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

TL;DR

Problem addressed: representing and implementing qudit multi-controlled unitaries using N-body angular momentum couplings, with a focus on odd dimensions and Pegg-Barnett phase connections. Method and main results: develop an angular momentum formalism via the Jordan-Schwinger map, express C^(n)Z^k and X^k as N-body J_z interactions, decompose tripartite qutrit gates into two-body couplings, and propose angular momentum hypergraph states with a concrete optical implementation using single-photon sources, cross-Kerr interactions, and linear optics. Contributions: explicit j=1 simplifications, Clifford+T gate constructions in qutrits, bipartite decompositions, and a new angular momentum hypergraph state class; plus a practical photonic protocol. Significance: enables cheaper or more direct realizations of high-order qudit gates, clarifies the role of angular momentum couplings in quantum information, and broadens the toolbox for quantum optics and many-body state engineering.

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.

Figures (5)

• Figure 1: Diagram for the realization of the qutrit gates $F$ (left to right) and $F^{\dagger}$ (right to left).
• Figure 2: Diagram for the realization of the qutrit gates $X$ (left to right) and $X^{\dagger}=X^2$ (right to left).
• Figure 3: Diagram for the realization of the qutrit gate $X_{12}=X_{12}^{\dagger}$.
• Figure 4: Diagram for the implementation of the three-qutrit hard-controlled phase gates $|1\rangle-CZ$ (left to right) and $|1\rangle-CZ^2$ (right to left).
• Figure 5: Schematic representation of multiple three-qubit interactions in an octahedral disposition ($j=1$ case).

Theorems & Definitions (1)

• Definition 1