Unstructured randomness, small gaps and localization

Authors:
Edward Farhi;Jeffrey Goldstone;David Gosset;Sam Gutmann;Peter Shor
Affiliations:
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA;Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA;Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA;Center for Theoretical Physics and Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA;Center for Theoretical Physics and Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA
Venue:
Quantum Information & Computation
Year:
2011

Citing 3
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Abstract

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.