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

TL;DR

This work reframes simple linear-algebra iterations as continuous-time dynamical systems and applies Schrödingerisation to map them into quantum dynamics, enabling quantum simulation of Jacobi- and power-method–like processes. By handling both Hermitian and non-Hermitian cases and exploring discrete, CV, and hybrid CV-DV architectures, it presents quantum algorithms for solving linear systems and for estimating maximum eigenvalues and eigenvectors via quantum state preparation. Key contributions include explicit continuous-time encodings, dilation-based treatments for non-Hermitian operators, and complexity analyses that contrast with conventional quantum linear-system solvers like HHL. The approach broadens the toolkit for quantum linear algebra, offering potentially favorable scaling when spectral gaps are favorable and overlaps with target states are large, and it suggests practical pathways for implementing these methods on hybrid quantum hardware.

Abstract

Quantum simulation is known to be capable of simulating certain dynamical systems in continuous time -- Schrodinger'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 Schrodinger's equations via a method called Schrodingerisation. Building on the observation that iterative methods in linear algebra, and more generally discrete linear dynamical systems, can be viewed as discrete time approximations of dynamical systems which evolve continuously in time, we can apply the Schrodingerisation technique. Thus quantum simulation can be directly applied to the continuous-time limits of some of the simplest iterative methods. This applies to general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we introduce the quantum Jacobi and quantum power methods for solving the quantum linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix respectively. The proposed quantum simulation can be performed on either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative method to solve linear algebra problems using quantum simulation.