Unbounded-error quantum computation with small space bounds

TL;DR

This paper explores the computational power of quantum Turing machines and quantum finite automata in unbounded error settings, revealing their advantages over classical models under specific space constraints and characterizing their language recognition capabilities.

Contribution

It establishes new bounds on the power of QTMs with small space and characterizes the language recognition power of real-time QFAs with restricted measurements.

Findings

QTMs are more powerful than probabilistic Turing machines for space bounds s(n)=o(log log n).

Real-time QFAs with restricted measurements are equivalent in power to probabilistic automata.

Allowing stationary head moves in QFAs increases the class of recognizable languages.

Abstract

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound $ s $ satisfying $ s(n)=o(\log \log n) $. For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for $s(n)=o(\log n) $. We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.