Succinctness of two-way probabilistic and quantum finite automata

TL;DR

This paper demonstrates that two-way probabilistic and quantum finite automata are significantly more concise than their one-way and nondeterministic counterparts, recognizing certain regular languages with fewer states and supporting efficient probability amplification.

Contribution

It introduces the superior succinctness of 2PFA's and 2QFA's over other automata models and presents infinite language families recognized with fixed probability using constant states.

Findings

2QFA's can recognize certain regular languages with fixed probability and constant states.

2QFA's with mixed states enable efficient probability amplification.

2QFA's outperform 1PFA's, 1QFA's, and 2NFA's in recognizing some regular languages.

Abstract

We prove that two-way probabilistic and quantum finite automata (2PFA's and 2QFA's) can be considerably more concise than both their one-way versions (1PFA's and 1QFA's), and two-way nondeterministic finite automata (2NFA's). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than $ {1/2} $ by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFA's, 1QFA's and 2NFA's grow without bound. We also show that 2QFA's with mixed states can support highly efficient probability amplification. The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is introduced.