### 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). 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.

Original language | English (US) |
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Title of host publication | Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007 |

Pages | 373-383 |

Number of pages | 11 |

DOIs | |

State | Published - 2007 |

Event | 48th Annual Symposium on Foundations of Computer Science, FOCS 2007 - Providence, RI, United States Duration: Oct 20 2007 → Oct 23 2007 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 48th Annual Symposium on Foundations of Computer Science, FOCS 2007 |
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Country | United States |

City | Providence, RI |

Period | 10/20/07 → 10/23/07 |

### ASJC Scopus subject areas

- Engineering(all)

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## Cite this

*Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007*(pp. 373-383). [4389508] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2007.4389508