Unstructured Randomness, Small Gaps and Localization

Details Unstructured Randomness, Small Gaps and Localization Unstructured Randomness, Small Gaps and Localization

2010-09-30

arxiv.org/abs/1010.0009v1

Quantum Information & Computation, Volume 11 number 9&10, 2011, pages 840-854

Physics

Quantum Physics

Scientific paper

We study the Hamiltonian associated with the quantum adiabatic algorithm with a random cost function. Because the cost function lacks structure we can prove results about the ground state. We find the ground state energy as the number of bits goes to infinity, show that the minimum gap goes to zero exponentially quickly, and we see a localization transition. We prove that there are no levels approaching the ground state near the end of the evolution. We do not know which features of this model are shared by a quantum adiabatic algorithm applied to random instances of satisfiability since despite being random they do have bit structure.

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