Sampling random quantum circuits: a pedestrian's guide

Recent experiments completed by collaborating research groups from Google, NASA Ames, UC Santa Barbara, and others provided compelling evidence that quantum supremacy has finally been achieved on a superconducting quantum processor. The theoretical basis for these experiments depends on sampling the output distributions of random quantum circuits; unfortunately, understanding how this theoretical basis can be used to define quantum supremacy is an extremely difficult task. Anyone attempting to understand how this sampling task relates to quantum supremacy must study concepts from random matrix theory, mathematical analysis, quantum chaos, computational complexity, and probability theory. Resources connecting these concepts in the context of quantum supremacy are scattered and often difficult to find. This article is an attempt to alleviate this difficulty in those who wish to understand the theoretical basis of Google's quantum supremacy experiments, by carefully walking through a derivation of their precise mathematical definition of quantum supremacy. It's designed for advanced undergraduate or graduate students who want more information than can be provided in popular science articles, but who might not know where to begin when tackling the many research papers related to quantum supremacy.