Lieb-Robinson correlation functions in 1D, 2D, and 3D qubit networks

The Lieb-Robinson correlation function provides a state-independent measure of quantum entanglement between two qubits. An important and well-known result is that this quantum correlation between qubits is local and its spatial spread in a network of interacting qubits is limited by a finite velocity, the Lieb-Robinson velocity. We consider qubits inspired by quantum-dot cellular automata whose parameters can be electronically tuned. The resulting Hamiltonian is of the transverse-field Ising model form, which has been realized in multiple systems including superconducting qubit arrays. Our focus is on the early-time behavior which grows with a power-law dependence on time. The power-law exponent increases linearly with distance--here characterized by the number of interacting qubit links connecting the two correlated qubits. We deduce an analytic form for the early-time correlation function for a network of qubits with arbitrary connectivity in one, two, and three dimensions.  For regular square arrays we calculate the dependence of the Lieb-Robinson velocity on the direction of propagation.