Quantum Simulation of Partial Differential Equations via Schrödingerization

We present a novel new way-called Schrödingerization-to simulate general (quantum and nonquantum) systems of linear ordinary and partial differential equations (PDEs) via quantum simulation. We introduce a new transform, referred to as the warped phase transformation, where any linear-including nonautonamous-system of ordinary or partial differential equation can be recast into a system of Schrödinger's equations, in real time, in a straightforward way. This approach is not only applicable to PDEs for classical problems but is also useful for quantum problems, including the preparation of quantum ground states and Gibbs thermal states, the simulation of quantum states in random media in the semiclassical limit, simulation of Schrödinger's equation in a bounded domain with artificial boundary conditions, and other non-Hermitian physics. This formulation is versatile enough to be applicable in a simple way to both digital quantum simulation as well as to analog quantum simulation, and using either qubits or continuous-variable quantum systems (qumodes).