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.
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.
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 pnoisy of a generic noisy circuit instance and the output distribution pideal of the corresponding noiseless instance shrink exponentially with the expected number of gate-level errors, as F=exp(−2sε±O(sε2)), 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 punif at precisely the same rate and can be approximated as 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), so the "white-noise approximation" is meaningful when εs≪1, a quadratically weaker condition than the εs≪1 requirement to maintain high fidelity. The bound applies when the circuit size satisfies s≥Ω(nlog(n)) and the inverse error rate satisfies 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.
Random circuit sampling (RCS) remains one of the most competitive frameworks for demonstrating quantum advantage in near-term noisy intermediate-scale quantum (NISQ) hardware. Unfortunately, absent error-correction, existing benchmarks to characterize these experiments, like linear cross-entropy, have been classically spoofed due to noise. Because of this, there are interesting regimes, like shallow-depth random quantum circuits, where sampling is plausibly classically intractable, but no existing benchmark can distinguish between a noisy quantum computer and an adversarial classical spoofer. In this paper, we demonstrate that the nonlinear cross-entropy provides a sample-efficient benchmark for shallow-depth all-to-all random quantum circuits whose score cleanly separates noisy quantum computers from state-of-the-art classical spoofers, even in the presence of depolarizing noise. Further, we develop a binary classifier based on the notion of heavy output generation that features logarithmic sample complexity at short depth. Our evidence comes from exact analytic expressions for all-to-all Brownian circuit ensembles derived using replica tricks, and numerical simulations that corroborate these results for discrete Haar-random unitary circuits.
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 pnoisy p noisy and the corresponding noiseless output distribution 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 = exp ( - 2 s ϵ ± O ( s ϵ 2 ) ) , where 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 pnoisy p noisy and the uniform distribution punif p unif decays at precisely the same rate. Consequently, the noisy output distribution can be approximated as 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−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 ϵ s ) . Thus, the “white-noise approximation” is meaningful when ϵs≪1 ϵ s ≪ 1 , a quadratically weaker condition than the ϵs≪1 ϵ s ≪ 1
Quantum technologies work by utilizing properties inherent in quantum systems such as quantum coherence and quantum entanglement and are expected to be superior to classical counterparts for solving certain problems in science and engineering. The quantum technologies are, however, fragile against an interaction with an environment (decoherence) and in order to utilize them with high accuracy we need to develop error mitigation techniques which reduce decoherence effects. In this work, we analyze quantum error mitigation (QEM) protocol for quantum metrology in the presence of quantum noise. We demonstrate the effectiveness of our QEM protocol by analyzing three types of quantum Fisher information (QFI), ideal (error-free) QFI, noisy (erroneous) QFI, and quantum-error-mitigated QFI, and show both analytically and numerically that the scaling behaviors of quantum-error-mitigated QFI with respect to the number of probes become restored to the those exhibited in the ideal quantum metrology. Our QEM protocol is constructed by an ensemble of quantum circuits, namely QEM circuit groups, and has advantages such that it can be applied to noisy quantum metrology for any type of initial state as well as any type of the probe-system Hamiltonian, and it can be physically implemented in any type of quantum device. Furthermore, the quantum-error-mitigated QFI become approximately equal to the ideal QFI for almost any values of physical quantities to be sensed. Our protocol enables us to use quantum entanglement as a resource to perform high-sensitive quantum metrology even under the influence of quantum noise.
The quantum-trajectory description of a noisy quantum circuit is not unique: every Kraus representation of the noise channel defines a different unraveling, i.e., a different ensemble of pure states that averages to the same density matrix. While all unravelings share the same expectation values of observables, they generally differ in the amount of quantum resources carried by the individual trajectories. Here we study non-stabilizerness, or magic, quantified by the stabilizer Rényi entropy, in quantum-trajectory unravelings of Haar-random and T-doped random Clifford brickwork circuits subject to local dephasing noise of strength p. The canonical unraveling in terms of Pauli jumps is Clifford and leaves magic untouched, so that trajectories retain the Haar-random value of non-stabilizerness. Exploring the full two-parameter family of minimal unravelings, we find that the trajectory-averaged magic can be substantially reduced by a state-independent choice of Kraus operators, and that the optimal choice is not in general the projective-measurement unraveling, but a two-outcome generalized measurement in which the Kraus operators correspond to a weak measurement of the noisy qubit. The optimal unraveling reduces the extensive magic density of the trajectories by a p-dependent factor without altering its linear scaling with system size. At the same time, it produces a very broad distribution of magic over trajectories, with a sizable fraction of trajectories retaining Haar-random magic. Since the cost of stabilizer-based classical simulation grows exponentially with the magic of the state, these high-magic trajectories, rather than the average, control the simulation cost. Our results delineate the potential and the limitations of unraveling optimization as a route to the classical simulation of noisy quantum dynamics with stabilizer methods.
Cross-entropy benchmarking is a central technique adopted to certify a quantum chip in recent investigations. To better understand its mathematical foundation and develop new benchmarking schemes, the concept of ergodicity was introduced to random circuit sampling and it was found that the Haar random quantum circuit could satisfy an ergodicity condition—the average of certain types of postprocessing function over the output bit strings is close to the average over the unitary ensemble. For noiseless random circuits, it was proven that the ergodicity holds for polynomials of degree t with positive coefficients when the random circuits form a unitary 2 t -design. For strong enough noise, the ergodicity condition is violated, which suggests that ergodicity is a property that can be exploited to certify a quantum chip. The deviation of ergodicity was formulated as a measure for quantum chip benchmarking, and it was demonstrated that it can be used to estimate the circuit fidelity for global depolarizing noise and weakly correlated noise. For a quadratic postprocessing function, our framework recovered Google's result on estimating the circuit fidelity via linear cross-entropy benchmarking (XEB), and we gave a sufficient condition on the noise model characterizing when such estimation is valid. The results establish an interesting connection between ergodicity and noise in random circuits and provide new insights into designing quantum benchmarking schemes.