Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

We address the problem of approximating numerically the solutions $\left( X_{t}:t\in \left[ 0,T\right] \right)$ of stochastic evolution equations on Hilbert spaces $\left( \mathfrak{h},\left\langle \cdot ,\cdot \right\rangle \right)$, with respect to Brownian motions, arising in the unraveling of backward quantum master equations. In particular, we study the computation of mean values of $\left\langle X_{t},AX_{t}\right\rangle$, where $A$ is a linear operator. First, we introduce estimates on the behavior of $X_{t}$. Then we characterize the error induced by the substitution of $X_{t}$ with the solution $X_{t,n}$ of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of $\mathbf{E} \left\langle \tilde{X}_{t,n},A\tilde{X}_{t,n}\right\rangle$ to $\mathbf{E} \left\langle X_{t},AX_{t}\right\rangle$, where $\tilde{X}_{t,n}$ 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.