Random and Small-scale Quantum Ergodicity

Understanding how isolated quantum many-body systems thermalize remains a central question in modern physics. We study the onset of ergodicity in a two-dimensional disordered Heisenberg Floquet model using digital quantum simulation on IBM's Nighthawk superconducting processor, reaching system sizes of up to $10\times10$ qubits. We probe ergodicity across different length scales by coarse-graining the system into spatial patches of varying sizes and introducing a measure based on the collision entropy of each patch, enabling a detailed study of when ergodic behavior emerges across scales. The high sampling rate of superconducting quantum processing units, together with an optimal sample estimator, allow us to access patches of sizes up to $3\times3$ . We observe that as the Heisenberg coupling $J$ increases, the noiseless system undergoes a smooth crossover from subergodic to ergodic behavior, with smaller patches approaching their random-matrix-theory values first, thereby revealing a hierarchy across scales. In the region of parameter space where classical tensor-network simulations are reliable, small patches or small values of $J$ , we find excellent agreement with the error-mitigated quantum simulation. Beyond this regime, volume-law entanglement and contraction complexity growth causes the cost of classical methods to rise sharply. Our results open new directions for the use of quantum computers in the study of quantum thermalization.Understanding how isolated quantum many-body systems thermalize remains a central question in modern physics. We study the onset of ergodicity in a two-dimensional disordered Heisenberg Floquet model using digital quantum simulation on IBM's Nighthawk superconducting processor, reaching system sizes of up to $10\times10$ qubits. We probe ergodicity across different length scales by coarse-graining the system into spatial patches of varying sizes and introducing a measure based on the collision entropy of each patch, enabling a detailed study of when ergodic behavior emerges across scales. The high sampling rate of superconducting quantum processing units, together with an optimal sample estimator, allow us to access patches of sizes up to $3\times3$. We observe that as the Heisenberg coupling $J$ increases, the noiseless system undergoes a smooth crossover from subergodic to ergodic behavior, with smaller patches approaching their random-matrix-theory values first, thereby revealing a hierarchy across scales. In the region of parameter space where classical tensor-network simulations are reliable, small patches or small values of $J$, we find excellent agreement with the error-mitigated quantum simulation. Beyond this regime, volume-law entanglement and contraction complexity growth causes the cost of classical methods to rise sharply. Our results open new directions for the use of quantum computers in the study of quantum thermalization.

quant-pharXiv:2603.12236Mar 13, 20260 citations

Quantum Ergodicity and Chaos

Göran Lindblad

NATO ASI series. Series B : PhysicsJan 1, 19868 citations

Quantum theory of measurement and ergodicity conditions

A. Daneri, A. Loinger, Giovanni Maria Prosperi

Nuclear PhysicsMay 1, 1962347 citations

General relation between quantum ergodicity and fidelity of quantum dynamics

Tomaž Prosen

A general relation is derived, which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the Hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time scale proportional to delta(-2), delta approximately strength of perturbation, whereas faster, typically Gaussian decay on shorter time scale proportional delta(-1) is predicted for integrable, or generally nonergodic dynamics. This result needs the perturbation delta to be sufficiently small such that the fidelity decay time scale is larger than any (quantum) relaxation time, e.g., mixing time for mixing dynamics, or averaging time for nonergodic dynamics (or Ehrenfest time for wave packets in systems with chaotic classical limit). Our surprising predictions are demonstrated in a quantum Ising spin-(1/2) chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of nonergodic and nonintegrable motion in the thermodynamic limit.A general relation is derived, which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the Hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time scale proportional to delta(-2), delta approximately strength of perturbation, whereas faster, typically Gaussian decay on shorter time scale proportional delta(-1) is predicted for integrable, or generally nonergodic dynamics. This result needs the perturbation delta to be sufficiently small such that the fidelity decay time scale is larger than any (quantum) relaxation time, e.g., mixing time for mixing dynamics, or averaging time for nonergodic dynamics (or Ehrenfest time for wave packets in systems with chaotic classical limit). Our surprising predictions are demonstrated in a quantum Ising spin-(1/2) chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of nonergodic and nonintegrable motion in the thermodynamic limit.

arXiv:quant-ph/0106149Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsFeb 11, 2002187 citations

Implications of classical ergodicity in quantum mechanics

Annals of PhysicsApr 1, 19720 citations

Time-inhomogeneous Quantum Markov Chains with Decoherence on Finite State Spaces

Chia-Han Chou, Wei-Shih Yang

We introduce and study time-inhomogeneous quantum Markov chains with parameter $ζ\ge 0$ and decoherence parameter $0 \leq p \leq 1$ on finite spaces and their large scale equilibrium properties. Here $ζ$ resembles the inverse temperature in the annealing random process and $p$ is the decoherence strength of the quantum system. Numerical evaluations show that if $ ζ$ is small, then quantum Markov chain is ergodic for all $0 < p \le 1$ and if $ ζ$ is large, then it has multiple limiting distributions for all $0 < p \le 1$ . In this paper, we prove the ergodic property in the high temperature region $0 \le ζ\le 1$ . We expect that the phase transition occurs at the critical point $ζ_c=1$ . For coherence case $p=0$ , a critical behavior of periodicity also appears at critical point $ζ_o=2$ .We introduce and study time-inhomogeneous quantum Markov chains with parameter $ζ\ge 0$ and decoherence parameter $0 \leq p \leq 1$ on finite spaces and their large scale equilibrium properties. Here $ζ$ resembles the inverse temperature in the annealing random process and $p$ is the decoherence strength of the quantum system. Numerical evaluations show that if $ ζ$ is small, then quantum Markov chain is ergodic for all $0 < p \le 1$ and if $ ζ$ is large, then it has multiple limiting distributions for all $0 < p \le 1$. In this paper, we prove the ergodic property in the high temperature region $0 \le ζ\le 1$. We expect that the phase transition occurs at the critical point $ζ_c=1$. For coherence case $p=0$, a critical behavior of periodicity also appears at critical point $ζ_o=2$.

arXiv:2012.05449Dec 10, 20200 citations

10\times10

qubits. We probe ergodicity across different length scales by coarse-graining the system into spatial patches of varying sizes and introducing a measure based on the collision entropy of each patch, enabling a detailed study of when ergodic behavior emerges across scales. The high sampling rate of superconducting quantum processing units, together with an optimal sample estimator, allow us to access patches of sizes up to

3\times3

. We observe that as the Heisenberg coupling

J

increases, the noiseless system undergoes a smooth crossover from subergodic to ergodic behavior, with smaller patches approaching their random-matrix-theory values first, thereby revealing a hierarchy across scales. In the region of parameter space where classical tensor-network simulations are reliable, small patches or small values of

J

, we find excellent agreement with the error-mitigated quantum simulation. Beyond this regime, volume-law entanglement and contraction complexity growth causes the cost of classical methods to rise sharply. Our results open new directions for the use of quantum computers in the study of quantum thermalization.

We introduce and study time-inhomogeneous quantum Markov chains with parameter

ζ\ge 0

and decoherence parameter

0 \leq p \leq 1

on finite spaces and their large scale equilibrium properties. Here

ζ

resembles the inverse temperature in the annealing random process and

p

is the decoherence strength of the quantum system. Numerical evaluations show that if $ ζ$ is small, then quantum Markov chain is ergodic for all

0 < p \le 1

and if $ ζ$ is large, then it has multiple limiting distributions for all

0 < p \le 1

. In this paper, we prove the ergodic property in the high temperature region

0 \le ζ\le 1

. We expect that the phase transition occurs at the critical point

ζ_c=1

. For coherence case

p=0

, a critical behavior of periodicity also appears at critical point

ζ_o=2