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 prescribed $\epsilon$ error bounds with polynomial dependence on state dimension $d$ . Classical numerical methods scale exponentially with dimension, thus, our approach, under the aforementioned conditions, provides an exponential speed-up over traditional approaches.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 prescribed$\epsilon$error bounds with polynomial dependence on state dimension$d$. Classical numerical methods scale exponentially with dimension, thus, our approach, under the aforementioned conditions, provides an exponential speed-up over traditional approaches.

arXiv:2303.02463IEEE International Conference on Quantum Computing and Engineering (QCE)Sep 17, 20234 citations

Quasiprobability methods for nonlinear chemical and optical systems

P. D. Drummond, Crispin W. Gardiner, Daniel F. Walls

A review of quasiprobability methods for transforming chemical and quantum-optical master equations into Fokker-Planck equations is presented. For cases where conventional representations lead to Fokker-Planck equations with non-positive-definite diffusion coefficients; e.g., sub-Poissonian statistics, a generalization of the representation involving an extension to the complex plane enables analytic results to be obtained for certain nonlinear chemical and optical processes. Alternatively, a different integration measure may be chosen which ensures a positive distribution and Fokker-Planck equation with positive-semidefinite diffusion coefficients. This enables stochastic differential equations to be defined. These methods are applied to two-photon absorption and dispersive bistability in quantum optics where nonclassical photon statistics arise and to two models of nonlinear chemical reactions where sub-Poissonian statistics occur.A review of quasiprobability methods for transforming chemical and quantum-optical master equations into Fokker-Planck equations is presented. For cases where conventional representations lead to Fokker-Planck equations with non-positive-definite diffusion coefficients; e.g., sub-Poissonian statistics, a generalization of the representation involving an extension to the complex plane enables analytic results to be obtained for certain nonlinear chemical and optical processes. Alternatively, a different integration measure may be chosen which ensures a positive distribution and Fokker-Planck equation with positive-semidefinite diffusion coefficients. This enables stochastic differential equations to be defined. These methods are applied to two-photon absorption and dispersive bistability in quantum optics where nonclassical photon statistics arise and to two models of nonlinear chemical reactions where sub-Poissonian statistics occur.

Physical Review AAug 1, 198178 citations

\epsilon

error bounds with polynomial dependence on state dimension

d

. Classical numerical methods scale exponentially with dimension, thus, our approach, under the aforementioned conditions, provides an exponential speed-up over traditional approaches.