Quantum simulation of ODEs, PDEs and related problems. Part I: Schrodingerisation of linear systems

Abstract

One of the oldest and currently most promising application areas for quantum devices is quantum simulation. Popularised by Feynman in the early 1980s, it is important for the efficient simulation – compared to its classical counterpart – of one special partial differential equation (PDE): Schrodinger’s equation. This is possible because quantum devices themselves naturally obey Schrodinger’s equation. Just like with large-scale quantum systems, classical methods for other high-dimensional and large-scale PDEs often suffer from the curse-of-dimensionality, which a quantum treatment might in certain cases be able to mitigate. Aside from Schrodinger’s equation, can quantum simulators also efficiently simulate other PDEs? To enable the simulation of PDEs on quantum devices that obey Schrodinger’s equations, it is crucial to first develop good methods for mapping other PDEs onto Schrodinger’s equations. After a brief introduction to quantum simulation, I will address the above question in this talk by introducing a simple and natural method for mapping other linear PDEs onto Schrodinger’s equations. It turns out that by transforming a linear partial differential equation (PDE) into a higher-dimensional space, it can be transformed into a system of Schrodinger’s equations, which is the natural dynamics of quantum devices. This new method – called Schrodingerisation – thus allows one to simulate, in a simple way, any general linear partial differential equation and system of linear ordinary differential equations via quantum simulation. This simple methodology is also very versatile. It can be used directly either on discrete-variable quantum systems (qubits) or on analog/continuous quantum degrees of freedom (qumodes). The continuous representation in the latter case can be more natural for PDEs since, unlike most computational methods, one does not need to discretise the PDE first. In this way, we can directly map D-dimensional linear PDEs onto a (D + 1)-qumode quantum system where analog Hamiltonian simulation on (D + 1) qumodes can be used. It is the quantum version of analog computing and is more amenable to near-term realisation. I will show how this Schrodingerisation method can be applied to linear PDEs, systems of linear ODEs and also linear PDES with random coefficients, where the latter is important in the area of uncertainty quantification. Furthermore, these methods can be extended to solve problems in linear algebra by transforming iterative methods in linear algebra into the evolution of linear ODEs.