Qubit stabilizer states are complex projective 3-designs

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 2n. 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.