Perfect quantum state transfer using Hadamard diagonalizable graphs

Abstract

Quantum state transfer within a quantum computer can be achieved by using a network of qubits, and such a network can be modelled mathematically by a graph. Here, we focus on the corresponding Laplacian matrix, and those graphs for which the Laplacian can be diagonalized by a Hadamard matrix. We give a simple eigenvalue characterization for when such a graph has perfect state transfer at time $\pi /2$; this characterization allows one to choose the correct eigenvalues to build graphs having perfect state transfer. We characterize the graphs that are diagonalizable by the standard Hadamard matrix, showing a direct relationship to cubelike graphs. We then give a number of constructions producing a wide variety of new graphs that exhibit perfect state transfer, and we consider several corollaries in the settings of both weighted and unweighted graphs, as well as how our results relate to the notion of pretty good state transfer. Finally, we give an optimality result, showing that among regular graphs of degree at most $4$, the hypercube is the sparsest Hadamard diagonalizable connected unweighted graph with perfect state transfer.

29 pages, 1 figure, minor revisions, fixed typos

References in corpus (9)

• Perfect Transfer of Arbitrary States in Quantum Spin Networks
• Perfect State Transfer: Beyond Nearest-Neighbor Couplings
• From perfect to fractal transmission in spin chains
• Quantum Networks on Cubelike Graphs
• Perfect state transfer in cubelike graphs
• Pretty Good State Transfer in Qubit Chains - The Heisenberg Hamiltonian
• Effect of perturbations on information transfer in spin chains
• Perfect state transfer in products and covers of graphs
• Bounds on probability of state transfer with respect to readout time and edge weight

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