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

TL;DR

This paper investigates symmetric pseudo-Boolean functions, demonstrating their power series representation, kernel characterization via affine hyperplanes, and applications in spin glasses, quantum information, and tensor networks.

Contribution

It introduces a factorization and kernel analysis framework for symmetric pseudo-Boolean functions, linking them to affine hyperplanes and applying these insights to various scientific fields.

Findings

Any symmetric pseudo-Boolean function can be expressed as a power series.

The kernel corresponds to at least one affine hyperplane defined by a sum constraint.

Applications include analysis of spin glass energy functions, quantum information, and tensor networks.

Abstract

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 $f(x_1, x_2, \dots, x_n)$ has a kernel corresponding to at least one $n$-affine hyperplane, each hyperplane is given by a constraint $\sum_{l=1}^n x_l = \lambda$ for $\lambda\in \mathbb{C}$ 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.