Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
This chapter presents an approach to characterize a quantum version of cellular
automata which is based on continuous-variable systems and equipped with a quasifree
dynamics. For the general concept of quantum cellular automata we follow the
lines of Schumacher and Werner [70].
A cellular automaton (ca) is a discrete, regular, dynamical system with synchronous,
uniform time evolution generated by a local interaction. The dynamics
acts on an infinite lattice, exhibits translational symmetry and has finite propagation
speed. These characteristics render them a useful tool for the simulation of
dynamical systems of regularly arranged, discrete, identical constituents. Within
physics classical cas have been employed to study problems in particular from statistical
mechanics, e.g. Ising spin dynamics, point particle gases, percolation or annealing
[71]. Other problems include the dynamics of bacteria colony growth, forest
fires, sand piles or road traffic. Moreover, in classical information theory cas are
a model of universal computation, since a Turing machine can be simulated by
a ca. And finally, cas can provide diversion, e.g. in the form of John Conway’s
Game of Life[72]. Due to these applications the concept of a quantum cellular
automaton (qca), i.e. a quantum system with the above characteristics, seems to
promise exciting possibilities. In fact, such a quantum extension of cas has already
been considered by R. Feynman in his paper on the power of quantum computation
from 1982 [73]. Different notions of qcas were studied in the literature and found
to be capable of universal quantum computation [74, 75, 76, 77, 78]. And recently,
Vollbrecht et al. [79,80] have introduced a scheme for reversible, universal quantum
computing in translationally invariant systems which proved to be a qca.
While the development of a universal quantum computer is perhaps the most ambitious
aim of quantum information science, it is at the same time possibly the most
difficult undertaking (especially for interesting input problem size). However, specific
computational tasks might be more easy to accomplish but nevertheless very
useful from the point of view of general physics, e.g. the simulation of quantum
systems. Since Hilbert space dimension grows exponentially with the number of constituents,
classical computers face serious performance problems even for moderate
system sizes. This obstacle could be overcome by quantum computers which convert
the scaling into a feature. Even the simulation of quantum toy models with
moderate system size could provide valuable insight into real-world systems. The inherent
translational symmetry would make qcas especially suited for the simulation
of models in solid state physics.
In addition, the concept of a qca might prove useful for the realization of quantum
computing in optical lattices [81] and arrays of microtraps [82]. The experimental
technology of these systems is quite highly developed and they are promising
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