TY - GEN

T1 - The power of quantum systems on a line

AU - Aharonov, Dorit

AU - Gottesman, Daniel

AU - Irani, Sandy

AU - Kempe, Julia

PY - 2007

Y1 - 2007

N2 - 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 one-dimensional quantum system (with 9 states per particle). 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 QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the analogous classical problem, one dimensional MAX-2-SAT with nearest neighbor constraints, which is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal. Assuming BQP ≠ QMA, our construction gives a one-dimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a one-dimensional spin glass.

AB - 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 one-dimensional quantum system (with 9 states per particle). 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 QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the analogous classical problem, one dimensional MAX-2-SAT with nearest neighbor constraints, which is in P. The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal. Assuming BQP ≠ QMA, our construction gives a one-dimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a one-dimensional spin glass.

UR - http://www.scopus.com/inward/record.url?scp=46749142548&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2007.4389508

DO - 10.1109/FOCS.2007.4389508

M3 - Conference contribution

AN - SCOPUS:46749142548

SN - 0769530109

SN - 9780769530109

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 373

EP - 383

BT - Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007

T2 - 48th Annual Symposium on Foundations of Computer Science, FOCS 2007

Y2 - 20 October 2007 through 23 October 2007

ER -