Stable Adiabatic Times for Markov Chains

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In this paper we continue our work on adiabatic time of time-inhomogeneous Markov chains first introduced in Kovchegov (2010) and Bradford and Kovchegov (2011). 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)H₀ + sH₁. 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 ε-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., Pₜ = (1-t)P₀ + tP₁. In particular we show that the stable adiabatic time tsad(P₀, P₁, ε) = O (tmix⁴(ε 2) ε³), where tmix denotes the maximum mixing time over all Pₜ for 0 ≤ t ≤ 1.