The Power of a Single Qubit: Twoway Quantum Finite Automata and the Word Problem
Abstract
The twoway 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 contextfree 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 noncontextfree 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.09879
 Bibcode:
 2020arXiv200309879R
 Keywords:

 Computer Science  Computational Complexity;
 Quantum Physics
 EPrint:
 To appear in ICALP 2020