2015/10/09 by Richard Kueng, Kueng, Richard, David Groß +1 · 8 citations
Computer Science · Mathematics · #Coding theory and cryptography #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Information Theory (cs.IT) #Mathematical Approximation and Integration #Probability (math.PR) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1510.02767
openalex publication_date 2015/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.