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SUMMARY:Efficient quantum Gibbs samplers - Michael Kastoryano (Freie Unive
 rsitaet Berlin)
DTSTART:20141030T141500Z
DTEND:20141030T151500Z
UID:TALK54910@talks.cam.ac.uk
CONTACT:William Matthews
DESCRIPTION:We analyze the problem of preparing quantum Gibbs states of la
 ttice spin Hamiltonians with local and com- muting terms on a quantum comp
 uter and in nature.  We introduce a framework for analyzing the correlatio
 n and mixing characteristics of quantum Gibbs states and quantum Gibbs sam
 plers\, which is rooted in the theory of non-commutative Lp spaces. Our ce
 ntral result is an equivalence between the behavior of correlations in the
  Gibbs state and the mixing time of the semigroup which drives the system 
 to thermal equilibrium (the Gibbs sampler). We construct two distinct clas
 ses of Gibbs samplers\, one of which being the well-studied Davies generat
 ors modelling the dynamics on the system due to weak-coupling with a large
  Markovian environment. We show that their gap is independent of system si
 ze if\, and only if\, a certain strong form of clustering of correlations 
 holds in the Gibbs state. As concrete applications of our formalism\, we s
 how that for every one-dimensional lattice system\, or for systems in latt
 ices of any dimension at high enough temperatures\, the Gibbs samplers of 
 commuting Hamiltonians are always gapped\, giving an efficient way of pre
 paring these states on a quantum computer. We finally discuss the possible
  extensions to non-commuting Hamiltonian Gibbs samplers.
LOCATION:MR5\,  Centre for Mathematical Sciences\, Wilberforce Road\, Camb
 ridge
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