  Quantum phase transition in the one-dimensional period-two and uniform
quantum compass model are studied by using the pseudo-spin transformation
method and the trace map method. The exact solutions are presented, the
fidelity, the nearest-neighbor pseudo-spin entanglement, spin and pseudo-spin
correlation functions are then calculated. At the critical point, the fidelity
and its susceptibility change substantially, the gap of pseudo-spin concurrence
is observed, which scales as $1/N$ (N is system size). The spin correlation
functions show smooth behavior around the critical point. In the period-two
chain, the pseudo-spin correlation functions exhibit a oscillating behavior,
which is absent in the unform chain. The divergent correlation length at the
critical point is demonstrated in the general trend for both cases.
