Analytical technique for simplification of the encoder-decoder circuit for a perfect five-qubit error correction

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. In this work, the analytical method to simplify the encoder-decoder circuit is introduced and a circuit that is as simple as the existing 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 gates.