  Simpler encoding and decoding networks are necessary for more reliable
quantum error correcting codes (QECCs). The simplification of the
encoder-decoder circuit for a perfect five-qubit QECC can be derived
analytically if the QECC is converted from its equivalent one-way entanglement
purification protocol (1-EPP). In this work, the analytical method to simplify
the encoder-decoder circuit is introduced and a circuit that is as simple as
the existent simplest circuits is presented as an example. The encoder-decoder
circuit presented here involves nine single- and two-qubit unitary operations,
only six of which are controlled-NOT (CNOT) gates.
