Lecture Notes on Replica Tensor Networks for Random Quantum Circuits

We present a pedagogical, hands-on tutorial on replica tensor-network techniques for random quantum circuits. At its core, the method recasts circuit-averaged observables acting on multiple copies of the system as the contraction of a classical tensor network, equivalently the partition function of a statistical-mechanics model whose effective spins live in the commutant of the gate ensemble. The framework is general: changing the observable or the initial state modifies only the replica boundary conditions, while changing the ensemble modifies the bulk tensors. Focusing on quantum-information diagnostics, from metrics of wavefunction spreadings to entanglement quantifiers, we illustrate the approach in both clean and noisy random unitary circuits. We then briefly explain how the methodology extends to other ensembles, such as orthogonal or Clifford circuits. The lecture notes are accompanied by \texttt{ReplicaTN}, a self-contained C++/Python library and pedagogical notebooks.