Hy-Q Seminar - Vincent Elfving - Qu&Co

Title: Solving differential equations with differentiable quantum circuits

Abstract: With machine learning showing ever-increasing promise in wide variety of applications, recent developments of quantum computing hardware have allowed the realization of a relatively novel computational research field called Quantum Machine Learning (QML). We propose a new QML algorithm to solve systems of (nonlinear) differential equations, ubiquitous in fields ranging from thermodynamics, to fluid mechanics, to finance. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle. The results show an instance of QML applied to a practical problem, and we provide an outlook on what the future might bring, including applications with stochastic components, generative modelling, and general model discovery setting.

Bio: Vincent Elfving is co-founder and CTO at Qu&Co, a quantum-computational software startup based in Europe. His background is in theoretical applied physics, in particular in the areas of quantum information and quantum optics. He graduated with a PhD from the Niels Bohr Institute in Denmark, and worked at Google in Santa Barbara, USA in an experimental team realizing near-term variational quantum algorithms. In 2018 he co-founded Qu&Co together with CEO Benno Broer, and is now leading several multi-year collaborative research project with industrial clients, as well as building application-centric quantum computational software solutions in the areas of chemistry, multiphysics and finance.