Quantum Error Correction Fundamentals

Derive the conditions for quantum error correction using the Knill-Laflamme theorem

Implement the 3-qubit bit-flip code, including encoding, syndrome measurement, and correction for a single bit-flip error

Implement the 3-qubit phase-flip code and analyze its relationship to the bit-flip code via a change of basis

Define the stabilizer formalism and construct a stabilizer code from a set of commuting Pauli group generators

Implement syndrome measurement for the 5-qubit perfect code and use the syndrome to identify and correct single-qubit Pauli errors

Construct the encoding circuit for the 9-qubit Shor code and demonstrate that it is a stabilizer code

Implement the syndrome measurement circuits for the 9-qubit Shor code to extract bit-flip and phase-flip error syndromes

Using the extracted syndrome, implement the decoding logic and correction operations for the 9-qubit Shor code to recover the logical state from a single-qubit Pauli error

Construct a classical linear code from its generator matrix, verify its parameters, and derive the parity-check matrix of its dual code

Construct a Calderbank-Shor-Steane (CSS) code from a pair of classical linear codes and derive its distance and logical operators