Unitary Realizations of Synchronizing Automata in Quantum Systems

TL;DR

This paper develops a quantum version of a classical synchronizing automaton, enabling a unitary resetting protocol using auxiliary qubits and entanglement, bridging classical automata concepts with quantum mechanics.

Contribution

It introduces a novel quantum automaton model with a unitary resetting protocol based on quantum synchronizing words and auxiliary qubits.

Findings

A quantum resetting protocol driven by synchronizing words is proposed.

The automaton's state can be driven to a predetermined pure state unitarily.

Complex entangled states encoding automaton information can be generated.

Abstract

We introduce a quantum analogue of a classical synchronizing automaton. In classical case the state of a system evolves according to a set of rules forming an alphabet, and sequences of these rules, called words, govern its evolution. Certain special words, known as synchronizing words, drive the automaton into a predetermined state regardless of its initial configuration. Although such an apparently irreversible process seems incompatible with the unitarity of quantum mechanics, we present a resetting protocol based on quantum synchronizing words by incorporating auxiliary qubits whose states encode the rules of the automaton's alphabet. These qubits interact with the quantum automaton, whose state is encoded in a qudit, via a global unitary operation. When the qubit register is initially prepared in a state corresponding to a synchronizing word, the automaton evolves into a predetermined pure state independent of its initial state, while the qubit register is transformed into a complex, often entangled, state that encodes information about the automaton's original configuration. The resulting entanglement depends on both the rule set and the automaton's initial state, and we show how specific entangled states can be generated within this framework.