Lieb-Robinson correlation function in large qubit chains

The Lieb-Robinson (LR) correlation function is a commutator between local operators acting on separate subsystems at different times. This provides a useful state-independent measure for characterizing the specifically quantum entanglement between spatially separated qubits. The finite propagation velocity for this correlator defines a “light-cone” of quantum influence. We calculate the LR correlation function for one-dimensional qubit arrays described by the transverse field Ising model. Direct calculations of the LR correlation function have been limited by the exponential increase in the size of the state space with the number of qubits. We introduce a new technique that avoids this barrier by transforming the calculation to a sum over Pauli walks which results in linear scaling with system size. We can then explore propagation in arrays of many hundreds of qubits and observe the effects of the quantum phase transition in the system. We see the emergence of two velocities of propagation, one which is affected by the phase transition and one which is not. For the semi-infinite chain of qubits at the quantum critical point, we derive an analytical result for the correlation function.