2013

Quantum Systems on Non-$k$-Hyperfinite Complexes: A Generalization of Classical Statistical Mechanics on Expander Graphs

Freedman, M. H., Hastings, M. B.

Understand

We construct families of cell complexes that generalize expander graphs.

These families are called non-$k$-hyperfinite, generalizing the idea of a non-hyperfinite (NH) family of graphs.
Roughly speaking, such a complex has the property that one cannot remove a small fraction of points and be left with an object that looks $k-1$-dimensional at large scales.
We then consider certain quantum systems on these complexes.

Built on

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E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, J. Math. Phys. 43
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L. Bowen and C. Radin, “Optimally Dense Packings of Hyperbolic Space”, Geom. Dedicata 104
2004
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S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb-Robinson bounds and the generation of correlations and topological quantum order”, Phys. Rev. Lett. 97
2006
Earlier work this paper cites.

Similar

N. Linial and R. Meshulam, “Homological connectivity of random 2-complexes”, Combinatorica 26(4)
2006
Cited alongside, same era.
G. Elek, “The combinatorial cost”, arxiv:math/0608474, l’Enseignement Math. 53
2007
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J.-P. Tillich and G. Zémor, in IEEE International Symposium on Information Thoery
2009
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W. Breslin, “Thick Triangulations of Hyperbolic n n -manifolds”, Pac. J. Math. 24
2009
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D. Aharonov, I. Arad, Z. Landau, and U. Vazirani, The detectability lemma and quantum gap amplification
Original
2009
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M. B. Hastings, “Locality in Quantum Systems” in Lecture Notes of the Les Houches Summer School
2010
Cited alongside, same era.
In Ref. mixedtriv , states were considered on a two-dimensional lattice which were ( R , ϵ ) (R,\epsilon) trivial for an R R growing logarithmically in system size, which does not quite meet this requirement. However, Eq. (8) in that paper contains an error, and should instead read ł β = R i n t exp ⁡ ( ( 2 R i n t 2 β ) log ⁡ ( V / ϵ ) CLOSE \l_{\beta}=R_{int}\sqrt{\exp((2R_{int}^{2}\beta)\log(V/\epsilon)} as can be seen by the requirement on the length scale immediately below the equation. With this correction, the length scale indeed grows as the square-root of the log of the system size, allowing efficient evaluation of local observables. It is also worth pointing out that what we consider to be trivial depends upon the application: for computational complexity reasons as considered here we are interested in efficient evaluation, while for problems in topological order we are often instead interested in whether the length R R scales linearly in system size or instead scales slower
Cited in the paper.
N. Bergeron and A. Venkatesh, “The asymptotic growth of torsion homology for arithmetic groups”, arXiv:1004.1083
Original
Cited in the paper.
N. Bergeron, P. Linnell, W. Luck, and R. Sauer, “On the growth of Betti numbers in p p -adic analytic towers”, arXiv:1204.3298v2; M. Albert, N. Bergeron, and I. Biringer, T. Galander, N. Nikolov, J. Raimbault, I. Samet, “On the growth of Betti Numbers of locally Symmetric Spaces”, arXiv:1104.5559
Original
Cited in the paper.
M. B. Hastings, “Trivial Low Energy States for Commuting Hamiltonians, and the Quantum PCP Conjecture”, arXiv.org:1201.3387
Cited in the paper.
J. Fox, M.l Gromov, V. Lafforgue, A. Naor, and J. Pach, “Overlap properties for geometric expanders”, arXiv:1005.1392, to appear in Journal fur die reine und angewandte Mathematik
Original
Cited in the paper.

Then

S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: stability under local perturbations”, arXiv:1001.0344, J. Math. Phys. 51
Original
2010
Later among the works it cites.
M. B. Hastings, “Topological Order at Non-zero Temperature”, Phys. Rev. Lett. 107
2011
Later among the works it cites.
S. Bravyi and M. B. Hastings, “A short proof of stability of topological order under local perturbations”, arXiv:1001.4363, Commun. Math. Phys. 307
Original
2011
Later among the works it cites.
B. Yoshida, “Feasibility of self-correcting quantum memory and thermal stability of topological order”, Annals of Physics 326
2011
Later among the works it cites.
J. Haah, “ Local stabilizer codes in three dimensions without string logical operators”, Phys. Rev. A 83
2011
Later among the works it cites.
See for example M. B. Hastings, C. Nayak, and Z. Wang, Phys. Rev. B 87
Original
2013
Closest in time.

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