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Asian-European Journal of Mathematics

On the theory and application of one-step numerical schemes for solving quantum stochastic differential equation (QSDE)

T. O. Akinwumi, Bolujo Joseph Adegboyegun

Published Aug 29, 2014• 0 citations
Publisher

Abstract

This paper presents one-step numerical schemes for solving quantum stochastic differential equation (QSDE). The algorithms are developed based on the definition of QSDE and the solution techniques yield rapidly convergent sequences which are readily computable. As well as developing the schemes, we perform some numerical experiments and the solutions obtained compete favorably with exact solutions. The solution techniques presented in this work can handle all class of QSDEs most especially when the exact solution does not exist.

DOI
10.1142/s1793557114500375

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The classical stochastic differential equation given as dX(t,ω)=a(t,X) dt+b(t,X) dQ(t),X(t0)=X0,t∈[t0,T],dX(t, \omega) = a(t, X) \, dt + b(t, X) \, dQ(t), \quad X(t_0) = X_0, \quad t \in [t_0, T],dX(t,ω)=a(t,X)dt+b(t,X)dQ(t),X(t0​)=X0​,t∈[t0​,T], where the driving process Q(t) Q(t) Q(t) is a martingale, and a a a and b b b are adapted functionals has several numerical schemes developed for it, where each exhibits distinct stability and convergence properties depending on the nature of the driving process and the functional space of the solution. A generalization of this equation is the quantum stochastic differential equation introduced by Hudson and Parthasarathy, expressed as dX(t)=E(t,X(t)) dΛπ(t)+F(t,X(t)) dAf†(t)+G(t,X(t)) dAg(t)+H(t,X(t)) dt,dX(t) = E(t, X(t)) \, d\Lambda_\pi(t) + F(t, X(t)) \, dA_f^\dagger(t) + G(t, X(t)) \, dA_g(t) + H(t, X(t)) \, dt,dX(t)=E(t,X(t))dΛπ​(t)+F(t,X(t))dAf†​(t)+G(t,X(t))dAg​(t)+H(t,X(t))dt, where E,F,G, E, F, G, E,F,G, and H H H are stochastic operator-valued functions defined on a Hilbert space. In 1992, Ekhaquere reformulated this equation in an equivalent form involving the non-classical differential equation stated as; ddt⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0)=X0,t∈[t0,T],\frac{d}{dt} \langle \eta, X(t) \xi \rangle = P(t, X(t)) \langle \eta, \xi \rangle, \quad X(t_0) = X_0, \quad t \in [t_0, T],dtd​⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0​)=X0​,t∈[t0​,T], which represents an ordinary differential equation of non-classical type. The stochastic process X(t) X(t) X(t) acts as a densely defined linear operator on a tensor product of two Hilbert spaces, one of which is the Boson Fock space. The vectors η \eta η and ξ \xi ξ belong to a dense subset of this tensor product Hilbert space, and the mapping (η,ξ)↦P(t,X)(η,ξ)(\eta, \xi) \mapsto P(t, X)(\eta, \xi)(η,ξ)↦P(t,X)(η,ξ) defines a sesquilinear form for each fixed pair (t,X)(t, X)(t,X). We analyze the numerical approximation of the QSDE using its equivalent inner-product form. The numerical investigation employs three principal methods: the Lagrangian Quadrature Method, the Linear Multistep Scheme, and the Newton–Cotes Method. Each method is carefully designed to ensure the consistency, stability, and convergence of the approximate solution X(t)X(t)X(t) toward the exact stochastic process. These schemes are implemented using Python programming language, which provides a flexible and efficient computational environment for testing the proposed algorithms. Our computational results show that the developed schemes generalize Taylor-type approximations of Itô processes under weak convergence criteria. The proposed algorithms exhibit numerical stability and yield results that compare favorably with analytical benchmarks. Furthermore, the implementation avoids direct approximation of the driving quantum processes, thereby preserving the natural adaptation of the schemes to quantum dynamical models.
Multistep schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with the basic field operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to matrix element of solution being sufficiently differentiable. Results concerning convergence of explicit schemes of class A in the topology of the locally convex space of solution are presented.Numerical examples are given.
A continuous time Euler approximation scheme and a computational theorem for weak solutions of Lipschitzian quantum stochastic differential equations (QSDEs) are established. The work is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the equivalent forms of the equations satisfying the Caratheodory conditions. Our results generalize analogous results concerning classical Ito stochastic differential equations and those of differential equations in a Banach space context to the noncommutative quantum setting involving unbounded linear operators on a Hilbert space. Numerical examples are given.
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