Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

Mora, Carlos

doi:10.1090/S0025-5718-03-01595-3

Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups
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by Carlos M. Mora PDF

Math. Comp. 73 (2004), 1393-1415 Request permission

Abstract:

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.

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