Estudo de algoritmos quânticos para resolução de equações diferenciais parciais

Abstract

With great impact in various areas of engineering and mathematics, quantum computing is an emerging technology with one of its applications in resolving large-scale mathematical problems, such as optimization and simulation. Aiming to solve differential equations with quantum computers, we study quantum algorithms proposed to solve then and their implementations on state-of-the-art systems.

In particular, we study a quantum algorithm for solving nonhomogeneous linear partial differential equations proposed by J. M. Arrazola et al. in [Phys. Rev. A 100, 032306 (2019)]. By inverting the differential operator, it is possible to obtain one particular solution encoded on the wave function of a continuous-variables system, along with the preparation and measurement of special ancillary modes. Despite being a simple idea, running the algorithm on a physical quantum computer requires sophisticated elements, such as creating a large number of states in superposition and detecting states with low probability of success. In this work we suggest modifications in its structure to reduce the costs of preparing the initial ancillary states, increase the probability of success, and improve the precision of the algorithm for a specific set of inputs. These achievements enable easier experimental implementation of the quantum algorithm based on nowadays technology.