发表机构
Singapore Management University; Agency for Science, Technology and Research (A*STAR)（新加坡国立大学; 科学技术研究局（A*STAR））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出WCP-QAOA框架，利用Walsh-Hadamard变换将自定义惩罚函数展开为Pauli-Z字符串加权和，实现无松弛约束编码，并在6量子比特背包问题上验证了其有效性。
AI 中文摘要
组合优化中的不等式约束通常通过引入松弛变量和二次惩罚来处理，这会增加量子比特数量并扭曲优化景观。无松弛公式则直接将非线性自定义惩罚函数应用于约束，但这类函数难以在量子近似优化算法（QAOA）中实现。我们证明Walsh-Hadamard变换（WHT）对任何惩罚函数都能消除这一障碍：应用于线性约束的惩罚恰好是Pauli-$Z$字符串的加权和。我们提出这一框架，称为Walsh自定义惩罚QAOA（WCP-QAOA）。将展开截断至Walsh度$k$需要$\mathcal{O}(n^k)$个Pauli项，且量子比特数量保持不变。该截断是惩罚在$L^2$意义上的最优度-$k$近似，我们给出了被丢弃的Walsh系数的一个条件，在该条件下截断哈密顿量仍能将每个可行解排在每个不可行解之下。由于根据定义评估Walsh系数需要所有$2^n$个惩罚值，我们给出一种算法，能在次指数时间内返回任意系数，从而对于固定的$k$，度-$k$哈密顿量可在$n$的多项式时间内获得。我们证明对于指数惩罚，存在WHT的闭式乘积形式，因此存在计算Walsh系数的闭式表达式。我们识别出能缩短精确谱的结构性捷径。我们还测试了WCP-QAOA在求解6量子比特背包问题中的表现。
英文摘要
Inequality constraints in combinatorial optimization are conventionally handled by introducing slack variables and quadratic penalties, which increase the qubit count and distort the optimization landscape. Slack-free formulations instead apply a nonlinear custom penalty function directly to the constraint, which are difficult to be implemented in quantum approximate optimization algorithm (QAOA). We show that the Walsh--Hadamard transform (WHT) removes this obstruction for any penalty function: a penalty applied to a linear constraint is exactly a weighted sum of Pauli-$Z$ strings. We propose this framework as the Walsh custom penalty QAOA (WCP-QAOA). Truncating the expansion at Walsh degree $k$ costs $\mathcal{O}(n^k)$ Pauli terms and leaves the qubit count untouched. The truncation is the optimal degree-$k$ approximation of the penalty in $L^2$, and we give a condition on the discarded Walsh coefficients under which the truncated Hamiltonian still ranks every feasible solution below every infeasible one. Since evaluating a Walsh coefficient from its definition requires all $2^n$ penalty values, we give an algorithm that returns any coefficient in sub-exponential time, so that the degree-$k$ Hamiltonian is obtained in time polynomial in $n$ for fixed $k$. We show that for the exponential penalty, there exists a closed product form of WHT, and hence a closed form expression to calculate the Walsh coefficients. We identify structural shortcuts that shorten the exact spectrum. We also tested WCP-QAOA in solving a 6-qubit knapsack problem.
Comments16 pages, 4 figures. Submission to QIP 27. Subject to major changes
