Implementation and representation of qudit multi-controlled unitaries and hypergraph states by N-body angular momentum couplings

TL;DR

This paper presents a novel method to represent and implement qudit multi-controlled unitaries and hypergraph states using N-body angular momentum couplings, with practical optical system applications.

Contribution

It introduces a new angular momentum-based representation for qudit unitaries and hypergraph states, facilitating their physical realization and analysis.

Findings

Representation simplifies implementation in physical systems.

Connections established between qudit states and angular momentum interactions.

Concrete optical system example demonstrates practical feasibility.

Abstract

We construct a representation of qudit multi-controlled unitary operators in terms of N-body angular momentum interactions. The representation is particularly convenient for odd-dimensional systems, with interesting connections to the Pegg-Barnett phase formalism. We illustrate the main points in the special case of qutrits, where simplifications and connections to dipole-quadrupole and quadrupole-quadrupole interactions can be established. We describe the representation of the closely related set of qudit hypergraph states, identifying possible realizations and their main obstacles. Qutrit tripartite controlled unitaries are decomposed in terms of more familiar two-body angular momentum couplings, enabling their implementation in a variety of physical systems. We give then a concrete example of implementation of qutrit unitaries and hypergraph states in optical systems that employs single-photon sources, two-mode cross-Kerr interactions and linear optical operations. Moreover, we define a new set of states, called angular momentum hypergraph states, which are more directly related to the angular momentum representation.