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arXiv:0706.3821 · Physical Review A

Quantum Cayley networks of the hypercube

Christopher Facer, Jason Twamley, James D. Cresser

Published Jan 29, 2008• 29 citations
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Abstract

We develop the work of Christandl et al. [Phys. Rev. A 71, 032312 (2005)], to show how a $d$-hypercube homogenous network can be dressed by additional links to perfectly route quantum information between any given input and output nodes in a duration that is independent of the routing chosen and, surprisingly, the size of the network.

DOI
10.1103/physreva.77.012334

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We develop the work of Christandl et al. [Phys. Rev. A 71, 032312 (2005)], to show how a ddd-hypercube homogenous network can be dressed by additional links to perfectly route quantum information between any given input and output nodes in a duration that is independent of the routing chosen and, surprisingly, the size of the network.
Cubelike graphs are the Cayley graphs of the elementary Abelian group Z2n{\mathbb{Z}}_{2}^{n}Z2n​ (e.g., the hypercube is a cubelike graph). We study perfect state transfer between two particles in quantum networks modeled by a large class of cubelike graphs. This generalizes the results of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer et al. [Phys. Rev. A 92, 187902 (2008)].
We study the routing of quantum information in parallel on multidimensional networks of tunable qubits and oscillators. These theoretical models are inspired by recent experiments in superconducting circuits. We show that perfect parallel state transfer is possible for certain networks of harmonic oscillator modes. We extend this to the distribution of entanglement between every pair of nodes in the network, finding that the routing efficiency of hypercube networks is optimal and robust in the presence of dissipation and finite bandwidth.
A quantum network promises to enable long distance quantum communication, and assemble small quantum devices into a large quantum computing cluster. Each network node can thereby be seen as a small few qubit quantum computer. Qubits can be sent over direct physical links connecting nearby quantum nodes, or by means of teleportation over pre-established entanglement amongst distant network nodes. Such pre-shared entanglement effectively forms a shortcut - a virtual quantum link - which can be used exactly once. Here, we present an abstraction of a quantum network that allows ideas from computer science to be applied to the problem of routing qubits, and manage entanglement in the network. Specifically, we consider a scenario in which each quantum network node can create EPR pairs with its immediate neighbours over a physical connection, and perform entanglement swapping operations in order to create long distance virtual quantum links. We proceed to discuss the features unique to quantum networks, which call for the development of new routing techniques. As an example, we present two simple hierarchical routing schemes for a quantum network of N nodes for a ring and sphere topology. For these topologies we present efficient routing algorithms requiring O(log N) qubits to be stored at each network node, O(polylog N) time and space to perform routing decisions, and O(log N) timesteps to replenish the virtual quantum links in a model of entanglement generation.
Christandl et al. have noted that the ddd-dimensional hypercube can be projected to a linear chain with d+1d+1d+1 sites so that, by considering fixed but different couplings between the qubits assigned to the sites, the perfect state transfer (PST) can be achieved over arbitrarily long distances in the chain [Phys. Rev. Lett. 92, 187902 (2004); Phys. Rev. A 71, 032312 (2005)]. In this work we consider distance-regular graphs as spin networks and note that any such network (not just the hypercube) can be projected to a linear chain and so can allow PST over long distances. We consider some particular spin Hamiltonians which are the extended version of those of Christandl et al. Then, by using techniques such as stratification of distance-regular graphs and spectral analysis methods, we give a procedure for finding a set of coupling constants in the Hamiltonians so that a particular state initially encoded on one site will evolve freely to the opposite site without any dynamical control, i.e., we show how to derive the parameters of the system so that PST can be achieved. It is seen that PST is only allowed in distance-regular spin networks for which, starting from an arbitrary vertex as reference vertex (prepared in the initial state which we wish to transfer), the last stratum of the networks with respect to the reference state contains only one vertex; i.e., stratification of these networks plays an important role which determines in which kinds of networks and between which vertices of them, PST can be allowed. As examples, the cycle network with even number of vertices and ddd-dimensional hypercube are considered in details and the method is applied for some important distance-regular networks.