  The Lieb-Robinson correlation function is the norm of a commutator between
local operators acting on separate subsystems at different times. This provides
a useful state-independent measure for characterizing the specifically quantum
interaction between spatially separated qubits. The finite propagation velocity
for this correlator defines a "light-cone" of quantum influence. We calculate
the Lieb-Robinson correlation function for one-dimensional qubit arrays
described by the transverse field Ising model. Direct calculations of this
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 hundreds of qubits and observe the effects of the
quantum phase transition in the system. We observe the emergence of two
distinct velocities of propagation: a correlation front velocity, which is
affected by the phase transition, and the Lieb-Robinson velocity which is not.
The correlation front velocity is equal to the maximum group velocity of single
quasiparticle excitations. The Lieb-Robinson velocity describes the extreme
leading edge of correlations when the value of the correlation function itself
is still very small. For the semi-infinite chain of qubits at the quantum
critical point, we derive an analytical result for the correlation function.
