Classically Time-Controlled Quantum Automata: Definition and Properties

TL;DR

This paper introduces classically time-controlled quantum automata (CTQA), exploring how time-dependent evolution affects computational power, enabling recognition of non-regular languages and even solving the Halting problem under certain conditions.

Contribution

It defines CTQA as a novel quantum automaton model with time-dependent evolution and analyzes its computational capabilities and limitations.

Findings

Unrestricted CTQA can decide the Halting problem.

Restricted CTQA can recognize non-regular languages with cut-point.

CTQA can recognize non-regular promise languages with bounded-error.

Abstract

In this paper, we introduce classically time-controlled quantum automata or CTQA, which is a reasonable modification of Moore-Crutchfield quantum finite automata that uses time-dependent evolution and a "scheduler" defining how long each Hamiltonian will run. Surprisingly enough, time-dependent evolution provides a significant change in the computational power of quantum automata with respect to a discrete quantum model. Indeed, we show that if a scheduler is not computationally restricted, then a CTQA could even decide the Halting problem. In order to unearth the computational capabilities of CTQAs we study the case of a computationally restricted scheduler. In particular, we showed that depending on the type of restriction imposed on the scheduler, a CTQA can (i) recognize non-regular languages with cut-point, even in the presence of Karp-Lipton advice, and (ii) recognize non-regular promise languages with bounded-error. Furthermore, we study the cutpoint-union of cutpoint languages by introducing a new model of Moore-Crutchfield quantum finite automata with a rotating tape head. CTQA presents itself as a new model of computation that provides a different approach to a formal study of "classical control, quantum data" schemes in quantum computing.