Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

We address the problem of approximating numerically the solutions (Xt : t ∈ [0, T ]) of stochastic evolution equations on Hilbert spaces (h, 〈·, ·〉), with respect to Brownian motions, arising in the unraveling of backward quantum master equations. In particular, we study the computation of mean values of 〈Xt, AXt〉, where A is a linear operator. First, we introduce estimates on the behavior of Xt. Then we characterize the error induced by the substitution of Xt with the solution Xt,n of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of E 〈 X̃t,n, AX̃t,n 〉 to E 〈Xt, AXt〉, where X̃t,n denotes the explicit Euler method. Finally, we consider an extrapolation method based on the Euler scheme. An application to the quantum harmonic oscillator system is included.