, Volume 287, Issue 1, pp 41-65
Date: 13 Jan 2009

The Power of Quantum Systems on a Line

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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). 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 QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one-dimensional MAX-2-SAT with nearest neighbor constraints, is in P. The proof of the QMA-completeness 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 Church-Turing thesis and that quantum computers cannot efficiently solve QMA-complete problems) that there are one-dimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being one-dimensional spin glasses.

Communicated by M. B. Ruskai
Supported by Israel Science Foundation grant number 039-7549, Binational Science Foundation grant number 037-8404, and US Army Research Office grant number 030-7790.
Supported by CIFAR, by the Government of Canada through NSERC, and by the Province of Ontario through MRI.
Partially supported by NSF Grant CCR-0514082.
This work was mainly done while the author was at CNRS and LRI, University of Paris-Sud, Orsay, France. Partially supported by the European Commission under the Integrated Project Qubit Applications (QAP) funded by the IST directorate as Contract Number 015848, by an ANR AlgoQP grant of the French Research Ministry, by an Alon Fellowship of the Israeli Higher Council of Academic Research, by an Individual Research grant of the ISF, and by a European Research Council (ERC) Starting Grant.