Random quantum circuits transform local noise into global white noise

TL;DR

This work analyzes noisy random quantum circuits in the low-fidelity regime and proves that local unital noise is rapidly scrambled, effectively becoming white noise. It shows the noisy output distribution p_noisy is well approximated by a white-noise mixture p_wn = F p_ideal + (1−F) p_unif with F ≈ exp(−2sε), and provides an explicit bound on the average total-variation distance to p_wn that scales as O(Fε√s) under conditions s ≥ Ω(n log n) and ε^{-1} ≥ Ω(n). The study also proves exponential convergence of p_noisy to the uniform distribution for incoherent noise, and derives a quantitative bound on how close p_noisy is to p_wn (via δ and F) when the noise is incoherent; the latter supports the use of linear cross-entropy benchmarking and has implications for the hardness of sampling from noisy outputs. The methodology maps second-moment quantities to stochastic processes, enabling rigorous analysis of error scrambling, and numerical results for complete-graph architectures with parameters akin to recent experiments suggest a practical regime where the white-noise approximation is meaningful. Overall, the results provide a robust theoretical footing for salvaging signals from noisy random quantum circuits and for understanding the noise resilience of near-term quantum computational tasks.

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.

Figures (6)

• Figure 1: Example of a noisy quantum circuit diagram on $n=4$ qudits with $s=5$ two-qudit gates. A pair of single-qudit noise channels $\mathcal{N}$ follow each two-qudit gate. The circuit begins and ends with a layer of noiseless single-qudit gates.
• Figure 2: Toy example where global Haar-random gates $U^{(t)}$ act in between a depolarizing noise channel on a single qudit. In this model we can exactly compute quantities $Z_0$, $Z_1$, and $Z_2$ because the global Haar-random gates cause the probability mass in the stochastic process to fully re-equilibrate to one of the fixed points, $I^n$ or $S^n$.
• Figure 3: Plot of the numerically calculated upper bound on the expected total variation distance between $p_{\text{noisy}}$ and $p_{\text{wn}}$ divided by $F$ for a complete-graph version of recent random quantum circuit experiments by Google (53 qubits) Arute2019GoogleQuantumSupremacy and USTC (60 qubits) USTC2021Zhuchongzhi2.1. The large dots represent the circuit sizes (number of two-qubit gates) implemented in those experiments. The dotted black line is the function $2\epsilon\sqrt{s}/3$ for each experiment.
• Figure 4: Plot of the numerically calculated upper bound on the expected total variation distance between $p_{\text{noisy}}$ and $p_{\text{wn}}$ divided by $F$ for the complete-graph architecture at various values of $n$, $\epsilon$ and $s$. For each value of $n$, a threshold in $\epsilon$ is observed where error rates above the threshold lead to a bad approximation, while error rates below the threshold lead the approximation to become $O(F\epsilon \sqrt{s})$ once $s$ is sufficiently large. The threshold value of $\epsilon$ appears to be roughly $0.3/n$.
• Figure 5: Illustration of dynamics of coupled noiseless and noisy stochastic process. The gate at time step $t$ acts on sites $\{i_t,j_t\}$; the transition from time step $t-1$ to time step $t$ can modify the assignment only at these locations. In the example above, at time step $t-1$ (left), both the $X$ and $Y$ systems are assigned $I$ at position $i_t$ and $S$ at position $j_t$. Since $i_t$ and $j_t$ are assigned different values, the transformation $R_0^{(t)}$ forces a bit flip at one of the positions, but the same bit is flipped for the $X$ and $Y$ systems. In this example, the $I$ is flipped to $S$. Then the configuration at time step $t$ (right) is formed by applying noise operators $Q'^{(t)}_\sigma Q^{(t)}_\sigma$ only to the $Y$ copy, which results in a bit flip from $S$ to $I$ independently on each location with probability $\sigma$. In the example above, only the $i_t$ assignment is flipped. The system $W$ captures the difference between the $X$ and $Y$ copies; it is assigned $S$ wherever they agree and $I$ wherever they disagree. This formalism allows us to isolate the impact of the noise on a trajectory of the stochastic process compared to what "would have" happened had there been no noise.

Theorems & Definitions (50)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1: Regularly connected dalzell2020anticoncentration
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• proof