Quantum machines with classical control

TL;DR

This paper surveys quantum automata and machines with classical control, highlighting their expressivity, state efficiency, and deterministic termination, advancing understanding of quantum computational models with classical control mechanisms.

Contribution

It revises key results on the expressivity and state complexity of quantum automata and introduces a quantum Turing machine with classical control that has deterministic termination.

Findings

Quantum automata can recognize certain regular languages with exponentially fewer states.

Quantum Turing machines with classical control have deterministic termination.

These models retain the expressivity of quantum computation.

Abstract

Herein we survey the main results concerning quantum automata and machines with classical control. These machines were originally proposed by Sernadas et al in [37], during the FCT QuantLog project. First, we focus on the expressivity of quantum automata with both quantum and classical states. We revise the result obtained in [32] where it was proved that such automata are able to recognise, with exponentially less states than deterministic finite automata, a family of regular languages that cannot be recognised by other types of quantum automata. Finally, we revise the concept of quantum Turing machine with classical control introduced in [25]. The novelty of these machines consists in the fact that their termination problem is completely deterministic, in opposition to other notions in the literature. Concretely, we revisit the result that such machines fulfil the s-m-n property, while keeping the expressivity of a quantum model for computation.