Efficient quantum algorithm for dissipative nonlinear differential equations

Abstract

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log⁡T,log⁡n,log⁡1/ϵ)/ϵ, where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

Koopman von Neumann mechanics and the Koopman representation: A perspective on solving nonlinear dynamical systems with quantum computers

“…Both Koopman von Neumann wave mechanics [21,49,50] and Carleman linearization [24,37,38] have been proposed for the purpose of simulating classical nonlinear dynamics on a quantum computer, leveraging their linear representations. Towards the first aim, we present the duality between the forward representation and backward representation of dynamical systems.…”

“…. -see [24]. In that case the quantum algorithm prepares a quantum state whose amplitudes are proportional to all these observables.…”

“…Initial work attempting the integration of nonlinear, classical differential equations resulted in inefficient quantum algorithms requiring exponential resources [20]. However, it was recently realized that, for the nonlinear case, methods for lifting nonlinear equations to an infinite-dimensional Hilbert space, such as those using Koopman-von Neumann mechanics and the Koopman representation discussed below, were the appropriate setting [21][22][23][24][25][26].…”

“…The apparently disparate approaches that have been taken towards the lifting of nonlinear classical systems to a Hilbert space have involved using truncated Carleman linearization [24], simulating the dynamics of the nonlinear Schrodinger equation [25], and the Koopmanvon Neumann formulation of classical systems [21].…”

Towards Cosmological Simulations of Dark Matter on Quantum Computers

“…Thus something beyond these naive approaches is needed. Despite recent progress on solving nonlinear equations by direct simulation (Liu et al 2020;Lloyd et al 2020), we focus instead on the use of variational quantum computing (VQC; Cerezo et al 2020).…”