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GSoC2015: Quantum MCWF method and QuDynamics

Hello everyone,

During the past few weeks we have worked on integrating Quantum Monte-Carlo Wave Function Method as one of the solvers. We already had an initial version from QuDOS.jl and we had to set it up for QuDynamics. The setup of the alogrithm is as follows :

  1. For a single trajectory, we draw a state using draw from the iterator version of QuMCWFEnsemble(We will get back to the structure of QuMCWFEnsemble at a later stage) and for the first time step we draw a random number.
  2. We propagate the state from the previous step, and then we check certain conditions which involve the comparison of norm(state_after_propagation)^2, eps, jtol which is a constant. Here eps is a random number and is generated only for the first step and in certain cases again generated if one of the conditions from above gets satisfied. eps has to be stored from the current step to the use in next time step, for this we had to convert the QuMCF to a type from immutable which is different when compared to other solvers.
  3. Finally, as a result of the conditions, we end up with the propagated state, which is used to generate the density matrix.

Coming back to the structure of QuMCWFEnsemble this basically takes in a input of initial state, the number of trajectories (which is defaulted to 500), and also has decomp as a field which stores the eigen value decomposition of initial state, again we could have a vector passed, or we could have a density matrix for the initial state, for the former we pass nothing for decomp parameter but for the latter we store the decomposition. The iterator version of QuMCWFEnsemble generates a state using the draw function which again has a random value generated for the choice of eigen vector from the decomposition.

The type QuMCWF has the following fields : eps and options, the options is defaulted to an empty Dict, but can be used to provide a solver using the keyword :solver which is defaulted to use QuExpmV, the solver is required for propagating the state from draw. eps is the random number and rand() is used to generate eps.

The above implementation has been merged into the master and we have compared the results to QuTiP. Here is the notebook which present the QuTiP example. Also, we have worked on integrating some examples into the repo and this involves Jaynes-Cummings Model using MCWF method.

I would like to conclude by providing an example of how Quantum MCWF method works in QuDynamics :

# System construction
ad = raiseop(2)
hamiltonian = ad*ad'
cs = coherentstatevec(2, 1.)
c_ops = [lowerop(2)]
tlist = 0.:0.25:2*pi
rho = complex(cs*cs')
rhos = Array(typeof(rho), length(tlist)-1)
qumcwfen = QuMCWFEnsemble(complex(cs), 1000)

# rhos to store density matrices at every time step
for i=1:length(tlist)-1
    rhos[i] = complex(zeros(rho))
end

# using MCWF to propagate and storing states at every time step
# looped over the number of trajectories (length(qumcwfen))
for psi0 in qumcwfen
    i = 1
    for (t,psi) in QuPropagator(hamiltonian, c_ops, psi0, tlist, QuMCWF([:solver=>QuODE45()]))
        rhos[i] = rhos[i] + (psi*psi')/length(qumcwfen)/norm(psi)^2
        i = i + 1
    end
end

The next targets include working on the documentation and also the related scripts.