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arXiv:0709.0006 · Physical Review A

Local unitary quantum cellular automata

Carlos A. Pérez-Delgado, Donny Cheung

Published Sep 20, 2007• 45 citations
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Abstract

In this paper we present a quantization of cellular automata. Our formalism is based on a lattice of qudits and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. It is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains, and others. Some results that show the benefits of basing the model on local unitary operators are shown: universality, strong connections to the circuit model, simple implementation on quantum hardware, and a wealth of applications.

DOI
10.1103/physreva.76.032320

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math.ATCross-listedarXiv:2602.16572•Feb 19, 2026
In this paper we present a quantization of cellular automata. Our formalism is based on a lattice of qudits and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. It is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains, and others. Some results that show the benefits of basing the model on local unitary operators are shown: universality, strong connections to the circuit model, simple implementation on quantum hardware, and a wealth of applications.
This thesis presents a model of Quantum Cellular Automata (QCA). The presented formalism is a natural quantization of the classical Cellular Automata (CA). It is based on a lattice of qudits, and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. The main advantage that QCA have over quantum circuits is that QCA make considerably fewer demands on the underlying hardware. In particular, as opposed to direct implementations of quantum circuits, the global evolution of the lattice in the QCA model does not assume independent control over individual \emph{qudits}. Rather, all qudits are to be addressed collectively in parallel. The QCA model is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains and others. Some results that show the benefits of basing the model on local unitary operators are shown: computational universality, strong connections to the circuit model, simple implementation on quantum hardware, and a series of applications. A detailed discussion will be given on one particular application of QCA that lies outside either computation or simulation: single-spin measurement. This algorithm uses the techniques developed in this thesis to achieve a result normally considered hard in physics. It serves well as an example of why QCA are interesting in their own right.
A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas. KEY WORDS: quantum cellular automaton; quantum lattice gas; quantum computation. to appear in J. Stat. Phys.
Several proposed schemes for the physical realization of a quantum computer consist of qubits arranged in a cellular array. In the quantum circuit model of quantum computation, an often complex series of two-qubit gate operations is required between arbitrarily distant pairs of lattice qubits. An alternative model of quantum computation based on quantum cellular automata (QCA) requires only homogeneous local interactions that can be implemented in parallel. This would be a huge simplification in an actual experiment. We find some minimal physical requirements for the construction of unitary QCA in a one-dimensional Ising spin chain and demonstrate optimal pulse sequences for information transport and entanglement distribution. We also introduce the theory of nonunitary QCA and show by example that nonunitary rules can generate environment assisted entanglement.
Quantum cellular automata are alternative quantum-computing paradigms to quantum Turing machines and quantum circuits. Their working mechanisms are inherently automated, therefore measurement free, and they act in a translation invariant manner on all cells or qudits of a register, generating a global rule that updates cell states locally, i.e., based solely on the states of their neighbors. Although desirable features in many applications, it is generally not clear to which extent these fully automated discrete-time local updates can generate and sustain long-range order in the (noisy) systems they act upon. In particular, whether and how quantum cellular automata can perform quantum error correction remain open questions. We close this conceptual gap by proposing quantum cellular automata with quantum-error-correction capabilities. We design and investigate two (quasi)one dimensional quantum cellular automata based on known classical cellular-automata rules with density-classification capabilities, namely the local majority voting and the two-line voting. We investigate the performances of those quantum cellular automata as quantum-memory components by simulating the number of update steps required for the logical information they act upon to be afflicted by a logical bit flip. The proposed designs pave a way to further explore the potential of new types of quantum cellular automata with built-in quantum-error-correction capabilities.
Quantum cellular automata consist in arrays of identical finite-dimensional quantum systems, evolving in discrete-time steps by iterating a unitary operator G. Moreover the global evolution G is required to be causal (it propagates information at a bounded speed) and translation-invariant (it acts everywhere the same). Quantum cellular automata provide a model/architecture for distributed quantum computation. More generally, they encompass most of discrete-space discrete-time quantum theory. We give an overview of their theory, with particular focus on structure results; computability and universality results; and quantum simulation results.
Over an arbitrary commutative ring RRR, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space Q(X)\mathbf{Q}(X)Q(X) of quantum cellular automata (QCA) on a given metric space XXX. In most cases of interest, π0Q(X)π_0 \mathbf{Q}(X)π0​Q(X) classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences Q(∗)≃ΩnQ(Zn)\mathbf{Q}(*) \simeq Ω^n \mathbf{Q}(\mathbb{Z}^n)Q(∗)≃ΩnQ(Zn) for all nnn, which shows that the classification of QCA on Euclidean lattices is given by an ΩΩΩ-spectrum indexed by the dimension nnn. As a corollary, we also obtain a non-connective delooping of the K-theory of Azumaya RRR-algebras, which may be of independent interest. We also include a section leading to the ΩΩΩ-spectrum for QCA over C∗C^*C∗-algebras with unitary circuits.