Quantum Information Theory with Gaussian Systems

4 Gaussian quantum cellular automata
candidates for the successful realization of a quantum computing device; in particular,
they can be scaled to considerable systems sizes. However, most quantum
computing concepts today require the individual addressing of specific constituents,
e.g. qubits within the system, which is difficult in these approaches. It is much more
feasible to change external parameters for the whole system, which is exactly a characteristic
of a ca. As the essence of these arguments, we believe that quantum cellular
automata are a promising concept which should not be neglected in the process of
designing and developing systems capable of performing quantum computation.
We will in the following deal with Gaussian quantum cellular automata, i.e. a
continuous-variable quantum system with the above characteristics of a ca. As a
motivation to their study, consider the application of simulating a one-dimensional
quantum random walk [83] on a qca. In the most simple case of a random walk, a
singleparticleor excitation moves from a starting cell to one of the neighboring
sites. The direction of each step is determined randomly, e.g. byflipping a coin
in the one-dimensional case. This dynamics is perfectly suited for implementation
on a ca since the particle moves in steps within a finite neighborhood. From many
repetitions of the walk with identical initial conditions, one obtains a distribution of
final positions for the particle. In a quantum random walk, the states of the particle
and the coin can be coherent superpositions. A unitary evolution maps the state
of the coin onto the direction of the particle and moves it to the neighboring cell
on the left or right accordingly. The outcome of a single run over several steps is a
distribution of final positions of the particle in dependence of the initial conditions
and the number of steps. In a realization on a qca, each cell could correspond to the
combination of aslotto host the particle and acointo flip for the direction of
the next step. If a particle is present in the respective cell, the dynamics of the qca
unitarily maps the state of the coin onto the direction of the particle and moves it
to the neighboring cell on the left or right accordingly. Running the qca from an
initial state with one particle and the coins on every site in a coherent superposition
ofleftandrightthen results in a quantum random walk on the line.
An obvious extension of this model to quantum diffusion is to populate the lattice
with additional particles. However, in this case it is necessary to specify the
treatment of collisions between particles. One possible solution limits the number of
particles per site to a maximum of one particle moving left and one moving right.
This corresponds to ahard core interaction, i.e. particles are not allowed to share
sites but bounce off each other upon collision. Another solution allows for an arbitrary
number of particles per site by second quantization of the random walk. This
attaches to every cell a Fock space equipped with an occupation number state basis.
Equivalently, every cell can be described as a quantum harmonic oscillator in an
excited state according to the number of particles occupying the cell. The movement
of particles over the lattice corresponds to the exchange of excitations between the
oscillators. Together with a dynamics which can be implemented or approximated by
a quadratic Hamiltonian, this bosonic system naturally gives rise to Gaussian qcas,
i.e. continuous-variable qcas which map Gaussian states onto Gaussian states in
the Schrödinger picture and which start from a Gaussian initial state. Examples of
Gaussian qcas include the free evolution, theleft-andright-shifter, a contin-
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