A spherical t-design is a set of vectors S = { | ϕ ϕ k ⟩ ⟩ : | ϕ ϕ k ⟩ ⟩ ∈ ∈ S d } on the d-dimensional generalized hypersphere, such that the average value of any t t h -order polynomial f t ( ψ ψ ) over S is equal to the average of f t ( ψ ψ ) over all normalized vectors | ψ ψ ⟩ ⟩ . Defining H t = ⨂ ⨂ i = 1 t H as the t-fold tensor product of the Hilbert spaces, and