Two-tape finite automata with quantum and classical states

TL;DR

This paper introduces a new two-way two-tape automaton model with quantum and classical states, demonstrating its efficiency in recognizing complex languages and extending to k-tape automata with quantum capabilities.

Contribution

It proposes the 2TQCFA model, shows its efficiency for recognizing certain languages, and extends the framework to k-tape automata with quantum states.

Findings

2TFA algorithms recognize languages also recognized by 2QCFA.

2TQCFA efficiently recognize languages like L_{square} and others.

kTQCFA can recognize languages like a^n b^{n^k} for any k.

Abstract

{\it Two-way finite automata with quantum and classical states} (2QCFA) were introduced by Ambainis and Watrous, and {\it two-way two-tape deterministic finite automata} (2TFA) were introduced by Rabin and Scott. In this paper we study 2TFA and propose a new computing model called {\it two-way two-tape finite automata with quantum and classical states} (2TQCFA). First, we give efficient 2TFA algorithms for recognizing languages which can be recognized by 2QCFA. Second, we give efficient 2TQCFA algorithms to recognize several languages whose status vis-a-vis 2QCFA have been posed as open questions, such as $L_{square}=\{a^{n}b^{n^{2}}\mid n\in \mathbf{N}\}$. Third, we show that $\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\}$ can be recognized by {\it $(k+1)$-tape deterministic finite automata} ($(k+1)$TFA). Finally, we introduce {\it $k$-tape automata with quantum and classical states} ($k$TQCFA) and prove that $\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\}$ can be recognized by $k$TQCFA.