Les Houches lectures on random quantum circuits and monitored quantum dynamics

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.

💡 Research Summary

These lecture notes, based on talks given at the Les Houches 2025 summer school, present a unified statistical‑mechanics framework for studying information dynamics in random quantum circuits (RQCs) and their monitored extensions where projective measurements are interleaved with unitary evolution. The author begins by introducing the brick‑work architecture of one‑dimensional RQCs: a chain of L qudits (local dimension d) evolves under layers of two‑site Haar‑random gates arranged in an even‑odd pattern. Starting from a product state, the entanglement entropy of a subregion A grows linearly in time and eventually saturates to a volume‑law value, reflecting thermalization in the absence of conserved quantities.

To quantify entanglement, Rényi entropies (S^{(n)}_A) are expressed as expectation values of permutation operators acting on n‑fold replicated states. This representation makes the connection to tensor‑network language explicit: the circuit can be viewed as a network of four‑leg tensors, and the “minimal cut” construction provides a geometric upper bound on the entanglement, (S_A\le k\ln d), where k is the number of bonds intersected by the cut.

The core technical tool is Haar averaging over the random gates. Using Schur‑Weyl duality, the average of ((U\otimes U^\ast)^Q) reduces to a sum over permutations in the replica space, weighted by Weingarten functions. For Q=2 this yields a two‑state (↑,↓) Ising variable on each leg of the circuit. Replacing each gate by its averaged projector maps the entire circuit onto an anisotropic Ising model defined on a hexagonal lattice. The model possesses a global (\mathbb{Z}_2) symmetry and boundary conditions that encode the entanglement region. In the limit of large local dimension (d\to\infty) the Ising couplings simplify, and the model becomes equivalent to classical bond percolation on a square lattice. In this regime the entanglement growth rate is directly proportional to the length of the minimal cut, reproducing the well‑known “minimal‑cut picture”.

For finite d the same spin model remains valid but the Boltzmann weights retain a non‑trivial d‑dependence, defining a distinct “finite‑d universality class”. Critical exponents and the location of the entanglement transition shift with d, a fact that can be probed numerically and experimentally.

The notes then turn to monitored circuits, where each time step may also include local projective measurements with probability p. The competition between unitary scrambling (which creates entanglement) and measurements (which tend to collapse it) gives rise to measurement‑induced phase transitions (MIPTs). Three complementary viewpoints are discussed: (i) an entanglement transition from volume‑law to area‑law scaling, (ii) a purification transition where the mixed state of the whole system becomes pure after a finite time, and (iii) a learnability transition that quantifies how much information about the initial state can be inferred from measurement outcomes. The replica trick is again employed to handle the non‑linear dependence of Rényi entropies on the random circuit and measurement outcomes. After averaging, the replicated system maps onto a statistical‑mechanics model with domain‑wall free energies that depend on the measurement rate p. The model exhibits conformal invariance at the critical point, and its critical exponents match those of two‑dimensional percolation, confirming that the MIPT belongs to the percolation universality class in the large‑d limit. For finite d, deviations from pure percolation are captured by the same finite‑d Ising model introduced earlier.

A practical discussion of the “post‑selection problem” follows: although exact post‑selection of measurement outcomes is exponentially costly, the authors argue that typical experimental protocols that sample measurement records already reproduce the ensemble‑averaged quantities of interest, mitigating the computational obstacle.

Finally, the author outlines open directions, including extensions to higher spatial dimensions, non‑Haar gate ensembles, connections to quantum error correction, and the role of symmetries in modifying the universality class. Throughout, the emphasis is on exact analytical constructions—Haar averaging, replica mapping, and minimal‑cut geometry—that turn otherwise intractable quantum dynamics into solvable classical statistical‑mechanics problems. This synthesis provides a powerful toolbox for researchers studying scrambling, thermalization, and information flow in near‑term quantum devices.