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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. 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 .
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Keywords: Time-inhomogeneous Markov chain; adiabatic time; mixing time; stability

Document Type: Research Article

Affiliations: 1: Department of Mathematics, Oregon State University, Corvallis, OR, USA. 2: School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR, USA.

Publication date: 18 May 2016

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