the occurrence, location and type of errors. In most QEC codes, the type of error is either a bit flip, or a sign (of the phase) flip, or both (corresponding to the Pauli matrices X, Z, and Y). The measurement of the syndrome has the projective effect of a quantum measurement, so even if the error due to the noise was arbitrary, it can be expressed as a combination of basis operations called the error basis (which is given by the Pauli matrices and the identity). To correct the error, the Pauli operator corresponding to the type of error is used on the corrupted qubit to revert the effect of the error. The syndrome measurement provides information about the error that has happened, but not about the information that is stored in the logical qubit—as otherwise the measurement would destroy any quantum superposition of this logical qubit with other qubits in the quantum computer, which would prevent it from being used to convey quantum information. Bit flip code The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. This approach does not work for a quantum channel in which, due to the no-cloning theorem, it is not possible to repeat a single qubit three times. To overcome this, a different method has to be used, such as the three-qubit bit flip code first proposed by Asher Peres in 1985. This technique uses entanglement and syndrome measurements and is comparable in performance with the repetition code. Consider the situation in which we want to transmit the state of a single qubit through a noisy channel . Let us moreover assume that this channel either flips the state of the qubit, with probability , or leaves it unchanged. The action of on a general input can therefore be written as . Let be the quantum state to be transmitted. With no error correcting protocol in place, the transmitted state will be correctly transmitted with probability . We can however improve on this number by encoding the state into a greater number of qubits, in such a way that errors in the corresponding logical qubits can be detected and corrected. In the case of the simple three-qubit repetition code, the encoding consists in the mappings and . The input state is encoded into the state . This mapping can be realized for example using two