Encoding One Logical Qubit Into Six Physical Qubits

We discuss several methods to protect one qubit against single- qubit errors by encoding it into six physical qubits. We first present a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We then prove that a six-qubit code cannot simultaneously possess a Calderbank-Shor-Steane stabilizer and correct arbitrary single-qubit errors. We finally construct a six-qubit non-degenerate entanglement-assisted quantum error- correcting code that uses one bit of entanglement shared between the sender and the receiver. We discuss the advantages and disadvantages for each of our six-qubit quantum error-correcting codes.