The authors uncover here a universal functional form governing anticoncentration, which quantifies how broadly the output probabilities of a random quantum circuit are distributed, across architectures, depths, and even finite system sizes. Using analytical models and extensive simulations, they show that these distributions are characterized by just two parameters. Their results highlight finite-size and finite-depth effects and provide a practical framework for benchmarking quantum devices, enabling validation of much larger systems than previously possible.
The authors uncover here a universal functional form governing anticoncentration, which quantifies how broadly the output probabilities of a random quantum circuit are distributed, across architectures, depths, and even finite system sizes. Using analytical models and extensive simulations, they show that these distributions are characterized by just two parameters. Their results highlight finite-size and finite-depth effects and provide a practical framework for benchmarking quantum devices, enabling validation of much larger systems than previously possible.
We establish universal behavior in the anticoncentration properties of random quantum circuits, demonstrating its broad independence from the circuit architecture. Specifically, universality emerges in a certain scaling limit and extends beyond the leading order, incorporating subleading corrections arising from the finite system size N. We compute these corrections through exact calculations on ensembles of random tensor network states and corroborate the results with analytical findings in the random phase model. We then identify a heuristic framework for generic brickwork circuits, conjecturing the universality of these corrections. We further support our claim of anticoncentration universality through extensive numerical simulations, capturing the distribution of overlaps for systems up to N=64 qudits and computing collision probabilities for systems up to N≤1024. Collectively, our results highlight the critical role of finite-size corrections and lead to a thorough understanding of the core phenomenology governing anticoncentration in quantum circuits.
We present universal properties of anticoncentration in noisy quantum circuits at finite depth. We develop a generic framework for single- and multi-qubit noise channels in the weak-noise limit and introduce an effective description in terms of a . Within this weak-noise regime, we show that distinct noise mechanisms act in a quantitatively similar way, yielding a universal distribution of bit-string probabilities that is largely independent of the microscopic noise channel and of the circuit architecture. We identify three depth-dependent regimes, each characterized by a distinct scaling of cross-entropy benchmarking ( XEB ) with 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 anticoncentration in generic finite-depth circuits and benchmark them against numerical simulations, finding excellent agreement even at shallow depths. Moreover, we show that, contrary to previous expectations, the late-time value of XEB provides direct access to the global circuit fidelity, even at large noise strengths. Our results are directly applicable to current quantum processors and demonstrate universal behavior beyond the pure random-matrix-theory regime, which only emerges at asymptotically large depths.
Universal statistical laws are a hallmark of quantum chaos. A natural route to probing them in quantum circuits is offered by the statistics of output probabilities, which for sufficiently deep chaotic circuits approach the Porter--Thomas distribution. Here we show, theoretically and experimentally, that the approach to this limit is itself universal, and is organized into geometry-dependent universality classes. At shallow depths, growing only logarithmically with system size, we derive universal output-probability distributions that differ markedly from Porter--Thomas statistics, and we test them on a programmable superconducting processor. Two-dimensional lattices, open one-dimensional chains, and periodic rings exhibit distinct universal forms, retaining signatures of geometry and boundary topology within the same logarithmic-depth regime. Within each class, the distributions depend on only two parameters: an effective many-body Thouless scale and the accumulated number of incoherent errors. Extracting these directly from measured bitstring frequencies provides access to scrambling, entanglement growth of the target state, and global fidelity, without reconstructing or classically simulating the implemented circuit. We also identify a noise-driven transition between output statistics retaining system-wide quantum correlations and those reproducible by independent small patches. Our results establish universality beyond Porter--Thomas statistics as a quantitative tool for characterizing noisy quantum processors.
We consider quantum circuits consisting of randomly chosen two-local gates and study the number of gates needed for the distribution over measurement outcomes for typical circuit instances to be anticoncentrated, roughly meaning that the probability mass is not too concentrated on a small number of measurement outcomes.An understanding of the conditions for anticoncentration is important for determining which quantum circuits are difficult to simulate classically, as anticoncentration has been in some cases an ingredient of mathematical arguments that simulation is hard and in other cases a necessary condition for easy simulation.Our definition of anticoncentration is that the expected collision probability of the distribution-that is, the probability that two independently drawn outcomes will agree-is only a constant factor larger than the collision probability for the uniform distribution.We show that when the two-local gates are each drawn from the Haar measure (or any 2-design), at least (n log(n)) gates (and thus (log(n)) circuit depth) are needed for this condition to be met on an n-qudit circuit.In both the case where the gates are nearest neighbor on a one-dimensional ring and the case where gates are long range, we show that O(n log(n)) gates are also sufficient and we precisely compute the optimal constant prefactor for the n log(n).The technique we employ relies upon a mapping from the expected collision probability to the partition function of an Ising-like classical statistical-mechanical model, which we manage to bound using stochastic and combinatorial techniques.
Porter-Thomas statistics are a characteristic feature of the output distribution of random quantum states and, more broadly, chaotic quantum many-body systems. Convergence to Porter-Thomas plays a central role in random circuit sampling and experimental demonstrations of quantum advantage, where the output statistics of low-depth random quantum circuits are expected to be approximately Porter-Thomas, despite the absence of a rigorous proof of convergence. We show that the output distribution of polynomial-depth brickwork random circuits converges inverse-polynomially in total variation distance to the Porter-Thomas distribution. Specifically, consider the output probability distribution over a fixed bitstring of a local random quantum circuit, constructed from nearest-neighbor Haar random gates. Then, for any m≥0, the distribution corresponding to circuits of depth O(n2m+1log(n)) is at most O(1/nm) far in total variation distance from the Porter-Thomas distribution. Our proof uses moment bounds from approximate designs, analytic estimates for characteristic functions, and a local anticoncentration property for inverse moments.
In this work, drawing inspiration from the type of noise present in real hardware, we study the output distribution of random quantum circuits under practical nonunital noise sources with constant noise rates. We show that even in the presence of unital sources such as the depolarizing channel, the distribution, under the combined noise channel, never resembles a maximally entropic distribution at any depth. To show this, we prove that the output distribution of such circuits never anticoncentrates—meaning that it is never too “flat”—regardless of the depth of the circuit. This is in stark contrast to the behavior of noiseless random quantum circuits or those with only unital noise, both of which anticoncentrate at sufficiently large depths. As a consequence, our results shows that the complexity of random-circuit sampling under realistic noise is still an open question, since anticoncentration is a critical property exploited by both state-of-the-art classical hardness and easiness results. Published by the American Physical Society 2024