Quantum Logarithmic Space and Post-selection

TL;DR

This paper establishes that the class of problems solvable by logarithmic-space quantum algorithms with post-selection equals the classical class PL, extending Aaronson's results to space-bounded quantum complexity.

Contribution

It introduces the class PostBQL and proves it is equal to PL, providing a space-bounded analogue of the PostBQP=PP result.

Findings

PostBQL = PL

PL coincides with unbounded-time quantum algorithms in logarithmic space

Extends post-selection results to space-bounded quantum complexity

Abstract

Post-selection, the power of discarding all runs of a computation in which an undesirable event occurs, is an influential concept introduced to the field of quantum complexity theory by Aaronson (Proceedings of the Royal Society A, 2005). In the present paper, we initiate the study of post-selection for space-bounded quantum complexity classes. Our main result shows the identity $\sf PostBQL=PL$, i.e., the class of problems that can be solved by a bounded-error (polynomial-time) logarithmic-space quantum algorithm with post-selection ($\sf PostBQL$) is equal to the class of problems that can be solved by unbounded-error logarithmic-space classical algorithms ($\sf PL$). This result gives a space-bounded version of the well-known result $\sf PostBQP=PP$ proved by Aaronson for polynomial-time quantum computation. As a by-product, we also show that $\sf PL$ coincides with the class of problems that can be solved by bounded-error logarithmic-space quantum algorithms that have no time bound.