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arXiv:math/0508490 · The Annals of Applied Probability

Numerical solution of conservative finite-dimensional stochastic Schrödinger equations

Carlos M. Mora

Published Jul 15, 2005• 18 citations
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Abstract

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.

DOI
10.1214/105051605000000403

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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.
Abstract We address the numerical solution of the finite-dimensional non-linear stochastic Schrödinger equation, which is a locally Lipschitz stochastic differential equation modeling, for instance, quantum measurement processes. We study the rate of weak convergence of an exponential scheme that reproduces the norm of the desired solution by projecting onto the unit sphere. This justifies the use of the Talay-Tubaro extrapolation procedure in the numerical simulation of open quantum systems. In particular, we prove that an Euler-Exponential scheme converges with weak-order one, and we obtain the leading order term of its weak error expansion with respect to the step-size. Then, applying the Talay-Tubaro extrapolation procedure to the Euler-Exponential scheme under consideration we get a second-order method for computing the mean values of smooth functions of the solution of the non-linear stochastic Schrödinger equation. We also prove that the exponential scheme under study has order of strong convergence 1/2, which gives theoretical support to the use of the multilevel Monte Carlo method in simulating open quantum systems. We present a numerical experiment with a quantized electromagnetic field in interaction with a reservoir that illustrates the good performance of the weak second-order method, and the multilevel Monte Carlo method.
We present a universal framework for simulating NNN-dimensional linear Itô stochastic differential equations (SDEs) on quantum computers with additive or multiplicative noises. Building on a unitary dilation technique, we establish a rigorous mapping from the general linear SDEs dXt=A(t)Xt dt+∑j=1JBj(t)Xt dWtj dX_t = A(t) X_t\,dt + \sum_{j=1}^J B_j(t)X_t\,dW_t^j dXt​=A(t)Xt​dt+∑j=1J​Bj​(t)Xt​dWtj​ to stochastic Schrödinger equations (SSE) on a dilated Hilbert space. Crucially, this embedding is pathwise exact in that the classical solution is recovered as a projection of the dilated quantum state for each fixed noise realization. We demonstrate that the resulting SSEs are {naturally implementable} on digital quantum processors, where the stochastic Wiener increments are encoded directly by preparing the ancillary qubits. Exploiting this physical mapping, we develop two algorithmic strategies: (1) a trajectory-based approach that uses sequential weak measurements to realize efficient stochastic integrators, including a second-order scheme, and (2) an ensemble-based approach that maps moment evolution to a deterministic Lindblad quantum master equation, enabling simulation without Monte Carlo sampling. We provide error bounds based on a stochastic light-cone analysis and validate the framework with numerical experiments.
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.
In this paper, we propose efficient quantum algorithms for solving nonlinear stochastic differential equations (SDE) via the associated Fokker-Planck equation (FPE). We discretize the FPE in space and time using two well-known numerical schemes, namely Chang-Cooper and implicit finite difference. We then compute the solution of the resulting system of linear equations using the quantum linear systems algorithm. We present detailed error and complexity analyses for both these schemes and demonstrate that our proposed algorithms, under certain conditions, provably compute the solution to the FPE within prescribedϵ\epsilonϵerror bounds with polynomial dependence on state dimensionddd. Classical numerical methods scale exponentially with dimension, thus, our approach, under the aforementioned conditions, provides an exponential speed-up over traditional approaches.
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.
The paper is devoted to the study of nonlinear stochastic Schrödinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born–Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator.