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Abstract
In this paper we continue our work on adiabatic time of time-inhomogeneous Markov chains first introduced. Our study is an analog to the well-known quantum adiabatic (QA) theorem which characterizes the QA time for the evolution of a quantum system as a result of applying of a series of Hamilton operators, each is a linear combination of two given initial and final Hamilton operators, i.e. . Informally, the QA time of a quantum system specifies the speed at which the Hamiltonian operators changes so that the ground state of the system at any time s will always remain
-close to that induced by the Hamilton operator
at time s. Analogously, we derive a sufficient condition for the stable adiabatic time of a time-inhomogeneous Markov evolution specified by applying a series of transition probability matrices, each is a linear combination of two given irreducible and aperiodic transition probability matrices, i.e.
. In particular we show that the stable adiabatic time
where
denotes the maximum mixing time over all
for
.
Notes
No potential conflict of interest was reported by the authors.