Abstract
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. It is both a necessary milestone on the path to useful quantum computers as well as a test of quantum theory in the realm of high complexity. A leading nearterm 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, we give strong complexitytheoretic evidence for the classical hardness of RCS, placing it on par with the best theoretical proposals for supremacy. Specifically, we show that RCS satisfies an averagecase hardness condition  computing output probabilities of typical quantum circuits is as hard as computing them in the worstcase, and therefore #Phard. Our reduction exploits the polynomial structure in the output amplitudes of random quantum circuits, enabled by the Feynman path integral. In addition, it follows from known results that RCS also satisfies an anticoncentration property, namely that errors in estimating output probabilities are small with respect to the probabilities themselves. This makes RCS the first proposal for quantum supremacy with both of these properties. We also give a natural condition under which an existing statistical measure, crossentropy, verifies RCS, as well as describe a new verification measure which in some formal sense maximizes the information gained from experimental samples.
BibTeX  Entry
@InProceedings{bouland_et_al:LIPIcs:2018:10108,
author = {Adam Bouland and Bill Fefferman and Chinmay Nirkhe and Umesh Vazirani},
title = {{"Quantum Supremacy" and the Complexity of Random Circuit Sampling}},
booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
pages = {15:115:2},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770958},
ISSN = {18688969},
year = {2018},
volume = {124},
editor = {Avrim Blum},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10108},
URN = {urn:nbn:de:0030drops101084},
doi = {10.4230/LIPIcs.ITCS.2019.15},
annote = {Keywords: quantum supremacy, averagecase hardness, verification}
}
Keywords: 

quantum supremacy, averagecase hardness, verification 
Collection: 

10th Innovations in Theoretical Computer Science Conference (ITCS 2019) 
Issue Date: 

2018 
Date of publication: 

08.01.2019 