Search

We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the "quantum difference methods"). For the heat and linear hyperbolic equations we study the impact of explicit and implicit time discretizations on quantum advantages over the classical difference method. For the multiscale problem, we find the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as O (1 / ε) , where ε is the scaling parameter, while the scaling for the multiscale Asymptotic-Preserving (AP) schemes does not depend on ε. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing. • New quantum algorithm is proposed for a prototype multiscale problem. • Asymptotic-preserving schemes are shown to be important in quantum computation. • There is no difference in the cost of the quantum algorithm for the heat equation when using the forward-Euler compared to the Crank-Nicolson scheme. [ABSTRACT FROM AUTHOR]

This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretization of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [1,2] – it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions. • The first quantum simulation algorithms for PDEs with physical boundary conditions. • The first quantum simulation algorithms for PDEs incorporating physical interface conditions. • The first quantum algorithm for partial transmission and reflections from geometric optics with interfaces. [ABSTRACT FROM AUTHOR]