### Abstract

Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem (where QMA stands for Quantum Merlin Arthur) and initiate its study. We present two main results. The first shows that a nontrivial approximation ratio can be obtained in the class NP using product states. The second result (which builds on the first one) gives a polynomial time (classical) algorithm providing a similar approximation ratio for dense instances of the problem. The latter result is based on an adaptation of the "exhaustive sampling method" by Arora, Karger, and Karpinski [J. Comput. System Sci., 58(1999), p. 193] to the quantum setting and might be of independent interest.

Original language | English (US) |
---|---|

Pages (from-to) | 1028-1050 |

Number of pages | 23 |

Journal | SIAM Journal on Computing |

Volume | 41 |

Issue number | 4 |

DOIs | |

State | Published - Sep 24 2012 |

### Fingerprint

### Keywords

- Approximation algorithm
- Constraint satisfaction
- Local Hamiltonian
- QMA-complete

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

### Cite this

*SIAM Journal on Computing*,

*41*(4), 1028-1050. https://doi.org/10.1137/110842272

**Approximation algorithms for QMA-complete problems.** / Gharibian, Sevag; Kempe, Julia.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 41, no. 4, pp. 1028-1050. https://doi.org/10.1137/110842272

}

TY - JOUR

T1 - Approximation algorithms for QMA-complete problems

AU - Gharibian, Sevag

AU - Kempe, Julia

PY - 2012/9/24

Y1 - 2012/9/24

N2 - Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem (where QMA stands for Quantum Merlin Arthur) and initiate its study. We present two main results. The first shows that a nontrivial approximation ratio can be obtained in the class NP using product states. The second result (which builds on the first one) gives a polynomial time (classical) algorithm providing a similar approximation ratio for dense instances of the problem. The latter result is based on an adaptation of the "exhaustive sampling method" by Arora, Karger, and Karpinski [J. Comput. System Sci., 58(1999), p. 193] to the quantum setting and might be of independent interest.

AB - Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem (where QMA stands for Quantum Merlin Arthur) and initiate its study. We present two main results. The first shows that a nontrivial approximation ratio can be obtained in the class NP using product states. The second result (which builds on the first one) gives a polynomial time (classical) algorithm providing a similar approximation ratio for dense instances of the problem. The latter result is based on an adaptation of the "exhaustive sampling method" by Arora, Karger, and Karpinski [J. Comput. System Sci., 58(1999), p. 193] to the quantum setting and might be of independent interest.

KW - Approximation algorithm

KW - Constraint satisfaction

KW - Local Hamiltonian

KW - QMA-complete

UR - http://www.scopus.com/inward/record.url?scp=84866382049&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866382049&partnerID=8YFLogxK

U2 - 10.1137/110842272

DO - 10.1137/110842272

M3 - Article

VL - 41

SP - 1028

EP - 1050

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -