We give a construction of Quantum Low-Density Parity Check (QLDPC) codes with near-optimal rate-distance tradeoff and efficient list decoding up to the Johnson bound in polynomial time. Previous constructions of list decodable good distance quantum codes either required access to a classical side channel or were based on algebraic constructions that preclude the LDPC property. Our construction relies on new algorithmic results for codes obtained via the quantum analog of the distance amplification scheme of Alon, Edmonds, and Luby [FOCS 1995]. These results are based on convex relaxations obtained using the Sum-of-Squares hierarchy, which reduce the problem of list decoding the distance amplified codes to unique decoding the starting base codes. Choosing these base codes to be the recent breakthrough constructions of good QLDPC codes with efficient unique decoders, we get efficiently list decodable QLDPC codes.

翻译：我们提出了一种具有接近最优速率-距离权衡的量子低密度奇偶校验码构造，该码可在多项式时间内实现高达约翰逊界的高效列表解码。先前具有良好距离的列表可解码量子码构造要么需要依赖经典辅助信道，要么基于代数构造而无法满足LDPC特性。我们的构造依赖于通过量子版阿隆-埃德蒙兹-卢比距离放大方案获得的新算法结果。这些结果基于采用平方和层次结构获得的凸松弛方法，将距离放大码的列表解码问题简化为对初始基码的唯一解码。通过选择近期突破性构造的具有高效唯一解码器的优质量子低密度奇偶校验码作为基码，我们最终获得了高效列表可解码的量子低密度奇偶校验码。