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CATEGORIES:Applied and Computational Analysis
SUMMARY:Undecidability of the spectral gap - Toby Cubitt (
University College London)
DTSTART;TZID=Europe/London:20170216T150000
DTEND;TZID=Europe/London:20170216T160000
UID:TALK69729AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/69729
DESCRIPTION:The spectral gap - the difference between the smal
lest and\nsecond-smallest eigenvalue of a quantum
many-body Hamiltonian - is of\ncentral importance
to quantum many-body physics. It determines the ph
ase\ndiagram at low temperature\, with quantum pha
se transitions occurring when\nthe gap vanishes. S
ome of the most challenging and long-standing open
\nproblems in theoretical physics concern the spec
tral gap\, such as the\nfamous Haldane conjecture\
, or the infamous Yang-Mills gap conjecture (one\n
of the Millennium Prize problems). These problems
- and many others - are\nall particular cases of t
he general spectral gap problem: Given a quantum\n
many-body Hamiltonian\, is the system it describes
gapped or gapless?\n\nWe prove that this problem
is undecidable (in the Goedel and Turing\nsense).
Our results also extend to many other important ze
ro-temperature\nproperties of quantum many-body sy
stems\, such as correlation functions.\n\nThe proo
f is by reduction from the Halting problem. But th
e construction\nis complex and draws on a wide var
iety of techniques\, ranging from\nspectral theory
\, Hamiltonian complexity theory\, quantum algorit
hms\, and\nnew results on aperiodic tilings.\n\nI
will explain the result\, sketch the techniques in
volved in the proof at\nan accessible level\, disc
uss the striking implications this may have for\np
hysics\, and outline some interesting computabilit
y questions related to\nthis problem that remain o
pen.\n\nBased on the following papers:\n\nUndecida
bility of the Spectral Gap\nToby Cubitt\, David Pe
rez-Garcia and Michael Wolf\nNature\, 528\, p207-2
11\, (2015)\narXiv:1502.04135[quant-ph]\n\nUndecid
ability of the Spectral Gap (full version\, 143 pa
ges)\nToby Cubitt\, David Perez-Garcia and Michael
Wolf\narXiv:1502.04573[quant-ph]\n
LOCATION:MR 14\, CMS
CONTACT:Dr Hansen
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