Homological Connectivity Of Random 2-Complexes
Combinatorica
The detectability lemma and quantum gap amplification
Proceedings of the forty-first annual ACM symposium on Theory of computing
Quantum LDPC codes with positive rate and minimum distance proportional to n 1/2
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 2
Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture
Quantum Information & Computation
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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. A future goal is to construct a family of Hamiltonians such that every low energy state has topological order as part of an attempt to prove the quantum PCP conjecture. This goal is approached by constructing a toric code Hamiltonian with the property that every low energy state without vertex defects has topological order, a property that would not hold for any local system in any lattice Zd or indeed on any 1-hyperfinite complex. Further, such NH complexes find application in quantum coding theory. The hypergraph product codes[1] of Tillich and Zémor are generalized using NH complexes.