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.
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.
We present a class of exponential integrators to compute solutions of the stochastic Schrödinger equations arising from the modeling of open quantum systems. To be able to implement the methods within the same framework as the deterministic counterpart, we express the solution using Kunita's representation. With appropriate truncations, the solution operator can be written as matrix exponentials, which can be efficiently implemented by the Krylov subspace projection. The accuracy is examined in terms of the strong convergence by comparing trajectories, and in terms of the weak convergence by comparing the density-matrix operators. We show that the local accuracy can be further improved by introducing third-order commutators in the exponential. The effectiveness of the proposed methods is tested using the example from Di Ventra et al. [J. Phys.: Condens. Matter 16, 8025 (2004)JCOMEL0953-898410.1088/0953-8984/16/45/024].
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
N-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) and linear in the evolution time. Our main technical advance is a subroutine for simulating an auxiliary system of
N interacting quantum harmonic oscillators with cost polylogarithmic in
N. 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.
We address the problem of approximating numerically the solutions
(Xt:t∈[0,T]) of stochastic evolution equations on Hilbert spaces
(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⟩, where
A is a linear operator. First, we introduce estimates on the behavior of
Xt. Then we characterize the error induced by the substitution of
Xt with the solution
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⟩ to
E⟨Xt,AXt⟩, where
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.
We study an explicit exponential scheme for the time discretisation of stochastic Schrödinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schrödinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.
Addressing nonlinear numerical problems in quantum computing remains challenging, but several approaches, including Carleman linearization (CL), have been proposed to construct quantum algorithms for such problems. In this work, we develop a CL-based method for a high-dimensional nonlinear stochastic differential equation (SDE)
dXt=(A(t)Xt+F(t)Xt⊗s)dt+B(t)dWt for an
RN-valued random process
Xt, aiming at quantum speed-up with respect to the dimension
N. We introduce auxiliary processes
Ytj that approximate
Xt⊗j, derive a linearized system of SDEs for them, and obtain the linear ordinary differential equation (ODE) governing their expectations. We then apply a quantum ODE solver to it and, from the resulting quantum state, extract an estimate for the expectation of a linear function of
Xt by overlap estimation. We show that, provided the nonlinear and noise terms in the SDE are sufficiently small relative to the linear term, the CL error decreases exponentially with the truncation order
J in CL, and we can estimate the target expectation with a
polylog(N) number of queries to unitaries encoding coefficient matrices
A,
F, and
B. Finally, we also run a numerical demonstration for an SDE related to Langevin dynamics, observing the exponential decay of the CL error as
J increases.