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Non-Equilibrium Quantum Many-Body Physics with Quantum Circuits

Bruno Bertini

TL;DR

The notes introduce brickwork quantum circuits (BQCs) as a practical framework for studying non-equilibrium quantum many-body dynamics with local interactions, and connect BQCs to local-Hamiltonian dynamics via Suzuki–Trotter and Osborne constructions. They develop exact results for solvable entanglement dynamics in random and dual-unitary circuits, revealing universal features of finite-time relaxation, entanglement growth, and local operator spreading. A key focus is on dual-unitary circuits, where space-time interchange symmetry yields analytically tractable behavior for both entanglement growth and spectral statistics, including maximal LOE and random-matrix-like spectral form factors. The work provides concrete, solvable benchmarks for chaos versus integrability in many-body quantum dynamics, with implications for understanding thermalization, entanglement propagation, and the role of circuit architecture in non-equilibrium quantum physics.

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.

Non-Equilibrium Quantum Many-Body Physics with Quantum Circuits

TL;DR

The notes introduce brickwork quantum circuits (BQCs) as a practical framework for studying non-equilibrium quantum many-body dynamics with local interactions, and connect BQCs to local-Hamiltonian dynamics via Suzuki–Trotter and Osborne constructions. They develop exact results for solvable entanglement dynamics in random and dual-unitary circuits, revealing universal features of finite-time relaxation, entanglement growth, and local operator spreading. A key focus is on dual-unitary circuits, where space-time interchange symmetry yields analytically tractable behavior for both entanglement growth and spectral statistics, including maximal LOE and random-matrix-like spectral form factors. The work provides concrete, solvable benchmarks for chaos versus integrability in many-body quantum dynamics, with implications for understanding thermalization, entanglement propagation, and the role of circuit architecture in non-equilibrium quantum physics.

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.
Paper Structure (19 sections, 4 theorems, 103 equations, 6 figures)

This paper contains 19 sections, 4 theorems, 103 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Concrete Setting: Quantum Quench
  3. Brickwork quantum circuits
  4. Diagrammatic representation
  5. Basic physical properties of BQC
  6. Relation between BQC and Hamiltonian systems
  7. Suzuki--Trotter decomposition
  8. Osborne Construction
  9. General strategy of these lectures.
  10. Dynamics in BQC: a first look
  11. Finite-time evolution
  12. Rényi entropies in BQCs
  13. Circuits with solvable entanglement dynamics
  14. Random unitary circuits
  15. Dual-unitary circuits
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mathcal{O}_X$ be an operator acting non-trivially in the block $X$ and let $A$ and $B$ be two blocks at distance $\ell$. Then where $D$, $v$, and $\xi$ are system-dependent constants ($D$ also depends on $A$, $B$, and the operators).

Figures (6)

  • Figure 1.1: Setting in the one dimensional case.
  • Figure 1.2: Schematic depiction of the light cone of a local operator in a BQC (left) and local Hamiltonian system (right).
  • Figure 1.3: The light cone opens up when increasing $n$ for fixed $t$.
  • Figure 2.4: Leading-order evolution of the entanglement entropy in generic systems with local interactions.
  • Figure 3.5: Causal light cone.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem : Lieb and Robinson (1972)
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Theorem