Solving search problems by strongly simulating quantum circuits

TL;DR

This paper demonstrates that strong simulation of quantum circuits can be used to efficiently solve and count solutions to certain search problems, expanding the understanding of quantum-classical computational boundaries.

Contribution

It introduces a new strong simulation technique for restricted quantum circuits and links strong simulation to solving search problems efficiently.

Findings

Strong simulation can solve specific search problems efficiently.

Efficient strong simulation bounds the computational power of certain quantum circuits.

If all strongly simulable circuits could solve P problems, it would imply a collapse of the complexity hierarchy.

Abstract

Simulating quantum circuits using classical computers lets us analyse the inner workings of quantum algorithms. The most complete type of simulation, strong simulation, is believed to be generally inefficient. Nevertheless, several efficient strong simulation techniques are known for restricted families of quantum circuits and we develop an additional technique in this article. Further, we show that strong simulation algorithms perform another fundamental task: solving search problems. Efficient strong simulation techniques allow solutions to a class of search problems to be counted and found efficiently. This enhances the utility of strong simulation methods, known or yet to be discovered, and extends the class of search problems known to be efficiently simulable. Relating strong simulation to search problems also bounds the computational power of efficiently strongly simulable circuits; if they could solve all problems in $\mathrm{P}$ this would imply the collapse of the complexity hierarchy $\mathrm{P} \subseteq \mathrm{NP} \subseteq # \mathrm{P}$.