AI 中文总结
该研究提出基于沃尔什-傅里叶变换的无松弛惩罚框架，将受限二元优化重构为适配量子退火硬件的QUBO，所得硬件原生替代项在嵌入采样后样本目标间隙更低，性能优于现有方法。
AI 中文摘要
我们提出了一种新颖的无松弛、基于惩罚项的框架，用于将受限二元优化问题重构为适用于近期量子退火硬件的二次无约束二元优化（QUBO）问题。给定用户选定的最能自然表征约束条件的惩罚函数（通常为非二次函数，如海维赛德函数的替代形式），以及布尔超立方体上的目标概率测度，我们的方法会返回所选惩罚函数在由线性和二次沃尔什-傅里叶特征张成的子空间上的加权最小二乘投影，这些特征对应于目标硬件图上可物理实现的耦合。在这个受限族内，所得二次替代项由法方程唯一且最优地确定：与现有技术方法不同，它无需为每个约束调整惩罚系数，且通过构造避免了密集的全对耦合。这带来两个实际结果：其一，投影后的惩罚项符合器件连通性，在 minor embedding 后缩短了链长并降低了物理量子比特开销；其二，我们通过实验表明，这种硬件原生替代项虽来自严格更小的近似空间，却能优于更密集的全对投影。当QUBO在量子退火器上进行嵌入和采样后，该优势进一步扩大，与不平衡惩罚和面向所有二次项的硬件盲投影相比，所得样本具有最低的最坏情况和平均目标间隙。
英文摘要
We present a novel slack-free, penalty-based framework for reformulating constrained binary optimization as Quadratic Unconstrained Binary Optimization (QUBO) on near-term quantum annealing hardware. Given a user-chosen penalty function that most naturally captures a constraint---typically non-quadratic, such as a Heaviside-function surrogate---and a target probability measure over the Boolean hypercube, our method returns the weighted least-squares projection of the chosen penalty function onto the subspace spanned by linear and quadratic Walsh--Fourier characters that correspond to physically realizable couplings on the target hardware graph. Within this restricted family, the resulting quadratic surrogate is uniquely and optimally determined by the normal equations: unlike state-of-the-art approaches, it introduces no per-constraint penalty coefficients to tune and avoids dense all-pairs couplings by construction. Two practical consequences follow. First, the projected penalty respects device connectivity, reducing chain lengths and physical-qubit overhead after minor embedding. Second, we show empirically that this hardware-native surrogate can outperform denser full-pairwise projections, despite being drawn from a strictly smaller approximation space. This advantage widens once the QUBO is embedded and sampled on quantum annealers, yielding samples with the lowest worst-case and mean objective gaps compared to unbalanced penalization and a hardware-blind projection onto all quadratic terms.
Comments11 pages, 9 figures, 3 tables. Accepted at the IEEE International Conference on Quantum Computing and Engineering (QCE 2026). Code: https://github.com/lklee9/topology-aware-walsh-fourier-penalization
