Two-way affine automata can verify every language

TL;DR

This paper demonstrates that two-way affine automata can verify membership in any language with bounded error, showcasing their remarkable computational verification capabilities, surpassing classical automata in efficiency and scope.

Contribution

It introduces two protocols for two-way affine automata verifiers that can verify all languages, highlighting their superior verification power over classical automata.

Findings

Affine automata verify all languages with bounded error.

Two protocols: weak (single register, one read) and strong (two registers).

Affine automata outperform classical counterparts in language verification.

Abstract

When used as verifiers in Arthur-Merlin systems, two-way quantum finite automata can verify membership in all languages with bounded error with double-exponential expected running time, which cannot be achieved by their classical counterparts. We obtain the same result for affine automata with single-exponential expected time. We show that every binary (and r-ary) language is verified by some two-way affine finite automata verifiers by presenting two protocols: A weak verification protocol uses a single affine register and the input is read once; and, a strong verification protocol uses two affine registers. These results reflects the remarkable verification capabilities of affine finite automata.