Qubit phase-space: SU(n) coherent state P-representations
  We introduce a phase-space representation for qubits and spin models. The technique uses an SU(n) coherent state basis, and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive P-function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an Ising model of arbitrary structure can be efficiently simulated using SU(2) or atomic coherent states. The technique utilizes a transformation from a canonical (imaginary-time) weighted simulation to an equivalent unweighted real-time simulation. The results are compared to the exactly soluble two-dimensional case. We note that Ising models in one, two or three dimensions are potentially achievable experimentally as a lattice-gas of ultra-cold atoms in optical lattices. The technique is not restricted to canonical ensembles or to Ising-like couplings. It is also able to be used for real-time evolution, and for systems whose time-evolution follows a master-equation describing decoherence and coupling to external reservoirs. The case of SU(n) phase-space is used to describe n-level systems. In general, the requirement that time-evolution is stochastic corresponds to a restriction to Hamiltonians and master-equations that are quadratic in the group generators or generalized spin operators. 
D. W. Barry and P. D. Drummond