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

TL;DR

This paper introduces non-$k$-hyperfinite complexes that generalize expander graphs and explores their applications in quantum systems, topological order, and quantum coding theory, aiming to advance understanding of quantum PCP conjecture.

Contribution

It constructs non-$k$-hyperfinite complexes, generalizes hypergraph product codes, and investigates quantum systems with potential implications for the quantum PCP conjecture.

Findings

Construction of non-$k$-hyperfinite complexes.

Quantum systems on these complexes exhibit topological order.

Applications in quantum coding theory with generalized hypergraph codes.

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 Z\'{e}mor are generalized using NH complexes.