Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2405.09829 · Physical Review A

Quantum systems from random probabilistic automata

A. Kreuzkamp, CHRISTOF WETTERICH

Published Apr 18, 2025• 2 citations
arXivHTMLPDF

Abstract

Probabilistic cellular automata with deterministic updating are discrete quantum systems. They are characterized by a probability distribution over initial configurations. We employ the quantum formalism for an investigation of random probabilistic cellular automata for which the properties of the deterministic updating are randomly distributed over space and time. We are interested in a possible continuum limit for a very large number of cells or space points. We consider bits with two colors, moving to the left or right on a linear chain. At randomly distributed scattering points they change direction and color. A numerical simulation reveals the typical features of quantum systems. We find particular initial probability distributions which reemerge periodically after a certain number of time steps, as produced by the periodic evolution of energy eigenstates in quantum mechanics. Using a description in terms of wave functions allows us to introduce statistical observables for momentum and energy. They characterize the probabilistic information without taking definite values for a given bit configuration, with a conceptual status similar to temperature in classical statistical thermal equilibrium. Conservation of energy and momentum are essential ingredients for the understanding of the evolution of the stochastic probabilistic automata. This evolution resembles in some aspects a single Dirac fermion in two dimensions with a generalized random potential.

DOI
10.1103/physreva.111.042213

Related papers

Probabilistic cellular automaton for quantum particle in a potential

CHRISTOF WETTERICH

We propose that a quantum particle in a potential in one space dimension can be described by a probabilistic cellular automaton. While the simple updating rule of the automaton is deterministic, the probabilistic description is introduced by a probability distribution over initial conditions. The proposed automaton involves right- and left-movers, jumping from one cell to a neighboring one. They change their direction of motion at each randomly distributed disorder or scattering point. The continuum limit of an infinite number of cells yields a Dirac equation, and in the non-relativistic limit the familiar Schrödinger equation, with potential determined by the spacetime-distribution of scattering points. These equations describe the time evolution of the probabilistic information for the position of the particle. All quantum rules for observables, both for discrete possible measurement values and continuous expectation values, follow from the classical statistical laws.

arXiv:2211.17034•Nov 30, 2022•0 citations

Fermionic quantum field theories as probabilistic cellular automata

CHRISTOF WETTERICH

A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.

arXiv:2111.06728•Physical review. D/Physical review. D.•Apr 8, 2022•8 citations

Quantum mechanics for classical transport equations

Christof Wetterich

Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.

quant-pharXiv:2605.15969•May 18, 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 Cellular Automata Controlled Self-Organizing Networks

László Gyöngyösi, Sándor Imre

Every process of our universe is based on the fundamental elements of nature: information and computation.The basic motivation behind the study of quantum cellular automata (QCA) is the wish to analyze the processes of nature.QCA provide a natural framework within which to describe many classically undescribable and uncomputable physical phenomena, such as the properties of quantum physical systems and the complex background of quantum dynamics.Quantum cellular automata models are based on the working mechanism of classical cellular automata models and use the power of reversible quantum computation.Cellular automata can be used in many fields of science, such as parallel computation, artificial intelligence, image processing, biological systems, simulation of physics systems, hardware design, algorithm theory, and many more.In the first part of this chapter, we give a brief overview of the basic properties of quantum information processing and analyze the quantum versions of classical cellular automata models.We present all materials in a clear, perspicuous, and comprehensible manner, without using a complex mathematical background.After reviewing physical QCA implementations, we sketch future directions, and then conclude the first part.In the second part of the chapter, we examine one possible application of QCA, which uses quantum computing to realize real-life based, truly random network organization.This abstract machine is called a Quantum Cellular Machine (QCM), and we design it for controlling a truly random biologically-inspired network, and to integrate quantum learning algorithms and quantum searching into a controlled, self-organizing system.The selforganizing processes in classical systems cannot be truly random.Using our quantum probabilistic QCM unit, we can add truly random behavior to the self-organizing processes of biological networks.A quantum mechanical-based quantum cellular machine (QCM) controls the self-organizing processes of the network and uses a closed, non-classical quantum mechanical-based language inside the QCM.The proposed QCM solution has deep relevance in the evolution of truly random quantum probabilistic self-organizing network structures.The QCM controls the evolution of the system, changes its environment and creates plans without any human interaction, using truly random quantum probabilistic decisions.In a classical system, the classical circuits can only exhibit deterministic behavior.In a quantum probabilistic control system, the quantum circuits can follow both deterministic and quantum probabilistic quantum cellular machine control behaviors.The QCM has classical and quantum communication layers, it uses the classical layer to detect www.intechopen.comCellular Automata -Innovative Modelling for Science and Engineering 114 the network environment.The lower layer of the quantum cellular machine contains the non-deterministic quantum probabilistic decisions and interacts with the classical level.The quantum cellular machine model with the power of quantum computing can be used for the development of a real-life based network organism.We show, that a real, biologically inspired, non-deterministic, truly random network model can be achieved by the discussed QCM model.In the third part of this chapter, we show that a very efficient quantum searching algorithm can be integrated into the QCM, to find the best solution to a given network input command.We present a quantum searching based method specially designed for quantum probabilistic self-organizing networks, to reduce the complexity of the classical traditional search in the network.The proposed QCM can process both quantum and classical information, and accomplish both deterministic and quantum probabilistic tasks.The information unit is a quantum bit, which can lie in a coherent superposition state of logical states zero and one, and can thus simultaneously store zero and one.Using quantum bits, we can speed up the solutions of classical problems, and even solve some hard problems that classical computers can't solve.The key aspect behind the optimal decisions of a QCM is to design a high-efficiency searching algorithm.A QCM updates the probability amplitudes of its quantum register, according to a given reward value, derived from the network environment.The QCM repeatedly applies a unitary transformation to the quantum states, thus it can enhance, for example, the probability amplitude of the optimal path in the self-organizing network environment, while suppressing the amplitude of all other solutions.The QCM's quantum searching algorithm can help resolve many hard tasks, for example it could be applied to find an optimal logical path, using the effects of quantum mechanics.In the numerical analysis we will show that the quantum communication layer could improve the performance of classical systems.

InTech eBooks•Apr 11, 2011•5 citations

Quantum fermions from classical bits

CHRISTOF WETTERICH

A simple probabilistic cellular automaton is shown to be equivalent to a relativistic fermionic quantum field theory with interactions. Occupation numbers for fermions are classical bits or Ising spins. The automaton acts deterministically on bit configurations. The genuinely probabilistic character of quantum physics is realized by probabilistic initial conditions. In turn, the probabilistic automaton is equivalent to the classical statistical system of a generalized Ising model. For a description of the probabilistic information at any given time quantum concepts as wave functions and non-commuting operators for observables emerge naturally. Quantum mechanics can be understood as a particular case of classical statistics. This offers prospects to realize aspects of quantum computing in the form of probabilistic classical computing. This article is part of the theme issue 'Quantum technologies in particle physics'.

arXiv:2106.15517•Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Dec 20, 2021•7 citations
Probabilistic cellular automata with deterministic updating are discrete quantum systems. They are characterized by a probability distribution over initial configurations. We employ the quantum formalism for an investigation of random probabilistic cellular automata for which the properties of the deterministic updating are randomly distributed over space and time. We are interested in a possible continuum limit for a very large number of cells or space points. We consider bits with two colors, moving to the left or right on a linear chain. At randomly distributed scattering points they change direction and color. A numerical simulation reveals the typical features of quantum systems. We find particular initial probability distributions which reemerge periodically after a certain number of time steps, as produced by the periodic evolution of energy eigenstates in quantum mechanics. Using a description in terms of wave functions allows us to introduce statistical observables for momentum and energy. They characterize the probabilistic information without taking definite values for a given bit configuration, with a conceptual status similar to temperature in classical statistical thermal equilibrium. Conservation of energy and momentum are essential ingredients for the understanding of the evolution of the stochastic probabilistic automata. This evolution resembles in some aspects a single Dirac fermion in two dimensions with a generalized random potential.
We propose that a quantum particle in a potential in one space dimension can be described by a probabilistic cellular automaton. While the simple updating rule of the automaton is deterministic, the probabilistic description is introduced by a probability distribution over initial conditions. The proposed automaton involves right- and left-movers, jumping from one cell to a neighboring one. They change their direction of motion at each randomly distributed disorder or scattering point. The continuum limit of an infinite number of cells yields a Dirac equation, and in the non-relativistic limit the familiar Schrödinger equation, with potential determined by the spacetime-distribution of scattering points. These equations describe the time evolution of the probabilistic information for the position of the particle. All quantum rules for observables, both for discrete possible measurement values and continuous expectation values, follow from the classical statistical laws.
A class of fermionic quantum field theories with interactions is shown to be equivalent to probabilistic cellular automata, namely cellular automata with a probability distribution for the initial states. Probabilistic cellular automata on a one-dimensional lattice are equivalent to two-dimensional quantum field theories for fermions. They can be viewed as generalized Ising models on a square lattice and therefore as classical statistical systems. As quantum field theories they are quantum systems. Thus quantum mechanics emerges from classical statistics. As an explicit example for an interacting fermionic quantum field theory we describe a type of discretized Thirring model as a cellular automaton. The updating rule of the automaton is encoded in the step evolution operator that can be expressed in terms of fermionic annihilation and creation operators. The complex structure of quantum mechanics is associated to particle-hole transformations. The naive continuum limit exhibits Lorentz symmetry. We exploit the equivalence to quantum field theory in order to show how quantum concepts as wave functions, density matrix, noncommuting operators for observables and similarity transformations are convenient and useful concepts for the description of probabilistic cellular automata.
Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.
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.
Every process of our universe is based on the fundamental elements of nature: information and computation.The basic motivation behind the study of quantum cellular automata (QCA) is the wish to analyze the processes of nature.QCA provide a natural framework within which to describe many classically undescribable and uncomputable physical phenomena, such as the properties of quantum physical systems and the complex background of quantum dynamics.Quantum cellular automata models are based on the working mechanism of classical cellular automata models and use the power of reversible quantum computation.Cellular automata can be used in many fields of science, such as parallel computation, artificial intelligence, image processing, biological systems, simulation of physics systems, hardware design, algorithm theory, and many more.In the first part of this chapter, we give a brief overview of the basic properties of quantum information processing and analyze the quantum versions of classical cellular automata models.We present all materials in a clear, perspicuous, and comprehensible manner, without using a complex mathematical background.After reviewing physical QCA implementations, we sketch future directions, and then conclude the first part.In the second part of the chapter, we examine one possible application of QCA, which uses quantum computing to realize real-life based, truly random network organization.This abstract machine is called a Quantum Cellular Machine (QCM), and we design it for controlling a truly random biologically-inspired network, and to integrate quantum learning algorithms and quantum searching into a controlled, self-organizing system.The selforganizing processes in classical systems cannot be truly random.Using our quantum probabilistic QCM unit, we can add truly random behavior to the self-organizing processes of biological networks.A quantum mechanical-based quantum cellular machine (QCM) controls the self-organizing processes of the network and uses a closed, non-classical quantum mechanical-based language inside the QCM.The proposed QCM solution has deep relevance in the evolution of truly random quantum probabilistic self-organizing network structures.The QCM controls the evolution of the system, changes its environment and creates plans without any human interaction, using truly random quantum probabilistic decisions.In a classical system, the classical circuits can only exhibit deterministic behavior.In a quantum probabilistic control system, the quantum circuits can follow both deterministic and quantum probabilistic quantum cellular machine control behaviors.The QCM has classical and quantum communication layers, it uses the classical layer to detect www.intechopen.comCellular Automata -Innovative Modelling for Science and Engineering 114 the network environment.The lower layer of the quantum cellular machine contains the non-deterministic quantum probabilistic decisions and interacts with the classical level.The quantum cellular machine model with the power of quantum computing can be used for the development of a real-life based network organism.We show, that a real, biologically inspired, non-deterministic, truly random network model can be achieved by the discussed QCM model.In the third part of this chapter, we show that a very efficient quantum searching algorithm can be integrated into the QCM, to find the best solution to a given network input command.We present a quantum searching based method specially designed for quantum probabilistic self-organizing networks, to reduce the complexity of the classical traditional search in the network.The proposed QCM can process both quantum and classical information, and accomplish both deterministic and quantum probabilistic tasks.The information unit is a quantum bit, which can lie in a coherent superposition state of logical states zero and one, and can thus simultaneously store zero and one.Using quantum bits, we can speed up the solutions of classical problems, and even solve some hard problems that classical computers can't solve.The key aspect behind the optimal decisions of a QCM is to design a high-efficiency searching algorithm.A QCM updates the probability amplitudes of its quantum register, according to a given reward value, derived from the network environment.The QCM repeatedly applies a unitary transformation to the quantum states, thus it can enhance, for example, the probability amplitude of the optimal path in the self-organizing network environment, while suppressing the amplitude of all other solutions.The QCM's quantum searching algorithm can help resolve many hard tasks, for example it could be applied to find an optimal logical path, using the effects of quantum mechanics.In the numerical analysis we will show that the quantum communication layer could improve the performance of classical systems.
A simple probabilistic cellular automaton is shown to be equivalent to a relativistic fermionic quantum field theory with interactions. Occupation numbers for fermions are classical bits or Ising spins. The automaton acts deterministically on bit configurations. The genuinely probabilistic character of quantum physics is realized by probabilistic initial conditions. In turn, the probabilistic automaton is equivalent to the classical statistical system of a generalized Ising model. For a description of the probabilistic information at any given time quantum concepts as wave functions and non-commuting operators for observables emerge naturally. Quantum mechanics can be understood as a particular case of classical statistics. This offers prospects to realize aspects of quantum computing in the form of probabilistic classical computing. This article is part of the theme issue 'Quantum technologies in particle physics'.