The question of computability is at the center of most modern problems of multi-particle quantum mechanics. The key question here is: "What can we learn about the system using a classical or quantum computer?" In this course, we will try to understand how much information we can obtain, having access only to the computing power of a regular laptop. During the course, we will consider a number of routine questions that constantly arise in modern problems of multi-particle quantum physics. 

In particular, we will learn to effectively use exact diagonalization and integrability for analyzing properties of a multi-particle system, estimate the computational complexity of arising tasks, use spectral statistics and random matrix theory, as well as apply the quasi-classical approximation for quantum dynamics. We will also study classical and modern results in this field. Lectures will be conducted in the format of theory presentation, followed by the analysis of a specific problem from start to finish. Upon completion of the course, participants will receive a set of tools for continuing independent research. The course is designed for students starting from the 2nd year and above.  

Preliminary program

1. Introduction.
1.1 Basic concepts. (Schrodinger's equation, qubit, basis, quantum computations, quantum simulations, quasi-classical approximation. Various models of multi-particle physics).
1.2 Computational complexity of multi-particle problems. 
1.3 The problem of instability of quasi-classical solutions.

2. Low-dimensional systems.
2.1 Two-level systems (Bose-Hubbard dimer, Tavis-Cummings model). 
2.2 Coherent states, collapses and revivals.
2.3 Algebraic Bethe ansatz.
2.4* Dynamical Bethe equations.

3. Multi-particle systems.
3.1 Examples of multi-particle systems (spin chains, interacting electrons). 
3.2 Exact diagonalization, use of symmetries (translation invariance, reflection, etc.).
3.3 Application of random matrix theory, spectral statistics, $〈r〉$-value, Wigner-Dyson statistics, Poisson statistics.
3.4 Thouless time, entropy growth. 
3.5 Density of states, Gibbs ensemble, generalized Gibbs ensemble, average force. 
3.6 Quantum typicality. 
3.7 Quantum integrability and dynamic integrability. Onsager's algebra and closed hierarchy of Heisenberg equations. Different representations of Onsager's algebra.

4. Quasiclassical approximation.
4.1 Quasiclassical equations of motion, their instability, KAM theorem, Poincare section. 
4.2 Lyapunov spectrum, dependence of Lyapunov exponent on initial conditions.

5. Thermalization and its violations.
5.1 Classical chaotic billiards and quantum scars.
5.2 Quantum multi-particle scars. Onsager's scars.
5.3 Eigenstate Thermalization Hypothesis, Gauge adiabatic potential, Quantum butterfly effect.

Lecturer
Ermakov Igor Vladimirovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center