correlations completely invisible to RB. We further provide operational criteria for witnessing temporal correlations due to quantum memory through RB experiments. Importantly, while classical correlations may remain undetectable in the ASF data, they can nonetheless significantly affect worst-case errors quantified by the diamond norm, a metric central to fault-tolerant quantum computing. In particular, we demonstrate that temporal correlations may suppress worst-case errors, highlighting that temporal correlations may not always have detrimental effects on gate performance.
Quantum error correction (QEC) can strongly suppress noise, but predicting whether a computation succeeds requires understanding not only how much effective noise persists after QEC, but what form that noise takes. To thoroughly characterize this noise, we introduce logical gate set tomography (``logical GST''). Logical GST produces detailed representations of noisy logical operations that can be used to derive higher-level metrics, such as gate fidelity and diamond distance, and to test for non-Markovian and coherent logical errors. We use logical GST to obtain the first full, self-consistent characterization of a universal single-qubit logical gate set, implemented in the seven-qubit Steane code using Quantinuum System Model H1, a 20-qubit trapped-ion quantum processor. In the experimental data from H1, logical GST identified predominantly logical Pauli-stochastic errors, found no evidence of logical non-Markovianity, and detected weak potential signatures of coherent logical error. By contrast, higher-shot datasets generated with Quantinuum's detailed system emulator, identify mechanisms for weak but statistically significant logical non-Markovianity and strong evidence of coherent logical error. Finally, we use logical GST to perform error attribution with the system emulator, revealing how specific physical error sources shape logical error mechanisms.
As quantum devices make steady progress towards intermediate scale and fault-tolerant quantum computing, it is essential to develop rigorous and efficient measurement protocols that account for known sources of noise. Most existing quantum characterization protocols such as gate-set tomography and randomized benchmarking assume the noise acting on the qubits is Markovian. However, this assumption is often not valid, as for the case of 1/f charge noise or hyperfine nuclear spin noise. Here, we present a general framework for quantum process tomography (QPT) in the presence of time-correlated noise. We further introduce fidelity benchmarks that quantify the relative strength of different sources of Markovian and non-Markovian noise. As an application of our method, we perform a comparative theoretical and experimental analysis of silicon spin qubits. We first develop a detailed noise model that accounts for the dominant sources of noise and validate the model against experimental data. Applying our framework for time-correlated QPT, we find that the number of independent parameters needed to characterize one- and two-qubit gates can be compressed by 10× and 100× , respectively, when compared to the fully generic case. These compressions reduce the amount of tomographic measurements needed in experiment, while also significantly speeding up numerical simulations of noisy quantum circuit dynamics compared to time-dependent Hamiltonian simulation. Using this compressed noise model, we find good agreement between our theoretically predicted process fidelities and two-qubit interleaved randomized benchmarking fidelities of 99.8% measured in recent experiments on silicon spin qubits. More broadly, our formalism can be directly extended to develop efficient and scalable tuning protocols for high-fidelity control of large arrays of quantum devices with non-Markovian noise. Published by the American Physical Society 2024
Abstract Gate set tomography (GST) allows for a self-consistent characterization of noisy quantum information processors (QIPs). The standard device-agnostic approach treats the QIPs as black boxes that are only constrained by the laws of physics, attaining full generality at aconsiderable resource cost: numerous circuits built from the gate set must be run in order to amplify each of the gate set parameters. In this work, we show that a microscopic parametrization of quantum gates under time-correlated noise on the driving phase, motivated by recent experiments with trapped-ion gates, reduces the required resources enabling a more efficient version of GST. By making use of the formalism of filter functions over the noise spectral densities, we discuss the minimal parametrizations of the gate set that include the effect of finite correlation times and non-Markovian quantum evolutions during the individual gates. We compare the estimated gate sets obtained by our method and the standard long-sequence GST, discussing their accuracies in terms of established metrics, as well as showcasing the advantages of the parametrized approach in terms of the sampling complexity for specific examples.
Quantum error correction provides a path to large-scale quantum computers, but is built on challenging assumptions about the characteristics of the underlying errors. In particular, the mathematical assumption of independent errors in quantum logic operations is at odds with realistic environments where error-sources may exhibit strong temporal correlations. We present experiments enabling the identification of error correlations between operations in quantum circuits, using only projective measurements at the end of the circuit. Using a single trapped ion qubit and engineered noise with tunable temporal correlations, we identify a clear signature of error correlations between sequential gates in randomly composed quantum circuits, and extract quantitative measures linked to the underlying noise correlation length. By replacing all gates in these circuits with "virtual" dynamically corrected gates (DCGs), we demonstrate that even in the presence of strongly correlated noise the signatures of error correlations between sequential gates appear similar to standard gates exposed to uncorrelated noise. A theoretical model applied to our experiments reveals that common DCGs suppress the correlated error component by over 270×270\times270× with 95%95\%95% confidence. Using block-correlated noise, we explore the scaling of the effective error correlation length at the virtual level, and show that DCGs exhibit error correlations indistinguishable from those arising from uncorrelated noise.