Universality in the Anticoncentration of Noisy Quantum Circuits at Finite Depths

TL;DR

This work addresses anticoncentration in noisy quantum circuits at finite depth by establishing a universal PoP distribution governed by two scaling parameters x and η. Using a RMPO mapping and Weingarten calculus, the authors derive a transferable transfer-matrix form that yields universal moments I_k and the full PoP P_{x,η}(w), capturing the quantum to classical crossover as depth and noise vary. They identify three dynamical regimes with distinct XEB scalings, and show that late-time XEB can still reveal the global circuit fidelity F = e^{−η}, enabling practical benchmarking for near-term devices. The results hold across noise channels and circuit architectures, and are validated by extensive numerics, providing a scalable framework for studying anticoncentration in realistic noisy quantum processors with direct experimental relevance.

Abstract

We present universal properties of the anticoncentration of noisy quantum circuits at finite depth. By employing an effective model of random matrix product operators, we show that in the weak-noise regime different types of noise act in a similar fashion, leading to a universal distribution of bit-string probabilities, largely independent of the specific noise channel or circuit architecture. We identify three distinct depth-dependent regimes, each signaled by a different scaling of cross-entropy benchmarking (XEB) over rescaled depth. In the shallow-depth regime, noise effects are perturbatively small; in the intermediate regime, circuit-induced fluctuations and noise compete on equal footing; and in the deep-depth regime, the output distribution becomes effectively classical, up to corrections that are exponentially small in the noise strength. We provide quantitative predictions for the anticoncentration of generic circuits at finite depth, which we benchmark with numerical simulations displaying perfect agreement even for shallow circuits. Moreover, we show that, contrary to previous belief, the late-time XEB does give access to the global circuit fidelity, even for large noise strengths. Our findings are directly applicable to current quantum processors and demonstrate universal behavior beyond pure random-matrix-theory regimes which are only applicable at large depths.

Figures (8)

• Figure 1: Sketch of the work. (a) We consider the task of random circuit sampling. The circuit is composed by random unitary gates in the usual brickwall geometry, and it is affected by noise represented by a quantum channel $\mathcal{N}$. We explore noise acting either on single or two-adjacent qubits, all yielding similar conclusions. (b) Theoretical arguments show that the gate average of the replicated circuit gives rise to an effective one-dimensional statistical model of permutations. In this model, noise acts as an external field that biases the system toward permutations close to the identity. (c) As a result, the original spatio-temporal circuit becomes equivalent to a one-dimensional random Matrix Product Operator (MPO) which incorporate noise.
• Figure 2: Plot of the $\text{XEB}$ for noisy RMPO (a), and random brickwall circuits (b)-(c). Different noise models are considered: RMPO with depolarizing channel see Eq. \ref{['eq:mps_circuit']}, (a), random brickwall circuits with two-qubit depolarizing noise (b), or single-qubit amplitude damping noise (c). The noise strength is $\gamma \propto 1/(N r)$ for RMPO (a), or $\gamma \propto 1/(N t)$ for the circuit (b)-(c). The proportionality constant is indicated by the horizontal color bar at the top. Note that in panel (a), the label is $\gamma N r$ rather than $\gamma N t$. We plot $\Delta\log(1+\text{XEB})=\log(1+\text{XEB})-\lim_{x \to 0^+}\log(1+\text{XEB})$ as a function of the rescaled circuit depth $x \sim N e^{- t/ \tau}$. We identify three different scaling regimes fixed by the fitting parameter $\eta = - \log F$. For $x \gg \eta$, the scaling is with $x$ and is independent of $\eta$, showing that the system is effectively noiseless. Then for $x^2\sim \eta$, the scaling is with $x^2$, characteristic of strong noise. For $x<1$, it goes to its asymptotic value as a linear function of $x$. In panel (a) we also plot the prediction of Eq. \ref{['eq:xeb_sl']} (dashed black line).
• Figure 3: Plot of the PoP for a Haar-random brickwall circuit subject to two-qubit depolarizing noise with $\gamma N t = 2$ and system size $N = 32$. Different depths $t$ are explored (see values reported in the plots). At the shortest depth, the distribution is well approximated by the noiseless distribution (dotted red curve) $P_{x,0} \simeq P_{x,\eta}$, whereas the latter fails to accurately describe the data at larger depths. In contrast, the theoretically predicted distribution $P_{x,\eta}$ (black line) accurately reproduces the data. The Shifted Porter-Thomas (SPT), Eq. \ref{['eq:shiftedPT']}, corresponding to the distribution at infinite times, $x=0$, is also plotted for reference in the last plot. The values of $x$ and $\eta$ are obtained by the fitting procedure involving the $\text{XEB}$ data explained in the main text (sec. \ref{['sec:pop']}).
• Figure 4: Plot of the PoP for a Haar-random brickwall circuit subject to two-qubit depolarizing noise with two different values of $\gamma N$ and $N = 32$. We present both $\gamma N=0.1$ in (a),(b), and $\gamma N=1.0$ (c),(d). For each value of $\gamma N$, we consider two different depths $t$, and consequently $x$ (see values reported in the plots), corresponding to two different regimes of the $\text{XEB}$ (see Eq. \ref{['eq:xeb_sl_new']}), (a) $\lambda < 1/\tau$ and (b) $\lambda > 1/\tau$. The values of $x$ and $\lambda$ are obtained by the fitting procedure involving the $\text{XEB}$ data explained in the main text (sec. \ref{['sec:strong_noise']}).
• Figure 5: Plot of the $\text{XEB}$ for the brickwall circuit with (a) two-qubits depolarizing (b) single-qubit amplitude damping and depolarizing channels, with different $\gamma N$, and $N=1024$. We observe the transition at large times in the scaling of $\log(1+\text{XEB})$, from $-\lambda t$ to $-t/\tau$ as the noise strength $\gamma N$ is increased. The crossing happens at $\lambda=1/\tau$ as predicted by Eq. \ref{['eq:xeb_sl_new']} ($\tau= \tau_{\text{bw}}(2)$ in this model). In the regime $\lambda> 1/\tau$, while the slope of the decay is fixed by the noise-independent $\tau$, a shift proportional to $1/\lambda$ as predicted by Eq. \ref{['eq:xeb_sl_new']} is clearly visible, the latter allowing to determine the global fidelity.