© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2601.22375 · SciPost Physics Lecture Notes

Non-Equilibrium Quantum Many-Body Physics with Quantum Circuits

Bruno Bertini

cond-mat.stat-mechquant-phSubmitted Jan 29, 2026• Announced Feb 2, 2026• 0 citations• Revised May 20, 2026 (v2)

v1 · Jan 29, 2026 v2 · May 20, 2026

Abstract

These are the notes for the 4.5-hour course with the same title that I delivered in August 2025 at the Les Houches summer school ``Exact Solvability and Quantum Information''. In these notes I pedagogically introduce the setting of brickwork quantum circuits and show that it provides a useful framework to study non-equilibrium quantum many-body dynamics in the presence of local interactions. I first show that brickwork quantum circuits evolve quantum correlations in a way that is fundamentally similar to local Hamiltonians, and then present examples of brickwork quantum circuits where, surprisingly, one can compute exactly several relevant dynamical and spectral properties in the presence of non-trivial interactions.

Journal reference: SciPost Phys. Lect. Notes 124 (2026)
DOI: 10.21468/SciPostPhysLectNotes.124
Comments: 49 pages, 6 figures; v2 minor changes

Related papers

Les Houches lectures on random quantum circuits and monitored quantum dynamics

Romain Vasseur

These lecture notes are based on lectures given by the author at the Les Houches 2025 summer school on "Exact Solvability and Quantum Information". The central theme of these notes is to apply the philosophy of statistical mechanics to study the dynamics of quantum information in ideal and monitored random quantum circuits -- for which an exact description of individual realizations is expected to be generically intractable.

quant-pharXiv:2602.17258•SciPost Physics Lecture Notes•Feb 20, 2026•1 citation

Entanglement spreading in non-equilibrium integrable systems

Pasquale Calabrese

These are lecture notes for a short course given at the Les Houches Summer School on ``Integrability in Atomic and Condensed Matter Physics'', in summer 2018. Here, I pedagogically discuss recent advances in the study of the entanglement spreading during the non-equilibrium dynamics of isolated integrable quantum systems. I first introduce the idea that the stationary thermodynamic entropy is the entanglement accumulated during the non-equilibrium dynamics and then join such an idea with the quasiparticle picture for the entanglement spreading to provide quantitive predictions for the time evolution of the entanglement entropy in arbitrary integrable models, regardless of the interaction strength.

arXiv:2008.11080•SciPost Physics Lecture Notes•Dec 1, 2020•77 citations

Random quantum circuits, chaos and quantum thermalization

J. T. Chalker

These notes accompany lectures given in June 2025 at the summer school \emph{Fundamental Problems in Statistical Physics XVI}. They offer a short introduction to random quantum circuits as simple models for generic many-body quantum systems. They give an outline of the motivation for introducing these models, starting from ideas of random matrix theory. They also provide a sketch of calculations of some of the quantities of most physical interest, based on an average over an ensemble of systems. These quantities give insights into operator spreading, entanglement dynamics and spectral correlations.

quant-pharXiv:2608.21303•Journal of Statistical Mechanics Theory and Experiment•Aug 24, 2026•0 citations

Many-body localization: an introduction for mathematicians

François Huveneers

These notes are based on a mini-course delivered at the Les Houches Summer School, Quantum Theory on All Scales, in August 2026. They introduce many-body localization to mathematicians, emphasizing the physical mechanisms that support or destabilize it. The mathematical discussion deals with how perturbative rotations, renormalization, and probabilistic estimates are used in a rigorous approach to many-body localization.

arXiv:2609.39462•Sep 30, 2026•0 citations

Nonequilibrium quantum fields: from cold atoms to cosmology

Jürgen Berges

Lecture notes of the Les Houches Summer School on 'Strongly interacting quantum systems out of equilibrium'.

arXiv:1503.02907•Oxford University Press eBooks•Jun 23, 2016•80 citations

Asymptotically Solvable Quantum Circuits

Samuel H. Pickering, Bruno Bertini

The discovery of chaotic quantum circuits with (partially) solvable dynamics has played a key role in our understanding of non-equilibrium quantum matter and, at the same time, has helped the development of concrete platforms for quantum computation. It was shown that solvability does not prevent the generation of chaotic dynamics, however, it imposes non-trivial constraints on the generated correlations. A natural question is then whether it is possible to gain insight into the generic case despite the latter being very hard to access. To address this question here we introduce a family of 'asymptotically solvable' quantum circuits where the solvability constraints only affect correlations on length scales beyond a tuneable threshold. This means that their dynamics are only solvable for long enough times: for times shorter than the threshold they are generic. We show this by computing both their dynamical correlations on the equilibrium (infinite temperature) state and their thermalisation dynamics following quantum quenches from compatible (asymptotically solvable) non-equilibrium initial states. The class of systems we introduce is generically ergodic but contains a non-interacting point, which we use to provide exact analytical results, complementing those of numerical experiments, on the non-solvable early time regime.

cond-mat.stat-mechCross-listedarXiv:2602.24276•PRX Quantum•Mar 2, 2026•0 citations

These are the notes for the 4.5-hour course with the same title that I delivered in August 2025 at the Les Houches summer school ``Exact Solvability and Quantum Information''. In these notes I pedagogically introduce the setting of brickwork quantum circuits and show that it provides a useful framework to study non-equilibrium quantum many-body dynamics in the presence of local interactions. I first show that brickwork quantum circuits evolve quantum correlations in a way that is fundamentally similar to local Hamiltonians, and then present examples of brickwork quantum circuits where, surprisingly, one can compute exactly several relevant dynamical and spectral properties in the presence of non-trivial interactions.

These are lecture notes for a short course given at the Les Houches Summer School on ``Integrability in Atomic and Condensed Matter Physics'', in summer 2018. Here, I pedagogically discuss recent advances in the study of the entanglement spreading during the non-equilibrium dynamics of isolated integrable quantum systems. I first introduce the idea that the stationary thermodynamic entropy is the entanglement accumulated during the non-equilibrium dynamics and then join such an idea with the quasiparticle picture for the entanglement spreading to provide quantitive predictions for the time evolution of the entanglement entropy in arbitrary integrable models, regardless of the interaction strength.

These notes accompany lectures given in June 2025 at the summer school \emph{Fundamental Problems in Statistical Physics XVI}. They offer a short introduction to random quantum circuits as simple models for generic many-body quantum systems. They give an outline of the motivation for introducing these models, starting from ideas of random matrix theory. They also provide a sketch of calculations of some of the quantities of most physical interest, based on an average over an ensemble of systems. These quantities give insights into operator spreading, entanglement dynamics and spectral correlations.

The discovery of chaotic quantum circuits with (partially) solvable dynamics has played a key role in our understanding of non-equilibrium quantum matter and, at the same time, has helped the development of concrete platforms for quantum computation. It was shown that solvability does not prevent the generation of chaotic dynamics, however, it imposes non-trivial constraints on the generated correlations. A natural question is then whether it is possible to gain insight into the generic case despite the latter being very hard to access. To address this question here we introduce a family of 'asymptotically solvable' quantum circuits where the solvability constraints only affect correlations on length scales beyond a tuneable threshold. This means that their dynamics are only solvable for long enough times: for times shorter than the threshold they are generic. We show this by computing both their dynamical correlations on the equilibrium (infinite temperature) state and their thermalisation dynamics following quantum quenches from compatible (asymptotically solvable) non-equilibrium initial states. The class of systems we introduce is generically ergodic but contains a non-interacting point, which we use to provide exact analytical results, complementing those of numerical experiments, on the non-solvable early time regime.