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Mathematics of Computation

Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups

Carlos M. Mora

Published Aug 4, 2003• 15 citations
Publisher

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.

DOI
10.1090/s0025-5718-03-01595-3

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We address the problem of approximating numerically the solutions (Xt:t∈[0,T])\left ( X_{t}:t\in \left [ 0,T\right ] \right )(Xt​:t∈[0,T]) of stochastic evolution equations on Hilbert spaces (h,⟨⋅,⋅⟩)\left ( \mathfrak {h},\left \langle \cdot ,\cdot \right \rangle \right )(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⟩\left \langle X_{t},AX_{t}\right \rangle⟨Xt​,AXt​⟩, where AAA is a linear operator. First, we introduce estimates on the behavior of XtX_{t}Xt​. Then we characterize the error induced by the substitution of XtX_{t}Xt​ with the solution Xt,nX_{t,n}Xt,n​ of a convenient stochastic ordinary differential equation. It allows us to establish the rate of convergence of E⟨X~t,n,AX~t,n⟩\mathbf {E} \left \langle \tilde {X}_{t,n},A\tilde {X}_{t,n}\right \rangleE⟨X~t,n​,AX~t,n​⟩ to E⟨Xt,AXt⟩\mathbf {E} \left \langle X_{t},AX_{t}\right \rangleE⟨Xt​,AXt​⟩, where X~t,n\tilde {X}_{t,n}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.
The paper deals with the numerical solution of the nonlinear Itô stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finite-dimensional nonlinear stochastic Schrödinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme.
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.
Stochastic nonlinear dynamics underlie many models in engineering and computational physics, yet accurate high-dimensional simulation remains challenging. We present a quantum algorithm for a broad class of NNN-dimensional stochastic differential equations with dissipation and quadratic drift. The algorithm applies to strongly nonlinear systems with all-to-all interactions, thereby extending the scope of previously known quantum algorithms that were limited to weak nonlinearity and sparse systems. For norm-preserving drifts, a condition satisfied by key fluid dynamics discretizations, our method approximates expectation values of low-order correlation functions with rigorous error bounds at a cost polynomial in log⁡(N)\log{(N)}log(N) and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of NNN interacting quantum harmonic oscillators with cost polylogarithmic in NNN. Finally, we formulate turbulence models, including Navier-Stokes and damped Euler equations, within this framework, opening a route to quantum simulation of strongly nonlinear SDEs governing turbulence and nonlinear wave dynamics.
It is well known that the state operator of an open quantum system can be generically represented as the solution of a time-local equation — a quantum master equation. Unraveling in quantum trajectories offers a picture of open system dynamics dual to solving master equations. In the unraveling picture, physical indicators are computed as Monte Carlo averages over a stochastic process valued in the Hilbert space of the system. This approach is particularly adapted to simulate systems in large Hilbert spaces. We show that the dynamics of an open quantum system generically admits an unraveling in the Hilbert space of the system described by a Markov process generated by ordinary stochastic differential equations for which rigorous concentration estimates are available. The unraveling can be equivalently formulated in terms of norm-preserving state vectors or in terms of linear “ostensible” processes trace preserving only on average. We illustrate the results in the case of a two level system in a simple boson environment. Next, we derive the state-of-the-art form of the Diósi-Gisin-Strunz Gaussian random ostensible state equation in the context of a model problem. This equation provides an exact unraveling of open systems in Gaussian environments. We compare and contrast the two unravelings and their potential for applications to quantum error mitigation.
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.