  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.
