# QIP $ \subseteq $ AM(2QCFA)

## TL;DR

This paper demonstrates that the class PSPACE, including quantum interactive proof systems, can be contained within Arthur-Merlin systems with two-way finite automata equipped with quantum and classical states, using protocols with perfect completeness.

## Contribution

It establishes that PSPACE and QIP are subsets of AM(2QCFA), introducing protocols with rational quantum transitions and perfect completeness, expanding the understanding of quantum automata capabilities.

## Key findings

- PSPACE is contained in AM(2QCFA)
- Protocols use only rational-valued quantum transitions
- Achieve perfect completeness with double-exponential expected time

## Abstract

The class of languages having polynomial-time classical or quantum interactive proof systems ($\mathsf{IP}$ or $\mathsf{QIP}$, respectively) is identical to $\mathsf{PSPACE}$. We show that $\mathsf{PSPACE}$ (and so $\mathsf{QIP}$) is subset of $\mathsf{AM(2QCFA)}$, the class of languages having Arthur-Merlin proof systems where the verifiers are two-way finite automata with quantum and classical states (2QCFAs) communicating with the provers classically. Our protocols use only rational-valued quantum transitions and run in double-exponential expected time. Moreover, the member strings are accepted with probability 1 (i.e., perfect-completeness).