The reason is threefold: (1) the gates used for encoders and decoders are composed of imperfect gates, including controlled imperfect gates (2) syndrome extraction applies unitary operators to entangle ancilla qubits with code block and (3) the error recovery action requires the use of a controlled operation to correct for the errors. Because of this fact, the reliability of data processed by quantum computer is not a priori guaranteed by QECC. The quantum error correction coding (QECC) scheme now needs to deal not only with errors introduced by the quantum channel by also with errors introduced by imperfect quantum gates during the encoding/decoding process. Moreover, the imperfect control gates introduce errors in processed sequence since wrong operations are applied. Imperfect quantum gates affect quantum computation by introducing errors in computed data. One of the most powerful applications of quantum error correction is the protection of quantum information as it dynamically undergoes quantum computation. Djordjevic, in Quantum Information Processing, Quantum Computing, and Quantum Error Correction (Second Edition), 2021 11.1 Fault-Tolerance Basics These two conditions can be formulated as a theorem that provides the necessary and sufficient conditions to be satisfied for a quantum code C q to be able to correct a given set of errors. The following two conditions should be satisfied during the design of a quantum code: (1) the encoded CB states must be chosen carefully so that the environment is not able to distinguish among different CBs, and (2) the corrupted images of codewords must be orthogonal among each other. If the actual error differs from true error E S, the resulting state E † S E|s〉 will be from C q but different from the uncorrupted state |s〉. We then perform a recovery operation by simply applying E † S. Among the different candidate errors, we choose for the expected error the most probable one, say E S. In relation to classical error correction, this set of errors E S having the same syndrome S can be called the coset. The measured value of syndrome S determines a set of errors E S = ( E 1, E 2,…) with syndromes being equal to the measured value: S( E i) = S, ∀ E i ∈ E S.
Djordjevic, in Quantum Information Processing, Quantum Computing, and Quantum Error Correction (Second Edition), 2021 8.3.4 Necessary and Sufficient Conditions for Quantum Error Correction CodingĪ quantum error correction circuit determines the error syndrome with the help of ancilla qubits.