"Quantum Supremacy" and the Complexity of Random Circuit Sampling

A critical goal for the field of quantum computation is quantum supremacy -- a demonstration of any quantum computation that is prohibitively hard for classical computers. Besides dispelling any skepticism about the viability of quantum computers, quantum supremacy also provides a test of quantum theory in the realm of high complexity. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, called Random Circuit Sampling (RCS).

While RCS was defined with experimental realization in mind (the first results are expected later this year), we give the first complexity-theoretic evidence of classical hardness of RCS, placing it on par with the best theoretical proposals for supremacy. Specifically, we show that RCS satisfies an average-case hardness condition -- computing output probabilities of typical quantum circuits is as hard as computing them in the worst-case, and therefore #P-hard. Our reduction exploits the polynomial structure in the output amplitudes of random quantum circuits, enabled by the Feynman path integral. We also describe a new verification measure which in some formal sense maximizes the information gained from experimental samples.

Based on joint work with Adam Bouland, Bill Fefferman and Chinmay Nirkhe.