Toward Quantum Advantage in Learning Parities with Structured Noise via Lower Bound Optimization of the Condition Number

Abstract

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.