# Mathematics Colloquia and Seminars

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### Stabilizer quantum states: Higher moments from finite geometries

**Mathematical Physics & Probability**

Speaker: | Michael Walter, Stanford |

Related Webpage: | http://web.stanford.edu/~waltemic/ |

Location: | 1147 MSB |

Start time: | Wed, Mar 8 2017, 4:10PM |

Stabilizer states are a ubiquitous tool in quantum information theory, used routinely for the construction of quantum error correcting codes. Moreover, they efficiently reproduce the lower moments of the Haar measure. This has important applications in a variety of statistical problems. In this talk, I will first introduce stabilizer states and their finite phase space formalism. I will then present a simple explicit formula for all their higher moments. Significantly, this formula is also applicable in the regime where the stabilizer states deviate from the Haar measure. This unifies many previous results on the statistical properties of stabilizer states and it enables new applications that I will sketch in the talk. Mathematically, our key result is a version of Schur-Weyl duality for the Clifford group. Whereas the commutant of the tensor power action of the unitary group is generated by permutations, we find that for the Clifford group the commutant has a natural description in terms of finite geometries, unraveling a new and surprising algebraic structure.