Quantum Resetting Protocols Based on Synchronizing Words

Resetting a system's state plays a fundamental role in physics, engineering, computer science, and many other fields. Here we focus on a method originally proposed in automata theory. The state of an automaton evolves according to a set of rules. These rules form an alphabet, and one can apply these rules one after another. Such a sequence of rules is known as a word. Some particular words, known as synchronizing words, enable a system to evolve into a predetermined state regardless of its initial configuration. This process, which is inherently irreversible, appears at first glance to be incompatible with the unitarity of quantum mechanics. Unitary evolution preserves information, hence forbids such classical resetting to a predetermined state. In this work, we introduce a novel resetting protocol based on quantum-synchronizing words by incorporating auxiliary qubits whose states encode rules of the automaton's alphabet. We further propose a quantum circuit that realizes this protocol and can be implemented on a quantum computer. Additionally, we establish a connection between this approach and Kraus channels, showing that quantum synchronizing words can be achieved without explicit reference to the states of ancillary qubits. Our results bridge classical and quantum notions of synchronizing words, shedding light on the interplay between quantum information processing and non-unitary dynamics.