Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture

@article{Hastings2013TrivialLE,
  title={Trivial low energy states for commuting Hamiltonians, and the quantum PCP conjecture},
  author={Matthew B. Hastings},
  journal={Quantum Inf. Comput.},
  year={2013},
  volume={13},
  pages={393-429}
}
  • M. Hastings
  • Published 16 January 2012
  • Mathematics
  • Quantum Inf. Comput.
We consider the entanglement properties of ground states of Hamiltonians which are sums of commuting projectors (we call these commuting projector Hamiltonians), in particular whether or not they have "trivial" ground states, where a state is trivial if it is constructed by a local quantum circuit of bounded depth and range acting on a product state. It is known that Hamiltonians such as the toric code only have nontrivial ground states in two dimensions. Conversely, commuting projector… 

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Lemma 5 Any set 1-localizable complex with range R is a cover 1-localizable complex with range R