Power of the interactive proof systems with verifiers modeled by semi-quantum two-way finite automata

TL;DR

This paper investigates the computational power of semi-quantum two-way finite automata within Arthur-Merlin proof systems, demonstrating they surpass classical probabilistic automata and can recognize complex languages with limited quantum resources.

Contribution

It introduces the model of QAM(2QCFA), showing it is more powerful than classical automata and can recognize NP-complete problems using minimal quantum states.

Findings

QAM(2QCFA) outperforms AM(2PFA) in polynomial expected time

Recognizes certain languages with exponential expected time

Can recognize NP-complete language L_{knapsack} with pure quantum states

Abstract

In this paper we explore the power of AM for the case that verifiers are {\em two-way finite automata with quantum and classical states} (2QCFA)--introduced by Ambainis and Watrous in 2002--and the communications are classical. It is of interest to consider AM with such "semi-quantum" verifiers because they use only limited quantum resources. Our main result is that such Quantum Arthur-Merlin proof systems (QAM(2QCFA)) with polynomial expected running time are more powerful than in the case verifiers are two-way probabilistic finite automata (AM(2PFA)) with polynomial expected running time. Moreover, we prove that there is a language which can be recognized by an exponential expected running time QAM(2QCFA), but can not be recognized by any AM(2PFA), and that the NP-complete language $L_{knapsack}$ can also be recognized by a QAM(2QCFA) working only on quantum pure states using unitary operators.