Qubit phase space: $\mathrm{SU}\left(n\right)$ coherent-state $P$ representations

We introduce a phase-space representation for qubits and spin models. The technique uses an $\mathrm{SU}\left(n\right)$ 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 ultracold 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 $\mathrm{SU}\left(n\right)$ phase space is used to describe $n$-level systems. In general, the requirement that time evolution be stochastic corresponds to a restriction to Hamiltonians and master equations that are quadratic in the group generators or generalized spin operators.

Figure 1

(Color online) Stochastic calculation of two-spin correlation: $J\beta =1$, 4000 trajectories, step size 0.05, semi-implicit method with three iterations.

Figure 2

(Color online) Stochastic calculation of two-spin correlations over a range of $J\beta$, comparison to exact results.

Figure 3

(Color online) Sampling error of two-spin correlations.

Figure 4

(Color online) Nearest-neighbor correlations for a $10\times 10$ lattice as a function of inverse temperature. Circles: results from stochastic calculations, 1000 trajectories. Solid line: exact solution in the limit of an infinite lattice.