Unstructured randomness, small gaps and localization

• Main Authors: Farhi, Edward (Contributor), Goldstone, Jeffrey (Contributor), Gosset, David Nicholas (Contributor), Gutmann, Sam (Author), Shor, Peter W. (Contributor)
• Other Authors: Massachusetts Institute of Technology. Center for Theoretical Physics (Contributor), Massachusetts Institute of Technology. Department of Mathematics (Contributor), Massachusetts Institute of Technology. Department of Physics (Contributor)
• Format: Article
• Language: English
• Published: Cornell University Library, 2011-06-29T15:32:10Z.
• Subjects: Article
• Online Access: Get fulltext

published in arxiv only per Mat Willmott.
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.
W. M. Keck Foundation Center for Extreme Quantum Information Theory
United States. Army Research Laboratory (ARO grant number W911NF-09-1-0438)
National Science Foundation (U.S.) (grant number CCF-0829421)
Natural Sciences and Engineering Research Council of Canada