Explicit decoders using fixed-point amplitude amplification based on QSVT

Reliably transmitting quantum information via a noisy quantum channel is a central challenge in quantum information science. While constructing a decoder is crucial to this goal, little was known about quantum circuit implementations of decoders that reach high communication rates. In this paper, we provide two decoders with explicit quantum circuits capable of recovering quantum information when the decoupling condition is satisfied, i.e., when quantum information is in principle recoverable. These are applicable to both entanglement-assisted and non-assisted settings. By developing a technique that relies on a symmetric structure of the decoders, we show that they are applicable to any noise model. As a consequence, for any noisy channel, our decoders can be used to achieve a communication rate arbitrarily close to the quantum capacity by increasing the number of channel uses. To construct the decoders, we employ the fixed-point amplitude amplification (FPAA) based on the quantum singular value transformation (QSVT), extending a previous approach applicable only to erasure noise. Our constructions offer advantages in the computational cost, largely reducing the circuit complexity compared to previous explicit decoders. Through an investigation of the decoding problem, unique advantages of the QSVT-based FPAA are highlighted.