Nana Liu: Analog quantum simulation of partial differential equations

Quantum simulators were originally proposed to be helpful for simulating one partial differential equation (PDE) in particular – Schrodinger’s equation. If quantum simulators can be useful for simulating Schrodinger’s equation, it is hoped that they may also be helpful for simulating other PDEs. As with large-scale quantum systems, classical methods for other high-dimensional and large-scale PDEs often suffer from the curse-of-dimensionality (costs scale exponentially in the dimension D of the PDE), which a quantum treatment might in certain cases be able to mitigate. To enable 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.

In this talk, I will introduce the notion of Schrodingerisation: a procedure for transforming non-Schrodinger PDEs into a Schrodinger-form. This simple methodology can be used directly on analog or continuous quantum degrees of freedom – called qumodes, and not only on qubits. This continuous representation 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.

I show how this method can also be applied to linear PDEs, certain nonlinear PDEs, nonlinear ODEs and also linear PDEs with random coefficients, which is important in uncertainty quantification.

This formulation makes it more amenable to more near-term quantum simulation methods and enables simulation of PDEs that are not possible with qubit-based formulations in the near-term.