Continuous Variables Quantum Algorithm for Solving Ordinary Differential Equations

Solving Ordinary Differential Equations (ODE) is important for a broad range of domains, such as engineering, weather forecast, and finance. Most quantum algorithms proposed to solve them revolve around transforming ODEs into a system of linear equations, in order to benefit from the exponential advantage promised by the HHL algorithm. However, there are known limitations to this subroutine such as the linear scaling with condition number. In particular, for HHL-based ODE solvers, this dependency yields a complexity that generally grows exponentially with the integration time. In this paper, we present a scheme that does not rely on the HHL subroutine. We use the Koopman Von Neumann (KvN) formalism that maps arbitrary nonlinear dynamics to a Hilbert space where the dynamics are unitary. This effectively reduces the ODE problem to a Hamiltonian time evolution, which is a well-known problem in quantum computing. However, this comes at a cost as the KvN formalism is expressed in an infinite-dimensional Hilbert space, and involves nonphysical states with infinite energies. Previous works have tackled this by truncating the Hilbert space, and by approximating all the relevant states and operations on qubit-based systems. Instead of mapping to qubit-based computations, in this paper, we investigate the direct use of continuous-variable quantum computers for this problem. We provide an algorithm to compile a sequence of Gaussian and non-Gaussian continuous variables gates to solve an arbitrary one-dimensional polynomial ODE. We analyze the algorithm and propose that it is intrinsically better suited for solving so-called initial distribution problems, rather than initial condition problems. We propose the first steps towards a comprehensive complexity and specify which steps need to be developed for a complete analysis.