  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.
