19 research outputs found
On the absence of homogeneous scalar unitary cellular automata
Failure to find homogeneous scalar unitary cellular automata (CA) in one
dimension led to consideration of only ``approximately unitary'' CA---which
motivated our recent proof of a No-go Lemma in one dimension. In this note we
extend the one dimensional result to prove the absence of nontrivial
homogeneous scalar unitary CA on Euclidean lattices in any dimension.Comment: 7 pages, plain TeX, 3 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages); minor changes (including
title wording) in response to referee suggestions, also updated references;
to appear in Phys. Lett.
Quantum mechanics of lattice gas automata. I. One particle plane waves and potentials
Classical lattice gas automata effectively simulate physical processes such
as diffusion and fluid flow (in certain parameter regimes) despite their
simplicity at the microscale. Motivated by current interest in quantum
computation we recently defined quantum lattice gas automata; in this paper we
initiate a project to analyze which physical processes these models can
effectively simulate. Studying the single particle sector of a one dimensional
quantum lattice gas we find discrete analogues of plane waves and wave packets,
and then investigate their behaviour in the presence of inhomogeneous
potentials.Comment: 19 pages, plain TeX, 14 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages), two additional large
figures available upon reques
A Quantum Game of Life
This research describes a three dimensional quantum cellular automaton (QCA)
which can simulate all other 3D QCA. This intrinsically universal QCA belongs
to the simplest subclass of QCA: Partitioned QCA (PQCA). PQCA are QCA of a
particular form, where incoming information is scattered by a fixed unitary U
before being redistributed and rescattered. Our construction is minimal amongst
PQCA, having block size 2 x 2 x 2 and cell dimension 2. Signals, wires and
gates emerge in an elegant fashion.Comment: 13 pages, 10 figures. Final version, accepted by Journ\'ees Automates
Cellulaires (JAC 2010)
Quantum Cellular Automata
Quantum cellular automata (QCA) are reviewed, including early and more recent
proposals. QCA are a generalization of (classical) cellular automata (CA) and
in particular of reversible CA. The latter are reviewed shortly. An overview is
given over early attempts by various authors to define one-dimensional QCA.
These turned out to have serious shortcomings which are discussed as well.
Various proposals subsequently put forward by a number of authors for a general
definition of one- and higher-dimensional QCA are reviewed and their properties
such as universality and reversibility are discussed.Comment: 12 pages, 3 figures. To appear in the Springer Encyclopedia of
Complexity and Systems Scienc
Quantum mechanics of lattice gas automata. II. Boundary conditions and other inhomogeneities
We continue our analysis of the physics of quantum lattice gas automata
(QLGA). Previous work has been restricted to periodic or infinite lattices;
simulation of more realistic physical situations requires finite sizes and
non-periodic boundary conditions. Furthermore, envisioning a QLGA as a
nanoscale computer architecture motivates consideration of inhomogeneities in
the `substrate'; this translates into inhomogeneities in the local evolution
rules. Concentrating on the one particle sector of the model, we determine the
various boundary conditions and rule inhomogeneities which are consistent with
unitary global evolution. We analyze the reflection of plane waves from
boundaries, simulate wave packet refraction across inhomogeneities, and
conclude by discussing the extension of these results to multiple particles.Comment: 24 pages, plain TeX, 9 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages), 3 additional large figures
available upon request or from
http://math.ucsd.edu/~dmeyer/papers/papers.htm
When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)?
Quantum cellular automata (QCA) are models of quantum computation of
particular interest from the point of view of quantum simulation. Quantum
lattice gas automata (QLGA - equivalently partitioned quantum cellular
automata) represent an interesting subclass of QCA. QLGA have been more deeply
analyzed than QCA, whereas general QCA are likely to capture a wider range of
quantum behavior. Discriminating between QLGA and QCA is therefore an important
question. In spite of much prior work, classifying which QCA are QLGA has
remained an open problem. In the present paper we establish necessary and
sufficient conditions for unbounded, finite Quantum Cellular Automata (QCA)
(finitely many active cells in a quiescent background) to be Quantum Lattice
Gas Automata (QLGA). We define a local condition that classifies those QCA that
are QLGA, and we show that there are QCA that are not QLGA. We use a number of
tools from functional analysis of separable Hilbert spaces and representation
theory of associative algebras that enable us to treat QCA on finite but
unbounded configurations in full detail.Comment: 37 pages, 7 figures, with changes to explanatory text and updated
figures, J. Math. Phys. versio
Quantum Computation
In the last few years, theoretical study of quantum systems serving as
computational devices has achieved tremendous progress. We now have strong
theoretical evidence that quantum computers, if built, might be used as a
dramatically powerful computational tool. This review is about to tell the
story of theoretical quantum computation. I left out the developing topic of
experimental realizations of the model, and neglected other closely related
topics which are quantum information and quantum communication. As a result of
narrowing the scope of this paper, I hope it has gained the benefit of being an
almost self contained introduction to the exciting field of quantum
computation.
The review begins with background on theoretical computer science, Turing
machines and Boolean circuits. In light of these models, I define quantum
computers, and discuss the issue of universal quantum gates. Quantum
algorithms, including Shor's factorization algorithm and Grover's algorithm for
searching databases, are explained. I will devote much attention to
understanding what the origins of the quantum computational power are, and what
the limits of this power are. Finally, I describe the recent theoretical
results which show that quantum computers maintain their complexity power even
in the presence of noise, inaccuracies and finite precision. I tried to put all
results in their context, asking what the implications to other issues in
computer science and physics are. In the end of this review I make these
connections explicit, discussing the possible implications of quantum
computation on fundamental physical questions, such as the transition from
quantum to classical physics.Comment: 77 pages, figures included in the ps file. To appear in: Annual
Reviews of Computational Physics, ed. Dietrich Stauffer, World Scientific,
vol VI, 1998. The paper can be down loaded also from
http://www.math.ias.edu/~doria