Quantum Systems on Non-k-Hyperfinite Complexes: a generalization of classical statistical mechanics on expander graphs

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{hpc} of Tillich and Z\'{e}mor are generalized using NH complexes.