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

Feynman integral and stochastic differential equations

Zbigniew Haba

Springer eBooksJan 1, 19990 citations

Simulating Stochastic Diffusions by Quantum Walks

Yan Wang

Stochastic differential equation (SDE) and Fokker-Planck equation (FPE) are two general approaches to describe the stochastic drift-diffusion processes. Solving SDEs relies on the Monte Carlo samplings of individual system trajectory, whereas FPEs describe the time evolution of overall distributions via path integral alike methods. The large state space and required small step size are the major challenges of computational efficiency in solving FPE numerically. In this paper, a generic continuous-time quantum walk formulation is developed to simulate stochastic diffusion processes. Stochastic diffusion in one-dimensional state space is modeled as the dynamics of an imaginary-time quantum system. The proposed quantum computational approach also drastically accelerates the path integrals with large step sizes. The new approach is compared with the traditional path integral method and the Monte Carlo trajectory sampling.Stochastic differential equation (SDE) and Fokker-Planck equation (FPE) are two general approaches to describe the stochastic drift-diffusion processes. Solving SDEs relies on the Monte Carlo samplings of individual system trajectory, whereas FPEs describe the time evolution of overall distributions via path integral alike methods. The large state space and required small step size are the major challenges of computational efficiency in solving FPE numerically. In this paper, a generic continuous-time quantum walk formulation is developed to simulate stochastic diffusion processes. Stochastic diffusion in one-dimensional state space is modeled as the dynamics of an imaginary-time quantum system. The proposed quantum computational approach also drastically accelerates the path integrals with large step sizes. The new approach is compared with the traditional path integral method and the Monte Carlo trajectory sampling.

ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE)Aug 4, 20136 citations

A General Solution of the Fokker-Planck Equation

Marcelo Tozo Araujo, Elso Drigo Filho

Journal of Statistical PhysicsDec 19, 201115 citations

Quantum-theoretical treatments of three-photon processes

M. K. Olsen, Lev I. Plimak, Michael Fleischhauer

We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency. As the generalized Fokker-Planck equation (GFPE) for the positive- $P$ function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency. As the generalized Fokker-Planck equation (GFPE) for the positive-$P$ function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.

Physical Review AApr 22, 200212 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.

P

function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.