The power of quantum systems on a line

Dorit Aharonov, Daniel Gottesman, Sandy Irani, Julia Kempe

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)41-65
Number of pages25
JournalCommunications in Mathematical Physics
Volume287
Issue number1
DOIs
StatePublished - Apr 1 2009

Fingerprint

Quantum Systems
Quantum Computer
quantum computers
Line
One-dimensional System
Configuration
ground state
Quantum Computation
theses
Ground State Energy
Exponential time
Turing
Spin Glass
completeness
quantum computation
configurations
spin glass
Dimensionality
Ground State
Completeness

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Aharonov, D., Gottesman, D., Irani, S., & Kempe, J. (2009). The power of quantum systems on a line. Communications in Mathematical Physics, 287(1), 41-65. https://doi.org/10.1007/s00220-008-0710-3

The power of quantum systems on a line. / Aharonov, Dorit; Gottesman, Daniel; Irani, Sandy; Kempe, Julia.

In: Communications in Mathematical Physics, Vol. 287, No. 1, 01.04.2009, p. 41-65.

Research output: Contribution to journalArticle

Aharonov, D, Gottesman, D, Irani, S & Kempe, J 2009, 'The power of quantum systems on a line', Communications in Mathematical Physics, vol. 287, no. 1, pp. 41-65. https://doi.org/10.1007/s00220-008-0710-3
Aharonov, Dorit ; Gottesman, Daniel ; Irani, Sandy ; Kempe, Julia. / The power of quantum systems on a line. In: Communications in Mathematical Physics. 2009 ; Vol. 287, No. 1. pp. 41-65.
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