### Lectures on many-body dynamics in quantum circuits

I will discuss out-of-equilibrium dynamics in chaotic quantum many-body systems, emphasising that some basic dynamical questions about correlations and entanglement can be reduced to effective *classical* stat mech problems (involving random walks, domain walls, percolation, traveling waves, etc.).

### 1. Many-body dynamics far from the ground state

Quantum many-body dynamics far from the ground state: the search for universality.
(Quantum chaos and complexity; entanglement entropy; decoherence and the emergence of hydrodynamics.)
Many-body dynamics in the NISQ era.
Random quantum circuits as simple models.

### 2. Quantum correlation functions and classical Markov processes

Lieb-Robinson and causality.
Correlations in quantum circuits: mapping to a classical Markov process.
Two-point functions and spacetime trajectories.
“Unbinding” phase transition for spacetime trajectories.
Questions about efficient simulation of quantum systems.

### 3. Mapping between quantum circuits and classical lattice models

Entanglement entropy as free energy of a membrane in an effective classical model.
Coarse-grained evolution equation for the entanglement entropy.

### 4. Decoherence

Motivations: quantum trajectories of continually measured systems.
Entanglement transitions: schematic picture.
Replica picture (overview)

### 5. Entanglement transitions in random (tree) tensor networks

The simplest example of an entanglement phase transition and reduction to a recursion relation on a tree.

### Outlook and other topics