# Notes on “Parallel State Transfer and Efficient Quantum Routing on Quantum Networks”

Published on: Sun Jan 09 2011

The paper “Parallel State Transfer and Efficient Quantum Routing on Quantum Networks” describes a theoretical model for the routing of quantum information in parallel on multi-dimensional networks. Routing on a multi-dimensional network can be used for the transfer of entanglement. (What is a multi-dimensional network? Is it the number of interconnected nodes? For instance, one node connected to three others, is that 3-d? Is 8 nodes all wired to each other an 8-d network?)

There are two conditions for efficient routing:

1.     Quantum states must be transferable between arbitrary nodes.

2.     The network should be able to transfer states between nodes in parallel.

A quantum network is made up of nodes. Each node is an oscillator with a tunable frequency. Different arrangements of coupling between oscillators with the same frequency and couplings between oscillators with different frequencies lead to different network schemes.

The paper considers parallel quantum networks by first considering the fidelity of parallel state transfer and next evaluating different entanglement distribution schemes. The fidelity of parallel state transfer shows that multiple nodes can accurately send entanglement through the network at the same time. This is important, because it allows for more entanglement schemes. An entanglement distribution scheme is the method of interconnection used in the network. Similar to how in a classical network, all of the nodes can take to each other, or one node can talk to the node before and after it etc… there are a variety of classical network schemes.

To begin the description of the network, it is described mathematically as a graph which is a function of vertices and edges, with a Hamiltonian that describes the frequencies of each node. Each node must be tunable, so it could be set to a particular frequency.

The next step is the analysis of the transfer of entanglement between nodes.  This is considered first as a process and then as a time-evolution of the initial state. Thinking about the transfer of entanglement as a time-evolution of the initial state allows for additional analysis, like calculating the fidelity of the final density.

Next parallel state transfer on a hypercube is considered. This is confusing to me, because the symbol capital omega, which I thought was being used to describe the frequency of each node is now being used to describe a coupling matrix. The coupling matrix is used to calculate the fidelity of the parallel state transfer, for qubit or oscillator networks. This is still on a hypercube though. By expanding parallel state transfer to a complete graph, the network becomes massively parallel. I think massively parallel means all nodes can send and receive with each other. This makes Fidelity not dependent on the number of nodes. The new calculation for the distribution rate is much faster.

Finally the effects of decoherence are briefly covered. Decoherence will reduce the performance of the network, and further study is needed to see the effects on the massively parallel scheme.