Pascal's pyramid and number projection operators for quantum computation

Abstract

The pursuit of quantum advantage in simulating many-body quantum systems on quantum computers has gained momentum with advancements in quantum hardware. This work focuses on leveraging the symmetry properties of these systems, particularly particle number conservation. We investigate the qubit objects corresponding to number projection operators in the standard Jordan-Wigner fermion-to-qubit mapping, and prove a number of their properties. This reveals connections between these operators and the generalised binomial coefficients originally introduced by Kravchuk in his research on orthogonal polynomials. The generalized binomial coefficients are visualized in a Pascal's pyramid structure.