  Iterative Refinement (IR) is a classical computing technique for obtaining
highly precise solutions to linear systems of equations, as well as linear
optimization problems. In this paper, motivated by the limited precision of
quantum solvers, we develop the first IR scheme for solving semidefinite
optimization (SDO) problems and explore two major impacts of the proposed IR
scheme. First, we prove that the proposed IR scheme exhibits quadratic
convergence toward an optimal solution without any assumption on problem
characteristics. We also show that using IR with Quantum Interior Point Methods
(QIPMs) leads to exponential improvements in the worst-case overall running
time of QIPMs, compared to previous best-performing QIPMs. We also discuss how
the proposed IR scheme can be used with classical inexact SDO solvers, such as
classical inexact IPMs with conjugate gradient methods.
