On Symmetric Pseudo-Boolean Functions: Factorization, Kernels and Applications

09/29/2022

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by   Richik Sengupta, et al.

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A symmetric pseudo-Boolean function is a map from Boolean tuples to real numbers which is invariant under input variable interchange. We prove that any such function can be equivalently expressed as a power series or factorized. The kernel of a pseudo-Boolean function is the set of all inputs that cause the function to vanish identically. Any n-variable symmetric pseudo-Boolean function f(x_1, x_2, …, x_n) has a kernel corresponding to at least one n-affine hyperplane, each hyperplane is given by a constraint ∑_l=1^n x_l = λ for λ∈ℂ constant. We use these results to analyze symmetric pseudo-Boolean functions appearing in the literature of spin glass energy functions (Ising models), quantum information and tensor networks.