Quantum Error Source and Channel Coding

Abstract

Quantum error correction draws many of its principles and constructions from classical coding theory, adapted to the unique aspects of quantum mechanics. Here we extend this correspondence to a block of logical qubits, each prepared in the same quantum code, by coupling the block through a classical error correcting code: syndrome qubits then carry error information from several logical qubits at once, and collective inference recovers the errors. Algebraically, the construction defines subgroups of the product stabilizer group corresponding to duals of classical codes, and we prove that a lookup table decoder corrects every error pattern respecting the correction radii of the two constituent codes. For classical algebraic code families, the number of syndrome qubits serving $L$ logical qubits scales as ${\cal O}(\log_2(L+1))$ at the code-capacity level, though at the price of stabilizer weight growing with the block length. An entropy bound shows that check weights need not grow with $L$. Under a phenomenological noise model, the same lookup table identifies measurement errors by nearest-neighbor post-processing, and classical algebraic decoders locate exactly the logical qubits carrying errors even with noisy syndromes. More broadly, we argue that quantum error correction, reduced to its functional core, is source compression in the sense of Shannon, whose source and channel coding theorems bound the overhead rates of quantum post-selection tasks.