Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2503.00119

Universality in the Anticoncentration of Chaotic Quantum Circuits

Anonymous, Beatrice Magni, Guglielmo Lami, Xhek Turkeshi, Jacopo De Nardis

Published Feb 28, 2025• 0 citations
arXivHTMLPDF

Abstract

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 $N \leq 1024$. 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.

DOI
10.48550/arxiv.2503.00119

Related papers

Universality in the anticoncentration of chaotic quantum circuits

Anonymous, Beatrice Magni, Guglielmo Lami +2 more

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.

Physical review. B./Physical review. B•Sep 25, 2025•9 citations

Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudomagic States

Beatrice Magni, Alexios Christopoulos, Andrea De Luca +1 more

Anticoncentration describes how an ensemble of quantum states spreads over the allowed Hilbert space, leading to statistically uniform output probability distributions. In this work, we investigate the anticoncentration of random Clifford circuits toward the overlap distribution of random stabilizer states. Using exact analytical techniques and extensive numerical simulations based on Clifford replica tensor networks, we demonstrate that random Clifford circuits fully anticoncentrate in logarithmic circuit depth; namely, higher-order moments of the overlap distribution converge to those of random stabilizer states. Moreover, we investigate the effect of introducing a controlled number of non-Clifford (magic) resources into Clifford circuits. We show that inserting a polylogarithmic in qudit number of T states is sufficient to drive the overlap distribution toward the Porter-Thomas statistics, effectively recovering full quantum randomness. In short, this fact presents doped tensor networks and shallow Clifford circuits as pseudomagic quantum states. Our results clarify the interplay between Clifford dynamics, magic-state injection, and quantum complexity, with implications for quantum circuit sampling, many-body quantum physics, and the benchmarking of quantum computational advantage.

arXiv:2502.20455•Physical Review X•Aug 15, 2025•24 citations

Universality in the Anticoncentration of Noisy Quantum Circuits at Finite Depths

Arman Sauliere, Guglielmo Lami, Corentin Boyer +2 more

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.

arXiv:2508.14975•PRX Quantum•Mar 17, 2026•4 citations

Phase transitions in sampling and error correction in local Brownian circuits

Subhayan Sahu, Shao-Kai Jian

We study the emergence of anticoncentration and approximate unitary design behavior in local Brownian circuits. The dynamics of circuit-averaged moments of the probability distribution and entropies of the output state can be represented as imaginary-time evolution with an effective local Hamiltonian in the replica space. This facilitates large-scale numerical simulation of the dynamics in $1+1$ dimensions of such circuit-averaged quantities using tensor network tools as well as identifying the various regimes of the Brownian circuit as distinct thermodynamic phases. In particular, we identify the emergence of anticoncentration as a sharp transition in the collision probability at $lnN$ timescale, where $N$ is the number of qubits. We also show evidence for a specific classical approximation algorithm undergoing a computational hardness transition at the same timescale. In the presence of noise, we show there is a noise-induced first-order phase transition in the linear cross entropy benchmark when the noise rate is scaled down as $1/N$. At longer times, the Brownian circuits approximate a unitary 2-design in $O(N)$ time. We directly probe the feasibility of quantum error correction by such circuits and identify a first-order transition at $O(N)$ timescales. The scaling behaviors for all these phase transitions are obtained from the large-scale numerics and corroborated by analyzing the spectrum of the effective replica Hamiltonian.

arXiv:2307.04267•Physical Review A•Apr 15, 2024•5 citations

Emergent universality in Kraus maps of quantum chaotic many-body dynamics

Qi Camm Huang, Wai-Keong Mok, Tobias Haug +1 more

Recent studies of "deep thermalization" have revealed universal physics in quantum many-body dynamics beyond equilibration towards Gibbs states: maximally random quantum state ensembles can emerge on local subsystems, generated by measurements on their complement. In this work, we further identify a new form of universality exhibited in the "finer fingerprints" of quantum dynamics for local subsystems, induced by global unitary time-evolution. Specifically, we consider the projected Kraus ensemble, an ensemble of Kraus operators obtained by unraveling the quantum channel on a small subsystem with respect to knowledge of the classical configurations of its complement. Our central result is a one-parameter random matrix Ansatz that captures the ensemble's emergent statistical behavior along particular spacetime scalings, valid for generic 1D circuit dynamics without conservation laws: the ensemble is described by the product of a complex Ginibre random matrix and an independent log-normal random real scalar. The former encodes scrambling within the subsystem, captured by rotational invariance of the Ginibre measure, whereas the latter encodes fluctuations in the Born probabilities, arising from locality of the underlying dynamics. Our Ansatz can be established in the special cases of dynamics generated by global Haar random unitaries and dual-unitary circuits, while we motivate it for generic circuits using arguments of spacetime duality and the multiplicative ergodic theorem on long products of spatial transfer matrices. Extensive numerical simulations for random and Floquet circuit models verify these predictions. This universality in Kraus operators also provides a microscopic mechanism for deep thermalization in generic 1D quantum circuit dynamics, and has implications for local quantum information recoverability involving classical side information tied to knowledge of the bath.

quant-pharXiv:2609.12689•Sep 14, 2026•0 citations

Random Quantum Circuits Anticoncentrate in Log Depth

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

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.

arXiv:2011.12277•PRX Quantum•Mar 1, 2022•81 citations
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 NNN. 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=64N = 64N=64 qudits and computing collision probabilities for systems up to N≤1024N \leq 1024N≤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.
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.
Anticoncentration describes how an ensemble of quantum states spreads over the allowed Hilbert space, leading to statistically uniform output probability distributions. In this work, we investigate the anticoncentration of random Clifford circuits toward the overlap distribution of random stabilizer states. Using exact analytical techniques and extensive numerical simulations based on Clifford replica tensor networks, we demonstrate that random Clifford circuits fully anticoncentrate in logarithmic circuit depth; namely, higher-order moments of the overlap distribution converge to those of random stabilizer states. Moreover, we investigate the effect of introducing a controlled number of non-Clifford (magic) resources into Clifford circuits. We show that inserting a polylogarithmic in qudit number of T states is sufficient to drive the overlap distribution toward the Porter-Thomas statistics, effectively recovering full quantum randomness. In short, this fact presents doped tensor networks and shallow Clifford circuits as pseudomagic quantum states. Our results clarify the interplay between Clifford dynamics, magic-state injection, and quantum complexity, with implications for quantum circuit sampling, many-body quantum physics, and the benchmarking of quantum computational advantage.
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.
We study the emergence of anticoncentration and approximate unitary design behavior in local Brownian circuits. The dynamics of circuit-averaged moments of the probability distribution and entropies of the output state can be represented as imaginary-time evolution with an effective local Hamiltonian in the replica space. This facilitates large-scale numerical simulation of the dynamics in 1+11+11+1 dimensions of such circuit-averaged quantities using tensor network tools as well as identifying the various regimes of the Brownian circuit as distinct thermodynamic phases. In particular, we identify the emergence of anticoncentration as a sharp transition in the collision probability at lnNlnNlnN timescale, where NNN is the number of qubits. We also show evidence for a specific classical approximation algorithm undergoing a computational hardness transition at the same timescale. In the presence of noise, we show there is a noise-induced first-order phase transition in the linear cross entropy benchmark when the noise rate is scaled down as 1/N1/N1/N. At longer times, the Brownian circuits approximate a unitary 2-design in O(N)O(N)O(N) time. We directly probe the feasibility of quantum error correction by such circuits and identify a first-order transition at O(N)O(N)O(N) timescales. The scaling behaviors for all these phase transitions are obtained from the large-scale numerics and corroborated by analyzing the spectrum of the effective replica Hamiltonian.
Recent studies of "deep thermalization" have revealed universal physics in quantum many-body dynamics beyond equilibration towards Gibbs states: maximally random quantum state ensembles can emerge on local subsystems, generated by measurements on their complement. In this work, we further identify a new form of universality exhibited in the "finer fingerprints" of quantum dynamics for local subsystems, induced by global unitary time-evolution. Specifically, we consider the projected Kraus ensemble, an ensemble of Kraus operators obtained by unraveling the quantum channel on a small subsystem with respect to knowledge of the classical configurations of its complement. Our central result is a one-parameter random matrix Ansatz that captures the ensemble's emergent statistical behavior along particular spacetime scalings, valid for generic 1D circuit dynamics without conservation laws: the ensemble is described by the product of a complex Ginibre random matrix and an independent log-normal random real scalar. The former encodes scrambling within the subsystem, captured by rotational invariance of the Ginibre measure, whereas the latter encodes fluctuations in the Born probabilities, arising from locality of the underlying dynamics. Our Ansatz can be established in the special cases of dynamics generated by global Haar random unitaries and dual-unitary circuits, while we motivate it for generic circuits using arguments of spacetime duality and the multiplicative ergodic theorem on long products of spatial transfer matrices. Extensive numerical simulations for random and Floquet circuit models verify these predictions. This universality in Kraus operators also provides a microscopic mechanism for deep thermalization in generic 1D quantum circuit dynamics, and has implications for local quantum information recoverability involving classical side information tied to knowledge of the bath.
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.