  In this paper, a quantum filter for estimating the states of a non-Markovian
qubit system is presented in an augmented Markovian system framework including
both the qubit system of interest and multi-ancillary systems for representing
the internal modes of the non-Markovian environment. The colored noise
generated by the multi-ancillary systems disturbs the qubit system via a direct
interaction. The resulting non-Markovian dynamics of the qubit is determined by
a memory kernel function arising from the dynamics of the ancillary system. In
principle, colored noise with arbitrary power spectrum can be generated by a
combination of Lorentzian noises. Hence, the quantum filter can be constructed
for the qubit disturbed by arbitrary colored noise and the conditional state of
the qubit system can be obtained by tracing out the multi-ancillary systems. An
illustrative example is given to show the non-Markovian dynamics of the qubit
system with Lorentzian noise.
