Measure Many Quantum Finite Automata on Infinite Words

TL;DR

This paper introduces Measure-Many Quantum B"uchi Automata (MMQBA), a quantum model for infinite words, analyzing their acceptance conditions, language properties, and decidability issues in quantum automata theory.

Contribution

It extends measure-many quantum finite automata to infinite words with B"uchi acceptance, providing formal semantics, language characterization, and decidability results.

Findings

MMQBA languages are exactly lim(L(M,p)) for MMQFA M

MMQBA is closed under union but not intersection or complement

Emptiness is semi-decidable; other problems are undecidable

Abstract

We define a quantum computational model over infinite words, called Measure-Many Quantum B\"uchi Automata (MMQBA), which extends Measure-many Quantum Finite automata (MMQFA) to the infinite word setting with B\"uchi acceptance condition. In MMQBA, the quantum state evolves through unitary transformations followed by repeated projective measurements. An infinite word is accearaq2ppted with respect to a cutpoint p is in (0, 1] if (i) the run visits accepting states infinitely often, (ii) the limiting cumulative acceptance probability is at least p, and (iii) the limiting cumulative rejection lprobability is strictly less than p. We formalize the semantics of MMQBA, establish a language-theoretic characterization showing that MMQBA languages are precisely of the form lim(L(M, p)) for MMQFA M , and develop a decomposition of the non-halting subspace. We prove that MMQBA is closed under union but not under intersection or complementation. On the algorithmic side, we show that the emptiness problem is semi-decidable, while universality, inclusion, equivalence, and membership remain undecidable.