  We describe a model for the interaction of the internal (spin) degree of
freedom 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 invariant. 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 constraint 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 preliminary simulation results illustrating the
intermediate behavior in the presence of weak quantum noise.
