The Power of Quantum Systems on a Line
Abstract
We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMAcomplete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, onedimensional MAX2SAT with nearest neighbor constraints, is in P. The proof of the QMAcompleteness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum ChurchTuring thesis and that quantum computers cannot efficiently solve QMAcomplete problems) that there are onedimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being onedimensional spin glasses.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 April 2009
 DOI:
 10.1007/s0022000807103
 arXiv:
 arXiv:0705.4077
 Bibcode:
 2009CMaPh.287...41A
 Keywords:

 Quantum Physics
 EPrint:
 21 pages. v2 has numerous corrections and clarifications, and most importantly a new author, merged from arXiv:0705.4067. v3 is the published version, with additional clarifications, publisher's version available at http://www.springerlink.com