Quantum Error Source and Channel Coding

A classical coding across a block of logical qubits is presented. We characterize subgroups of the product stabilizer group on a block of logical qubits corresponding to dual codes of classical error correcting codes. We prove conditions on the set of correctable error patterns allowing for unambiguous decoding based on a lookup table. For a large family of classical algebraic codes, we show that the qubit overhead required for syndrome extraction from $L$ logical qubits scales as ${\cal O}(\log_2(L+1)),$ asymptotically. The basic construction is adapted to account for two-qubit and measurement errors, while still employing a lookup table based decoder. Moreover, we characterize the set of detectable errors and show how classical algebraic decoders can unambiguously locate logical qubits with errors even in the presence of syndrome noise. We argue that quantum error correction is more aptly viewed as source compression in the sense of Shannon, and that Shannon's source and channel coding theorems provide bounds on the overhead rates of quantum post-selection tasks, such as quantum error correction, at the level of the encoded quantum register.