Efficient and optimal quantum state discrimination via quantum belief propagation

We present an efficient quantum algorithm for a structured state discrimination problem we call the subspace decoding task. Building on this, we show that the algorithm enables efficient and optimal decoding of certain families of structured classical linear codes transmitted over binary-input classical-quantum pure-state channels. Such decoders can substantially enhance the performance of quantum algorithms based on Regev's reduction, such as decoded quantum interferometry. In particular, we obtain optimal and efficient quantum decoders for all classical codes with efficient trellis representations. As an application, we design a quantum decoder for turbo codes and, through density evolution, demonstrate decoding thresholds that surpass the Shannon bound and closely approach the Holevo bound.