In this paper we continue our work on adiabatic time of time-inhomogeneous Markov chains first introduced in [9] and [3] . Our study is an analog to the well-known Quantum Adiabatic (QA) theorem which characterizes the quantum adiabatic 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. H(s) = (1 − s)H 0 + sH 1 . Informally, the quantum adiabatic 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 ǫ-close to that induced by the Hamilton operator H(s) 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., P t = (1 − t)P 0 + tP 1 . In particular we show that the stable adiabatic time t sad (P 0 , P 1 , ǫ) = O t 4 mix (ǫ/2)/ǫ 3 , where t mix denotes the maximum mixing time over all P t for 0 ≤ t ≤ 1.
Generally, Bradford et al. shows that the stable adiabatic time, which is a general concept of adiabatic time, is an order of the maximum mixing time to the power of four REF .
Stable Adiabatic Times for Markov Chains
arXiv: Probability
Mathematics