Southwest Quantum Information and Technology

Higher moments of stabilizer states

Presenting Author: Sepehr Nezami, Stanford University
Contributing Author(s): Michael Walter

Stabilizer states are a fundamental tool in quantum information theory. In the past years, there has been renewed interest in their statistical properties, motivated by a number of important applications. Celebrated results include a characterization of their third and fourth moments in the multiqubit case (e.g.,Zhu/Webb/Kueng&Gross, QIP 2016). In this work, we present a simple explicit expression for all higher moments of stabilizer states in odd prime power dimensions. Previously, it was only known that they form a 2-design but not a 3-design (i.e. that their second but not their third moments agree with the Haar measure). In contrast, and significantly for applications, our formula allows the computation of a t-th moment even when the stabilizer states fail to be a t-design. Our key technical 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 spanned by the permutation action, we show that for the Clifford group the commutant has a natural description in terms of discrete symplectic phase space, unraveling a new and surprising algebraic structure. We sketch possible applications of our result to quantum information theory and signal recovery.

(Session 5 : Thursday from 5:00pm - 7:00pm)