### Abstract

We consider coGapSVP_{√n}, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM ∩ coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first non-trivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover's possibility to cheat. We hope that these ideas will lead to further developments in the field.

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

Publisher | IEEE Computer Society |

Pages | 210-219 |

Number of pages | 10 |

Volume | 2003-January |

ISBN (Print) | 0769520405 |

DOIs | |

State | Published - 2003 |

Event | 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States Duration: Oct 11 2003 → Oct 14 2003 |

### Other

Other | 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 |
---|---|

Country | United States |

City | Cambridge |

Period | 10/11/03 → 10/14/03 |

### Fingerprint

### Keywords

- Algorithm design and analysis
- Autocorrelation
- Computer science
- Cryptography
- Lattices
- Polynomials
- Quantum computing
- Quantum mechanics
- Upper bound

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS*(Vol. 2003-January, pp. 210-219). [1238195] IEEE Computer Society. https://doi.org/10.1109/SFCS.2003.1238195

**A lattice problem in quantum NP.** / Aharonov, D.; Regev, O.

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

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS.*vol. 2003-January, 1238195, IEEE Computer Society, pp. 210-219, 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, Cambridge, United States, 10/11/03. https://doi.org/10.1109/SFCS.2003.1238195

}

TY - GEN

T1 - A lattice problem in quantum NP

AU - Aharonov, D.

AU - Regev, O.

PY - 2003

Y1 - 2003

N2 - We consider coGapSVP√n, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM ∩ coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first non-trivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover's possibility to cheat. We hope that these ideas will lead to further developments in the field.

AB - We consider coGapSVP√n, a gap version of the shortest vector in a lattice problem. This problem is known to be in AM ∩ coNP but is not known to be in NP or in MA. We prove that it lies inside QMA, the quantum analogue of NP. This is the first non-trivial upper bound on the quantum complexity of a lattice problem. The proof relies on two novel ideas. First, we give a new characterization of QMA, called QMA+ formulation allows us to circumvent a problem which arises commonly in the context of QMA: the prover might use entanglement between different copies of the same state in order to cheat. The second idea involves using estimations of autocorrelation functions for verification. We make the important observation that autocorrelation functions are positive definite functions and using properties of such functions we severely restrict the prover's possibility to cheat. We hope that these ideas will lead to further developments in the field.

KW - Algorithm design and analysis

KW - Autocorrelation

KW - Computer science

KW - Cryptography

KW - Lattices

KW - Polynomials

KW - Quantum computing

KW - Quantum mechanics

KW - Upper bound

UR - http://www.scopus.com/inward/record.url?scp=84943399777&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84943399777&partnerID=8YFLogxK

U2 - 10.1109/SFCS.2003.1238195

DO - 10.1109/SFCS.2003.1238195

M3 - Conference contribution

AN - SCOPUS:0345412697

SN - 0769520405

VL - 2003-January

SP - 210

EP - 219

BT - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

PB - IEEE Computer Society

ER -