Analog quantum simulation of partial differential equations

Abstract

Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular—Schrödinger’s equation. Can quantum simulators also efficiently simulate other PDEs? While most computational methods for PDEs—both classical and quantum—are digital (they must be discretised first), PDEs have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrödingerisation, we show how to directly map D-dimensional linear PDEs onto a ( D + 1 ) -qumode quantum system where analog or continuous-variable (CV) Hamiltonian simulation on D + 1 qumodes can be used. This very simple methodology does not require one to discretise PDEs first, and it is not only applicable to linear PDEs but also to some nonlinear PDEs and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker-Planck equation, Black-Scholes equations, wave equation and Maxwell’s equations. We also devise new protocols for linear PDEs with random coefficients, important in uncertainty quantification, where it is clear how the analog or CV framework is most natural. This also raises the possibility that some PDEs may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.