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arXiv:2111.14907

Random quantum circuits transform local noise into global white noise

Alexander M. Dalzell, Nicholas Hunter-Jones, Fernando G. S. L. Brandão

Published Nov 29, 2021• 12 citations
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Abstract

We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. We show that, for local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_{\text{noisy}}$ of a generic noisy circuit instance and the output distribution $p_{\text{ideal}}$ of the corresponding noiseless instance shrink exponentially with the expected number of gate-level errors, as $F=\text{exp}(-2sε\pm O(sε^2))$, where $ε$ is the probability of error per circuit location and $s$ is the number of two-qubit gates. Furthermore, if the noise is incoherent, the output distribution approaches the uniform distribution $p_{\text{unif}}$ at precisely the same rate and can be approximated as $p_{\text{noisy}} \approx Fp_{\text{ideal}} + (1-F)p_{\text{unif}}$, that is, local errors are scrambled by the random quantum circuit and contribute only white noise (uniform output). Importantly, we upper bound the total variation error (averaged over random circuit instance) in this approximation as $O(Fε\sqrt{s})$, so the "white-noise approximation" is meaningful when $ε\sqrt{s} \ll 1$, a quadratically weaker condition than the $εs\ll 1$ requirement to maintain high fidelity. The bound applies when the circuit size satisfies $s \geq Ω(n\log(n))$ and the inverse error rate satisfies $ε^{-1} \geq \tildeΩ(n)$. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; it was an underlying assumption in complexity-theoretic arguments that low-fidelity random quantum circuits cannot be efficiently sampled classically. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

DOI
10.48550/arxiv.2111.14907

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We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. We show that, for local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution pnoisyp_{\text{noisy}}pnoisy​ of a generic noisy circuit instance and the output distribution pidealp_{\text{ideal}}pideal​ of the corresponding noiseless instance shrink exponentially with the expected number of gate-level errors, as F=exp(−2sε±O(sε2))F=\text{exp}(-2sε\pm O(sε^2))F=exp(−2sε±O(sε2)), where εεε is the probability of error per circuit location and sss is the number of two-qubit gates. Furthermore, if the noise is incoherent, the output distribution approaches the uniform distribution punifp_{\text{unif}}punif​ at precisely the same rate and can be approximated as pnoisy≈Fpideal+(1−F)punifp_{\text{noisy}} \approx Fp_{\text{ideal}} + (1-F)p_{\text{unif}}pnoisy​≈Fpideal​+(1−F)punif​, that is, local errors are scrambled by the random quantum circuit and contribute only white noise (uniform output). Importantly, we upper bound the total variation error (averaged over random circuit instance) in this approximation as O(Fεs)O(Fε\sqrt{s})O(Fεs​), so the "white-noise approximation" is meaningful when εs≪1ε\sqrt{s} \ll 1εs​≪1, a quadratically weaker condition than the εs≪1εs\ll 1εs≪1 requirement to maintain high fidelity. The bound applies when the circuit size satisfies s≥Ω(nlog⁡(n))s \geq Ω(n\log(n))s≥Ω(nlog(n)) and the inverse error rate satisfies ε−1≥Ω~(n)ε^{-1} \geq \tildeΩ(n)ε−1≥Ω~(n). The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; it was an underlying assumption in complexity-theoretic arguments that low-fidelity random quantum circuits cannot be efficiently sampled classically. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.
Abstract We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution pnoisyp_{\text {noisy}}pnoisy​ p noisy and the corresponding noiseless output distribution pidealp_{\text {ideal}}pideal​ p ideal shrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmark F that measures this correlation behaves as F=exp(−2sϵ±O(sϵ2))F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))F=exp(−2sϵ±O(sϵ2)) F = exp ( - 2 s ϵ ± O ( s ϵ 2 ) ) , where ϵ\epsilon ϵ ϵ is the probability of error per circuit location and s is the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution pnoisyp_{\text {noisy}}pnoisy​ p noisy and the uniform distribution punifp_{\text {unif}}punif​ p unif decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as pnoisy≈Fpideal+(1−F)punifp_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}pnoisy​≈Fpideal​+(1−F)punif​ p noisy ≈ F p ideal + ( 1 - F ) p unif . In other words, although at least one local error occurs with probability 1−F1-F1−F 1 - F , the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by O(Fϵs)O(F\epsilon \sqrt{s})O(Fϵs​) O ( F ϵ s ) . Thus, the “white-noise approximation” is meaningful when ϵs≪1\epsilon \sqrt{s} \ll 1ϵs​≪1 ϵ s ≪ 1 , a quadratically weaker condition than the ϵs≪1\epsilon s\ll 1ϵs≪1 ϵ s ≪ 1
Random circuit sampling is a leading approach to demonstrating quantum computational advantage, but benchmarking noisy implementations through linear cross-entropy requires costly calculations of ideal output probabilities. We propose a metrological benchmark based on the response to controlled perturbations, characterized by quantum Fisher information (QFI). A relation between QFI and out-of-time-order correlators enables protocols with local or global control. For Haar-random circuits, the average QFI approaches its maximal value in the local protocol and grows linearly with the number of qubits under collective control. In contrast, Clifford circuits yield zero QFI despite extensive operator spreading, showing that the response probes dynamical properties beyond spreading alone. A butterfly protocol further uses system size to enhance sensitivity, yielding a mean inverse sensitivity proportional to the number of qubits with single-qubit readout. Its fluctuations distinguish the Haar and Clifford ensembles despite their identical mean responses. For noisy implementations, we derive an exact relation between the noisy and ideal QFI within a global white-noise model, providing a quantitative reference for the degradation of the benchmark. These protocols enable metrological benchmarking of noisy random quantum circuits without computing ideal output probabilities.
Present-day quantum processors are open systems in which dissipation and decoherence degrade the information carried by a quantum state as it propagates through a circuit. We study this degradation in dissipative random quantum circuits, modeling each two-qubit gate as a diluted unitary that interpolates between the intended unitary operation and a random dissipative quantum channel. Using the fidelity between the ideal and noisy output states as a diagnostic, we show that sufficiently random unitary gates or dissipative Kraus operators lead to a universal decay of the average fidelity. This decay is determined only by the dissipation strength, system size, and circuit depth, while microscopic details of the gates and noise, including the Kraus rank, appear only in subleading corrections to higher moments. We derive a closed-form expression for the average fidelity in terms of three physically meaningful error parameters, dissipation strength, coherent two-qubit gate-error strength, and connectivity-error probability. Finally, we identify the cases when dissipative effects can be reproduced by an unitary-noise model and when dissipation-induced decoherence remains distinguishable from coherent noise at the level of the average fidelity.
Noise is the defining feature of the NISQ era, but it remains unclear if noisy quantum devices are capable of quantum speedups. Quantum supremacy experiments have been a major step forward, but gaps remain between the theory behind these experiments and their actual implementations. In this work we initiate the study of the complexity of quantum random circuit sampling experiments with realistic amounts of noise. Actual quantum supremacy experiments have high levels of uncorrected noise and exponentially decaying fidelities. It is natural to ask if there is any signal of exponential complexity in these highly noisy devices. Surprisingly, we show that it remains hard to compute the output probabilities of noisy random quantum circuits without error correction. More formally, so long as the noise rate of the device is below the error detection threshold, we show it is #P-hard to compute the output probabilities of random circuits with a constant rate of noise per gate. This hardness persists even though these probabilities are exponentially close to uniform. Therefore the small deviations away from uniformity are hard to compute, formalizing an important intuition behind Google's supremacy claim. Interestingly these hardness results also have implications for the complexity of experiments in a low-noise setting. The issue here is that prior hardness results for computing output proba-bilities of random circuits are not robust enough to imprecision to connect with the Stockmeyer argument for hardness of sampling from circuits with constant fidelity. We exponentially improve the robustness of prior results to imprecision, both in the cases of Random Circuit Sampling and BosonSampling. In the latter case we bring the proven hardness within a constant factor in the exponent of the robustness required for hardness of sampling for the first time. We then show that our results are in tension with one another - the high-noise result implies the low-noise result is essentially optimal, even with generalizations of our techniques.
Random quantum circuits aim to efficiently reproduce the statistical properties of ideal random quantum evolution. One approach is to construct approximate unitary designs, which match the moments of Haar-random unitaries up to a prescribed order with controlled error. In this work, we establish quantitative guarantees for how accurately these designs reproduce the full distributions of individual output probabilities. We show that, for every strong ε\varepsilonε-approximate unitary kkk-design on nnn qubits, the distribution of each individual output probability is within O(2nk(k+2n)+ε)O\left(\sqrt{\frac{2^n}{k(k+2^n)}}+\varepsilon\right)O(k(k+2n)2n​​+ε) in Kolmogorov distance of the finite-dimensional Porter-Thomas distribution, a beta distribution with parameters 111 and 2n−12^n-12n−1. This bound is optimal up to constant factors, implying that a substantially better uniform bound requires additional structure. We further show that local invariance yields an exponential improvement in the dependence on kkk, with guarantees also in the stronger metric of total variation: every strong ε\varepsilonε-approximate unitary kkk-design invariant under unitary translations acting on log⁡k+O(1)\log k+O(1)logk+O(1) qubits achieves error 2−Ω(k)+O(ε)2^{-Ω(k)}+O(\varepsilon)2−Ω(k)+O(ε) in Kolmogorov distance and 2−Ω(k)+O(εlog⁡(2/ε))2^{-Ω(k)}+O(\varepsilon\log(2/\varepsilon))2−Ω(k)+O(εlog(2/ε)) in total variation distance. These results identify both the distributional accuracy guaranteed by moment matching and the additional structure that substantially improves it. As an application, we show that if the marginal distribution of each output probability of a strong approximate design is within Kolmogorov distance ηηη of its Haar counterpart, then the expected Shannon entropy of the output distribution differs from its Haar value by O(η)O(\sqrtη)O(η​).
Demonstrations of quantum computational advantage and benchmarks of quantum processors via quantum random circuit sampling are based on evaluating the linear cross-entropy benchmark (XEB). A key question in the theory of XEB is whether it approximates the fidelity of the quantum state preparation. Previous works have shown that the XEB generically approximates the fidelity in a regime where the noise rate per qudit ε\varepsilonε satisfies εN≪1\varepsilon N \ll 1εN≪1 for a system of NNN qudits and that this approximation breaks down at large noise rates. Here, we show that the breakdown of XEB as a fidelity proxy occurs as a sharp phase transition at a critical value of εN\varepsilon NεN that depends on the circuit architecture and properties of the two-qubit gates, including in particular their entangling power. We study the phase transition using a mapping of average two-copy quantities to statistical mechanics models in random quantum circuit architectures with full or one-dimensional connectivity. We explain the phase transition behavior in terms of spectral properties of the transfer matrix of the statistical mechanics model and identify two-qubit gate sets that exhibit the largest noise robustness.