We describe a quantum algorithm for the Planted Noisy $k$XOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic ($4$th power) speedup over the best known classical algorithm while also only using logarithmically many qubits. Our work generalizes and simplifies prior work of Hastings, by building on his quantum algorithm for the Tensor Principal Component Analysis (PCA) problem. We achieve our quantum speedup using a general framework based on the Kikuchi Method (recovering the quartic speedup for Tensor PCA), and we anticipate it will yield similar speedups for further planted inference problems. These speedups rely on the fact that planted inference problems naturally instantiate the Guided Sparse Hamiltonian problem. Since the Planted Noisy $k$XOR problem has been used as a component of certain cryptographic constructions, our work suggests that some of these are susceptible to super-quadratic quantum attacks.

翻译：我们提出了一种针对植入式噪声$k$XOR问题（亦称稀疏噪声学习奇偶性问题）的量子算法，该算法在仅使用对数级数量量子比特的同时，相比已知最佳经典算法实现了近四次方（$4$次幂）的加速。本研究基于Hastings针对张量主成分分析问题的量子算法进行推广与简化，通过构建基于Kikuchi方法的通用框架（恢复了张量主成分分析的四次方加速），我们预期该框架将为更多植入式推理问题带来类似的加速效果。这些加速依赖于植入式推理问题天然实例化引导稀疏哈密顿量问题的特性。鉴于植入式噪声$k$XOR问题已被用于某些密码学构造组件，我们的研究表明其中部分构造可能面临超二次量子攻击的威胁。