Proving the power of postselection

TL;DR

This paper demonstrates that postselection enhances the computational power of both classical and quantum polynomial-time machines, with quantum machines outperforming probabilistic ones under certain resource constraints, and provides classical characterizations of related complexity classes.

Contribution

It proves for the first time that postselection increases computational power for classical and quantum computers and shows quantum advantage under specific resource bounds.

Findings

Postselection augments classical and quantum computational power.

Quantum machines outperform probabilistic ones with postselection.

Classical characterizations of postselected space-bounded classes.

Abstract

It is a widely believed, though unproven, conjecture that the capability of postselection increases the language recognition power of both probabilistic and quantum polynomial-time computers. It is also unknown whether polynomial-time quantum machines with postselection are more powerful than their probabilistic counterparts with the same resource restrictions. We approach these problems by imposing additional constraints on the resources to be used by the computer, and are able to prove for the first time that postselection does augment the computational power of both classical and quantum computers, and that quantum does outperform probabilistic in this context, under simultaneous time and space bounds in a certain range. We also look at postselected versions of space-bounded classes, as well as those corresponding to error-free and one-sided error recognition, and provide classical characterizations. It is shown that $\mathsf{NL}$ would equal $\mathsf{RL}$ if the randomized machines had the postselection capability.