Qubit stabilizer states are complex projective 3-designs

TL;DR

This paper proves that n-qubit stabilizer states form a complex projective 3-design, extending their known properties from 2-designs and providing new tools for quantum information and signal processing.

Contribution

It establishes that stabilizer states form a 3-design, introduces a recursion formula for their frame potential, and solves a counting problem in symplectic vector spaces.

Findings

Stabilizer states form a 3-design in dimension 2^n.

A recursion formula for the frame potential is derived.

A counting formula for stabilizer states with prescribed inner product is provided.

Abstract

A complex projective $t$-design is a configuration of vectors which is ``evenly distributed'' on a sphere in the sense that sampling uniformly from it reproduces the moments of Haar measure up to order $2t$. We show that the set of all $n$-qubit stabilizer states forms a complex projective $3$-design in dimension $2^n$. Stabilizer states had previously only been known to constitute $2$-designs. The main technical ingredient is a general recursion formula for the so-called frame potential of stabilizer states. To establish it, we need to compute the number of stabilizer states with pre-described inner product with respect to a reference state. This, in turn, reduces to a counting problem in discrete symplectic vector spaces for which we find a simple formula. We sketch applications in quantum information and signal analysis.