### Abstract

We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates.

Original language | English (US) |
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Title of host publication | Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015 |

Publisher | IEEE Computer Society |

Pages | 1108-1122 |

Number of pages | 15 |

Volume | 2015-December |

ISBN (Print) | 9781467381918 |

DOIs | |

State | Published - Dec 11 2015 |

Event | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States Duration: Oct 17 2015 → Oct 20 2015 |

### Other

Other | 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 |
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Country | United States |

City | Berkeley |

Period | 10/17/15 → 10/20/15 |

### Fingerprint

### Keywords

- Grothendieck inequality
- hardness
- semidefinite programming

### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

*Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015*(Vol. 2015-December, pp. 1108-1122). [7354446] IEEE Computer Society. https://doi.org/10.1109/FOCS.2015.72

**Tight Hardness of the Non-commutative Grothendieck Problem.** / Briot, Jop; Regev, Oded; Saket, Rishi.

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

*Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015.*vol. 2015-December, 7354446, IEEE Computer Society, pp. 1108-1122, 56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, United States, 10/17/15. https://doi.org/10.1109/FOCS.2015.72

}

TY - GEN

T1 - Tight Hardness of the Non-commutative Grothendieck Problem

AU - Briot, Jop

AU - Regev, Oded

AU - Saket, Rishi

PY - 2015/12/11

Y1 - 2015/12/11

N2 - We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates.

AB - We prove that it is NP-hard to approximate the non-commutative Grothendieck problem to within any constant factor larger than one-half, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of finite-dimensional Hilbert spaces into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates.

KW - Grothendieck inequality

KW - hardness

KW - semidefinite programming

UR - http://www.scopus.com/inward/record.url?scp=84960423518&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960423518&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2015.72

DO - 10.1109/FOCS.2015.72

M3 - Conference contribution

SN - 9781467381918

VL - 2015-December

SP - 1108

EP - 1122

BT - Proceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015

PB - IEEE Computer Society

ER -