is used, which was shown before to equal . A little algebra then shows that the fidelity after error correction is greater than the one without for . Note that this is consistent with the working assumption that was made while deriving the protocol (of being small enough). Sign flip code Flipped bits are the only kind of error in classical computer, but there is another possibility of an error with quantum computers, the sign flip. Through the transmission in a channel the relative sign between and can become inverted. For instance, a qubit in the state may have its sign flip to The original state of the qubit will be changed into the state In the Hadamard basis, bit flips become sign flips and sign flips become bit flips. Let be a quantum channel that can cause at most one phase flip. Then the bit flip code from above can recover by transforming into the Hadamard basis before and after transmission through . Shor code The error channel may induce either a bit flip, a sign flip (i.e., a phase flip), or both. It is possible to correct for both types of errors on any one qubit using a QEC code, which can be done using the Shor code published in 1995. This is equivalent to saying the Shor code corrects arbitrary single-qubit errors. Let be a quantum channel that can arbitrarily corrupt a single qubit. The 1st, 4th and 7th qubits are for the sign flip code, while the three groups of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. With the Shor code, a qubit state will be transformed into the product of 9 qubits , where If a bit flip error happens to a qubit, the syndrome analysis will be performed on each block of qubits (1,2,3), (4,5,6), and (7,8,9) to detect and correct at most one bit flip error in each block. If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. This means that the Shor code can also repair a sign flip error for a single qubit. The Shor code also can correct for any arbitrary errors (both bit flip and sign flip) to a single qubit. If an error is modeled by a unitary transform U, which will act on