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SUMMARY:Quantum thermodynamics and semi-definite optimization - Mark M. Wi
 lde (Cornell University)
DTSTART:20250512T150000Z
DTEND:20250512T160000Z
UID:TALK231844@talks.cam.ac.uk
CONTACT:Bjarne Bergh
DESCRIPTION:In quantum thermodynamics\, a system is described by a Hamilto
 nian and a list of non-commuting charges representing conserved quantities
  like particle number or electric charge\, and an important goal is to det
 ermine the system's minimum energy in the presence of these conserved char
 ges. In optimization theory\, a semi-definite program involves a linear ob
 jective function optimized over the cone of positive semi-definite operato
 rs intersected with an affine space. These problems arise from differing m
 otivations in the physics and optimization communities and are phrased usi
 ng very different terminology\, yet they are essentially identical mathema
 tically. By adopting Jaynes' mindset motivated by quantum thermodynamics\,
  I'll discuss how minimizing free energy in the aforementioned thermodynam
 ics problem\, instead of energy\, leads to an elegant solution in terms of
  a dual chemical potential maximization problem that is concave in the che
 mical potential parameters. As such\, one can employ standard (stochastic)
  gradient ascent methods to find the optimal values of these parameters\, 
 and these methods are guaranteed to converge quickly. At low temperature\,
  the minimum free energy provides an excellent approximation for the minim
 um energy. I'll then show how this Jaynes-inspired gradient-ascent approac
 h can be used in both classical and quantum algorithms for minimizing ener
 gy\, and equivalently\, how it can be used for solving semi-definite progr
 ams\, with guarantees on the runtimes of the algorithms. The approach disc
 ussed here is well grounded in quantum thermodynamics and\, as such\, prov
 ides physical motivation underpinning why algorithms published fifty years
  after Jaynes' seminal work\, including the matrix multiplicative weights 
 update method\, the matrix exponentiated gradient update method\, and thei
 r quantum algorithmic generalizations\, perform well at semi-definite opti
 mization tasks. Joint work with Nana Liu\, Michele Minervini\, and Dhrumil
  Patel.
LOCATION:Center for Mathematical Sciences\, Lecture room MR2
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