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Quantum Cellular Automata

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Quantum cellular automata (QCA) area generalization of (classical) cellular automata (CA) and in particular of reversible CA. The latter arereviewed shortly. An overview is given over early attempts by various authors to define one-dimensionalQCA. These turned out to have serious shortcomings which are discussed as well. Various proposals subsequently putforward by a number of authors for a general definition of one- and higher-dimensional QCA arereviewed and their properties such as universality and reversibility are discussed.

Quantum cellular automata (QCA) are a quantization of classicalcellular automata (CA),d-dimensional arrays of cells with a finite-dimensional statespace and a local, spatially-homogeneous, discrete-time update rule. For QCA each cell isa finite-dimensional quantum system and the update rule is unitary. CA as well as some versions of QCAhave been shown to be computationally universal ....

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Notes

  1. 1.

    For details on these and other aspects of quantumcomputation see the article by Kendon in this Encyclopedia.

Abbreviations

Configuration:

The state of all cells at a given point in time.

Neighborhood:

All cells with respect to a given cell that can affect this cell's state at the next time step. A neighborhood always contains a finite number of cells.

Space-homogeneous:

The transition function / update table is the same for each cell.

Time-homogeneous:

The transition function / update table is time-independent.

Update table:

Takes the current state of a cell and its neighborhood as an argument and returns the cell's state at the next time step.

Schrödinger picture:

Time evolution is represented by a quantum state evolving in time according to a time-independent unitary operator acting on it.

Heisenberg picture:

Time evolution is represented by observables (elements of an operator algebra) evolving in time according to a unitary operator acting on them.

BQP complexity class:

Bounded error, quantum probabilistic, the class of decision problems solvable by a quantum computer in polynomial time with an error probability of at most 1/3.

QMA complexity class:

Quantum Merlin–Arthur, the class of decision problems such that a “yes” answer can be verified by a 1-message quantum interactive proof (verifiable in BQP).

Quantum Turing machine:

A  quantum version of a Turing machine – an abstract computational model able to compute any computable sequence.

Swap operation:

The one-qubit unitary gate \( { U = \small\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} } \)

Hadamard gate:

The one-qubit unitary gate

$$ U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $$
Phase gate:

The one-qubit unitary gate \( { U = \begin{pmatrix} 1 & 0 \\ 0 & \text{e}^{i\phi} \end{pmatrix} } \)

Pauli operator:

The three Pauli operators are

$$ \sigma_x = \small\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\:,\enskip \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}\:,\enskip \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
Qubit:

2-state quantum system, representable asvector \( { a \mathinner{|{0}\rangle} + b\mathinner{|{1}\rangle} } \) in complex space with \( { a^2 + b^2 = 1 } \).

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Author information

Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, UK

    Karoline Wiesner

  2. Centre for Complexity Sciences, University of Bristol, Bristol, UK

    Karoline Wiesner

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  1. Karoline Wiesner
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Wiesner, K. (2009). Quantum Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_426

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