Explicit decoders using fixed-point amplitude amplification based on QSVT (2405.06051v4)

Abstract: 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, which achieve the optimal rate, i.e., the quantum capacity. The decoders are constructed by extending a previous approach applicable only to erasure noise, using the fixed-point amplitude amplification (FPAA) based on the quantum singular value transformation (QSVT). By developing a proof technique that relies on a symmetric structure of the construction, we rigorously show that the proposed decoders are applicable to any noise model and successfully recover quantum information when the decoupling condition is satisfied. This implies that the proposed decoders can be used to achieve the quantum capacity. Our constructions have advantages also in terms of the computational cost, largely reducing the circuit complexity compared to previously known explicit decoders. Through an investigation of the decoding problem, unique advantages of the QSVT-based FPAA are highlighted.