  Established methods for characterizing quantum information processes do not
capture non-Markovian (history-dependent) behaviors that occur in real systems.
These methods model a quantum process as a fixed map on the state space of a
predefined system of interest. Such a map averages over the system's
environment, which may retain some effect of its past interactions with the
system and thus have a history-dependent influence on the system. Although the
theory of non-Markovian quantum dynamics is currently an active area of
research, a systematic characterization method based on a general
representation of non-Markovian dynamics has been lacking.
  In this article we present a systematic method for experimentally
characterizing the dynamics of open quantum systems. Our method, which we call
quantum process identification (QPI), is based on a general theoretical
framework which relates the (non-Markovian) evolution of a system over an
extended period of time to a time-local (Markovian) process involving the
system and an effective environment. In practical terms, QPI uses time-resolved
tomographic measurements of a quantum system to construct a dynamical model
with as many dynamical variables as are necessary to reproduce the evolution of
the system. Through numerical simulations, we demonstrate that QPI can be used
to characterize qubit operations with non-Markovian errors arising from
realistic dynamics including control drift, coherent leakage, and coherent
interaction with material impurities.
