Abstract

Random unitary dynamics are a toy model for chaotic quantum dynamics and also have applications to quantum information theory and computing. A basic question about them is whether their first few moments approximately match those of the Haar measure; if so, we call them approximate unitary designs. It is natural to conjecture that the time needed for quantum dynamics to yield an approximate design is given by the time for a signal to propagate from one side of the system to the other. I will describe the proof of this claim in one or more dimensions in Euclidean geometry and will give examples where this claim fails in more general geometries, including the Schwarzschild metric. I will briefly discuss two applications: (1) the proposal by Google and other groups to use random quantum circuits for “quantum supremacy,” meaning a quantum circuit performing a task that is hard for a classical computer to simulate; and (2) the question of how quickly information is scrambled in black holes.

This is based on the following three papers. arXiv:1208.0692 (with Fernando Brandao and Michal Horodeck) arXiv:1809.06957 (with Saeed Mehraban) arXiv:1906.02219 (with Linghang Kong, Zi-Wen Liu, Saeed Mehraban, and Peter Shor)