Communication in XYZ All-to-All Quantum Networks with a Missing Link

TL;DR

This paper explores quantum communication in all-to-all networks modeled by complete graphs, showing how removing a link enables perfect state transfer and relating the evolution to the Grover operator.

Contribution

It demonstrates that deleting an edge in a complete graph enables perfect state transfer between non-adjacent sites for certain sizes, and links network evolution to the Grover operator.

Findings

Deleting an edge in K_{n} enables PST when n is divisible by four.

The evolution in K_{n} can be equivalent to the Grover operator.

Routing qubits by switching off specific links is feasible.

Abstract

We explicitate the relation between Hamiltonians for networks of interacting qubits in the XYZ model and graph Laplacians. We then study evolution in networks in which all sites can communicate with each other. These are modeled by the complete graph K_{n} and called all-to-all networks. It turns out that K_{n} does not exhibit perfect state transfer (PST). However, we prove that deleting an edge in K_{n} allows PST between the two non-adjacent sites, when n is a multiple of four. An application is routing a qubit over n different sites, by switching off the link between the sites that we wish to put in communication. Additionally, we observe that, in certain cases, the unitary inducing evolution in K_{n} is equivalent to the Grover operator.