  It has previously been shown that probabilistic quantum logic operations can
be performed using linear optical elements, additional photons (ancilla), and
post-selection based on the output of single-photon detectors. Here we describe
the operation of several quantum logic operations of an elementary nature,
including a quantum parity check and a quantum encoder, and we show how they
can be combined to implement a controlled-NOT (CNOT) gate. All of these gates
can be constructed using polarizing beam splitters that completely transmit one
state of polarization and totally reflect the orthogonal state of polarization,
which allows a simple explanation of each operation. We also describe a
polarizing beam splitter implementation of a CNOT gate that is closely
analogous to the quantum teleportation technique previously suggested by
Gottesman and Chuang [Nature 402, p.390 (1999)]. Finally, our approach has the
interesting feature that it makes practical use of a quantum-eraser technique.
