## Abstract

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 Z^{d} 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.

Original language | English |
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Pages (from-to) | 144-180 |

Number of pages | 37 |

Journal | Quantum Information and Computation |

Volume | 14 |

Issue number | 1-2 |

State | Published - 1 Jan 2014 |

Externally published | Yes |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics
- General Physics and Astronomy
- Computational Theory and Mathematics