### 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 - Dec 1 2007 |

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

### 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 |

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### ASJC Scopus subject areas

- Engineering(all)

### Cite this

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

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007.*, 4389508, pp. 373-383, 48th Annual Symposium on Foundations of Computer Science, FOCS 2007, Providence, RI, United States, 10/20/07. https://doi.org/10.1109/FOCS.2007.4389508

}

TY - GEN

T1 - The power of quantum systems on a line

AU - Aharonov, Dorit

AU - Gottesman, Daniel

AU - Irani, Sandy

AU - Kempe, Julia

PY - 2007/12/1

Y1 - 2007/12/1

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

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

U2 - 10.1109/FOCS.2007.4389508

DO - 10.1109/FOCS.2007.4389508

M3 - Conference contribution

SN - 0769530109

SN - 9780769530109

SP - 373

EP - 383

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

ER -