Les Houches lectures on random quantum circuits and monitored quantum dynamics

TL;DR

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.

Abstract

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.

Figures (7)

• Figure 1: Random quantum circuit. Brick-work random quantum circuit evolution of a one dimensional system of qudits (horizontal direction), implementing a random discrete time evolution (vertical direction). Each green rectangle is a unitary gate acting on two nearest-neighbor qudits, drawn uniformly using the Haar measure. The blue dots represent potential measurements that will be discussed later in these notes. Reproduced from Ref. Jian2020.
• Figure 2: Minimal cut through a quantum circuit. A quantum circuit can be viewed as a tensor network representation of the state created by the circuit: each gate is a 4-legged tensor with a special structure that enforces unitarity. The quantum entanglement of a subregion $A$ is upper bounded by the number of links that need to be cut to isolate region $A$ times the logarithm of the bond dimension. Reproduced from Ref. PhysRevX.7.031016.
• Figure 3: Statistical mechanics model: geometry. (a) Geometry of the statistical mechanics model of $S_Q$ spins. The red sites corresponds to the boundary spins to be pinned by the boundary condition. (b) In the $d=\infty$ limit, the model reduces to a Potts model on a square lattice. Reproduced from Ref. Jian2020.
• Figure 4: Statistical mechanics mapping. The calculation of the average purity in random unitary circuit dynamics can be mapped onto an effective Ising model, where the two spins label different contractions of the two replicas. After averaging, the tensor network can be contracted exactly, giving local weights in terms of those Ising spins. Figure credits: Kabir Khanna, adapted from Ref. khanna2025randomquantumcircuitstimereversal.
• Figure 5: Learnability transition. Accuracy (probability of success) of distinguishing two random orthogonal initial states from the monitored dynamics (given by a random quantum circuit of depth $2L$) with measurement rate $\gamma$, for a system of $L$ qubits. (Figure credit: Abhishek Kumar, adapted from Ref. kim2025learningmeasurementinducedphasetransitions.)