Quantum Information Processing, Vol. 4, No. 4, October 2005 (© 2005)
From Dirac to Diffusion: Decoherence in Quantum
Peter J. Love
and Bruce M. Boghosian
Received February 7, 2005; accepted July 10, 2005
We describe a model for the interaction of the internal (spin) degree of free-
dom of a quantum lattice-gas particle with an environmental bath. We impose the
constraints that the particle-bath interaction be ﬁxed, while the state of the bath
is random, and that the effect of the particle-bath interaction be parity invari-
ant. The condition of parity invariance deﬁnes a subgroup of the unitary group
of actions on the spin degree of freedom and the bath. We derive a general con-
straint on the Lie algebra of the unitary group which deﬁnes this subgroup, and
hence guarantees parity invariance of the particle-bath interaction. We show that
generalizing the quantum lattice gas in this way produces a model having both
classical and quantum discrete random walks as different limits. We present pre-
liminary simulation results illustrating the intermediate behavior in the presence
of weak quantum noise.
KEY WORDS: Quantum lattice gas; decoherence; quantum random walk.
PACS: 03.67.Lx; 05.40.Ca.
Lattice gases are arguably the simplest models for the simulation of
classical physical systems. These models provide elementary microscopic
dynamics whose hydrodynamic limits are, inter alia, the diffusion equation,
Burgers’ equation and the Navier Stokes equations.
They also provide a
simple arena for the creation of new models of physical phenomena such
as multicomponent ﬂow and dynamical geometry.
Lattice gases pos-
sess deterministic, stochastic and quantum (unitary) formulations.
Department of Mathematics, Tufts University, Medford, Massachusetts, 02155, USA.
Current Address: D-Wave Systems Inc., Suite 100, 4401 Still Creek Drive, Burnaby, British
Columbia, Canada V5C 6G9.
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1570-0755/05/1000-0335/0 © 2005 Springer Science+Business Media, Inc.