Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra

Abstract

Quantum simulation is capable of simulating certain dynamical systems in continuous time—Schrödinger’s equations being the most direct and well known—more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger’s equations via a method called Schrödingerisation (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.