The Power of a Single Qubit: Two-way Quantum Finite Automata and the Word Problem

TL;DR

This paper demonstrates the enhanced computational power of two-way quantum finite automata with a single qubit, showing they can recognize complex languages and word problems of various groups efficiently.

Contribution

It extends the known capabilities of 2QCFA by proving they can recognize word problems of many groups, including finitely generated virtually abelian and linear groups, with improved parameters.

Findings

Recognize the word problem of finitely generated virtually abelian groups in polynomial time.

Recognize the word problem of many linear groups in exponential time.

Improved recognition parameters for the language $L_{eq}$.

Abstract

The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA, with a single qubit, can recognize, with bounded error, the language $L_{eq}=\{a^m b^m :m \in \mathbb{N}\}$ in expected polynomial time and the language $L_{pal}=\{w \in \{a,b\}^*:w \text{ is a palindrome}\}$ in expected exponential time. We further demonstrate the power of 2QCFA by showing that they can recognize the word problems of many groups. In particular 2QCFA, with a single qubit and algebraic number transition amplitudes, can recognize, with bounded error, the word problem of any finitely generated virtually abelian group in expected polynomial time, as well as the word problems of a large class of linear groups in expected exponential time. This latter class (properly) includes all groups with context-free word problem. We also exhibit results for 2QCFA with any constant number of qubits. As a corollary, we obtain a direct improvement on the original Ambainis and Watrous result by showing that $L_{eq}$ can be recognized by a 2QCFA with better parameters. As a further corollary, we show that 2QCFA can recognize certain non-context-free languages in expected polynomial time. In a companion paper, we prove matching lower bounds, thereby showing that the class of languages recognizable with bounded error by a 2QCFA in expected $\mathit{subexponential}$ time is properly contained in the class of languages recognizable with bounded error by a 2QCFA in expected $\mathit{exponential}$ time.