  In this article, we investigate the problem of engineering synchronization in
non-Markovian quantum systems. First, a time-convoluted linear quantum
stochastic differential equation is derived which describes the Heisenberg
evolution of a localized quantum system driven by multiple colored noise
inputs. Then, we define quantum expectation synchronization in an augmented
system consisting of two subsystems. We prove that, for two homogenous
subsystems, synchronization can always be synthesized without designing direct
Hamiltonian coupling given that the degree of non-Markovianity is below a
certain threshold. System parameters are explicitly designed to achieve quantum
synchronization. Also, a numerical example is presented to illustrate our
results.
