© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2604.20432 · New Journal of Physics

Unitary Realizations of Synchronizing Automata in Quantum Systems

Jȩdrzej Stempin, Jan Wójcik, Gabriela Banaszak, Andrzej Grudka, Marcin Karczewski, Paweł Kurzyński, Antoni Wójcik

quant-phSubmitted Apr 22, 2026• Announced Apr 23, 2026• 0 citations

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.

Comments: 11 pages, 11 figures, this manuscript is a modified and substantially extended version of arXiv:2504.01106

Related papers

Quantum synchronizing words

Andrzej Grudka, Marcin Karczewski, Paweł Kurzyński +3 more

Abstract Synchronizing words in classical automata theory provide a mechanism to reset any state of a deterministic automaton to a specific target state via a carefully chosen finite sequence of transition rules. In this work, we extend the concept of synchronizing words to quantum information theory. Specifically, we investigate whether the concatenation of two non-resetting quantum channels can give rise to a resetting channel. While we formulate the problem for general qudits, the complexity of the setting leads us to focus on particular resetting concatenations for qubits and qutrits. Furthermore, we demonstrate that following the reset, any pure real qutrit state can be closely approximated using the same two non-resetting channels. These findings establish a quantum analogue of synchronizing words, highlighting their potential applications in constructing minimal sets of quantum channels capable of both resetting and preparing arbitrary states.

New Journal of Physics•Oct 23, 2025•1 citation

Quantum Synchronizing Words: Resetting and Preparing Qutrit States

Andrzej Grudka, Marcin Karczewski, Paweł Kurzyński +3 more

Synchronizing words in classical automata theory provide a mechanism to reset any state of a deterministic automaton to a specific target state via a carefully chosen finite sequence of transition rules. In this work, we extend the concept of synchronizing words to quantum information theory. Specifically, we show that with only two quantum channels, it is possible to bring an arbitrary qutrit state close to a designated target state. Furthermore, we demonstrate that following this reset, any pure real qutrit state can be closely approximated using the same two channels. These findings establish a quantum analogue of synchronizing words, highlighting their potential applications in constructing minimal sets of universal quantum gates capable of both resetting and preparing arbitrary states.

arXiv:2502.09522•Feb 13, 2025•0 citations

Quantum Cellular Automata and Groups

Ruoli Bai

A quantum cellular automaton (QCA) is a unitary discrete-time evolution of a quantum many-body system, which can be described as a dynamics system , where A is any matrix and is a unitary operator representing the global evolution of QCA. And QCAs are characterized by locality (or causality). Among them, quantum circuits are the special classes of QCAs, which use unitary operators to control and transform the initial quantum state of qubit into a final quantum state, quantum circuits are special and widely studied in quantum information. A group is a set of elements combined with an operation, which satisfies associativity, identity and invertibility. This article shows that group theory provides the right language for studying the relation between QCAs and quantum circuits among them.

Theoretical and Natural Science•Apr 24, 2025•0 citations

Quantum State Collapse as a Cellular Automaton Algorithm

Jincheng Zhang

Quantum state collapse is a fundamental process in quantum mechanics, representing the irreversible loss of information upon measurement. This paper explores the potential of utilizing a cellular automata (CA) framework to model and simulate this collapse, offering a novel approach to understanding and potentially manipulating quantum evolution. We formalize collapse as a dynamic process governed by specific rules and feedback loops, enabling the simulation of complex quantum patterns. We propose a CA-based model that incorporates the probabilistic nature of quantum mechanics, allowing for a more intuitive and potentially controllable simulation of quantum phenomena. The core mechanism focuses on systematically evolving the system's state space, mirroring the collapse process and highlighting the interplay between initial conditions and emergent behavior. This work provides a foundational step in exploring the application of CA algorithms to complex quantum systems, opening avenues for future research in quantum simulation and understanding.

•Sep 6, 2026•0 citations

Reversible quantum cellular automata

Benjamin W. Schumacher, Reinhard F. Werner

We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.

arXiv:quant-ph/0405174•May 28, 2004•98 citations

Quantum homogenization

Mário Ziman, Peter Štelmachovič, Vladimír Bužek +3 more

We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state $ρ$ and a set of N reservoir qubits initially prepared in the same state $ξ$. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitratily small neighbourhood of the state $ξ$ irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state $ξ$. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in non-equilibrium states. In particular, the reversibility vs or irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination.We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.

arXiv:quant-ph/0110164•Oct 29, 2001•2 citations

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.

Abstract Synchronizing words in classical automata theory provide a mechanism to reset any state of a deterministic automaton to a specific target state via a carefully chosen finite sequence of transition rules. In this work, we extend the concept of synchronizing words to quantum information theory. Specifically, we investigate whether the concatenation of two non-resetting quantum channels can give rise to a resetting channel. While we formulate the problem for general qudits, the complexity of the setting leads us to focus on particular resetting concatenations for qubits and qutrits. Furthermore, we demonstrate that following the reset, any pure real qutrit state can be closely approximated using the same two non-resetting channels. These findings establish a quantum analogue of synchronizing words, highlighting their potential applications in constructing minimal sets of quantum channels capable of both resetting and preparing arbitrary states.

Synchronizing words in classical automata theory provide a mechanism to reset any state of a deterministic automaton to a specific target state via a carefully chosen finite sequence of transition rules. In this work, we extend the concept of synchronizing words to quantum information theory. Specifically, we show that with only two quantum channels, it is possible to bring an arbitrary qutrit state close to a designated target state. Furthermore, we demonstrate that following this reset, any pure real qutrit state can be closely approximated using the same two channels. These findings establish a quantum analogue of synchronizing words, highlighting their potential applications in constructing minimal sets of universal quantum gates capable of both resetting and preparing arbitrary states.

A quantum cellular automaton (QCA) is a unitary discrete-time evolution of a quantum many-body system, which can be described as a dynamics system , where A is any matrix and is a unitary operator representing the global evolution of QCA. And QCAs are characterized by locality (or causality). Among them, quantum circuits are the special classes of QCAs, which use unitary operators to control and transform the initial quantum state of qubit into a final quantum state, quantum circuits are special and widely studied in quantum information. A group is a set of elements combined with an operation, which satisfies associativity, identity and invertibility. This article shows that group theory provides the right language for studying the relation between QCAs and quantum circuits among them.

Quantum state collapse is a fundamental process in quantum mechanics, representing the irreversible loss of information upon measurement. This paper explores the potential of utilizing a cellular automata (CA) framework to model and simulate this collapse, offering a novel approach to understanding and potentially manipulating quantum evolution. We formalize collapse as a dynamic process governed by specific rules and feedback loops, enabling the simulation of complex quantum patterns. We propose a CA-based model that incorporates the probabilistic nature of quantum mechanics, allowing for a more intuitive and potentially controllable simulation of quantum phenomena. The core mechanism focuses on systematically evolving the system's state space, mirroring the collapse process and highlighting the interplay between initial conditions and emergent behavior. This work provides a foundational step in exploring the application of CA algorithms to complex quantum systems, opening avenues for future research in quantum simulation and understanding.

We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.

We design a universal quantum homogenizer, which is a quantum machine that takes as an input a system qubit initially in the state

ρ

and a set of N reservoir qubits initially prepared in the same state

ξ

. In the homogenizer the system qubit sequentially interacts with the reservoir qubits via the partial swap transformation. The homogenizer realizes, in the limit sense, the transformation such that at the output each qubit is in an arbitratily small neighbourhood of the state

ξ

irrespective of the initial states of the system and the reservoir qubits. This means that the system qubit undergoes an evolution that has a fixed point, which is the reservoir state

ξ

. We also study approximate homogenization when the reservoir is composed of a finite set of identically prepared qubits. The homogenizer allows us to understand various aspects of the dynamics of open systems interacting with environments in non-equilibrium states. In particular, the reversibility vs or irreversibility of the dynamics of the open system is directly linked to specific (classical) information about the order in which the reservoir qubits interacted with the system qubit. This aspect of the homogenizer leads to a model of a quantum safe with a classical combination.We analyze in detail how entanglement between the reservoir and the system is created during the process of quantum homogenization. We show that the information about the initial state of the system qubit is stored in the entanglement between the homogenized qubits.