通过条件数的下界优化实现带结构化噪声的奇偶学习的量子优势
Toward Quantum Advantage in Learning Parities with Structured Noise via Lower Bound Optimization of the Condition Number
AI总结：
本研究针对带结构化噪声的奇偶学习问题，提出优化Macaulay线性系统归约方法以降低条件数，改进量子算法并证实其在特定参数下优于经典算法，有望实现该问题的量子优势。
AI中文摘要：
带结构化噪声的奇偶学习（LPSN）可归约为求解非线性布尔系统。在量子计算中，此类系统通常被转化为Macaulay线性系统并通过量子线性系统算法求解，该过程受条件数的严重限制。为解决这一问题，我们提出了一种针对Macaulay线性系统的新型归约方法。在Ding等人的假设下，我们推导了包含缩放因子的条件数下界。该归约不仅保证了高效的量子态制备，且相对于归约后的右端向量，在条件数区间方面展现出显著优势，从而降低了条件数的下界，最终优化了求解布尔系统的量子算法的时间复杂度上界。此外，将此改进后的量子算法应用于LPSN时，可通过利用Macaulay系统的解结构大幅降低样本复杂度。我们还提供了具体的逻辑级量子资源估计，表明优化后的条件数可直接转化为电路宽度、深度及门数的减少。最后，我们通过在噪声模式适应性、样本复杂度和时间复杂度方面系统比较量子与经典方法，建立了算法选择策略。结果表明，我们的量子算法在特定参数范围内具备超越经典对应算法的潜力。
英文摘要：
Learning Parities with Structured Noise (LPSN) can be reduced to solving nonlinear Boolean systems. In quantum computing, such systems are typically transformed into Macaulay linear systems and solved via quantum linear system algorithms, a process severely limited by the condition number. To address this, we propose a novel reduction method for Macaulay linear systems. Under the assumptions of Ding et al., we derive a condition number lower bound incorporating a scaling factor. This reduction not only guarantees efficient quantum state preparation but also exhibits a distinct advantage regarding the condition number interval relative to the reduced right-hand side vector, thereby reducing the lower bound of the condition number and ultimately optimizing the upper bound on the time complexity of the quantum algorithm for solving Boolean systems. Furthermore, applying this improved quantum algorithm to LPSN significantly reduces sample complexity by exploiting the Macaulay system's solution structure. We further provide a concrete logical-level quantum resource estimate, demonstrating that the optimized condition number translates directly into a reduction in circuit width, depth, and gate count. Finally, we establish an algorithm selection strategy by systematically comparing quantum and classical approaches across noise pattern adaptability, sample complexity, and time complexity. Results demonstrate that our quantum algorithm exhibits the potential to outperform classical counterparts under specific parameter regimes.
