Pascal's pyramid and number projection operators for quantum computation

TL;DR

This paper explores the properties of number projection operators in quantum computing, revealing their connection to Pascal's pyramid and Kravchuk polynomials, which could enhance simulation of quantum many-body systems.

Contribution

It introduces a novel analysis of number projection operators using Pascal's pyramid and links them to Kravchuk polynomials, advancing understanding of symmetry in quantum simulations.

Findings

Number projection operators relate to generalized binomial coefficients.

Operators exhibit properties linked to Pascal's pyramid structure.

Connections to Kravchuk polynomials are established.

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.