Master course co-taught with Markus Heinrich: randomness in quantum systems as a toolbox for quantum information and many-body physics.
Quantum information asks how information is represented, processed, and transmitted when the underlying degrees of freedom obey quantum mechanics. Many-body physics asks how simple microscopic rules produce collective behaviour. The two meet because entanglement, scrambling, and computational resources are properties of an entire quantum state. Random unitaries, random circuits, and tensor networks provide controlled models in which one can compute ensemble averages and identify typical behaviour.
1. Quantum randomness I: Haar measure, permutations, Weingarten calculus, unitary designs, the Clifford group.
2. Classical shadows: randomized measurements as unbiased estimators and their sample complexity.
3. Quantum randomness II: finite-depth random circuits and moment-operator spectral gaps.
4. Random dynamics in many-body systems: anticoncentration, Pauli spectra and magic, entanglement growth, out-of-time-order correlators.
Prerequisites: a first course in quantum mechanics and first-year linear algebra and analysis.