发表机构
Quantum Computing Inc (QCi)（量子计算公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该框架通过正则化Motzkin-Straus团公式将离散优化问题编译为标准二次规划，覆盖Karp的21个NP问题，并保证每个赋值均为严格局部最小值。
AI 中文摘要
标准二次规划（StQP）在非负变量之和为1的单纯形上最小化二次型。我们将经典图归约与正则化的Motzkin-Straus团公式相结合，以在该连续域中表达离散优化问题。图矩阵的对角线元素为$\tau$，边上的元素为零，非边上的元素为一。对于$0<\tau<1$，其最小值为$\tau/\omega(G)$，其中$\omega(G)$是团数。其严格局部极小值恰好是最大团上的均匀分布，其全局极小值编码最大团。当$\tau=1/2$时，整数缩放给出系数属于$\{0,1,2\}$且最小值为$1/\omega(G)$，从而产生一个具有受限系数字母表的NP完全的StQP阈值问题。我们给出了可满足性、着色、哈密顿回路、独立集、顶点覆盖、集合打包、三维匹配和图同构的显式公式。将正则化加权团公式与局部状态兼容图相结合，可得到一个精确的编译器，用于由完整局部表指定的有限域因子模型，包括QUBO，每个二元对因子至多使用四个单纯形坐标。该目录涵盖Karp的21个问题：十二个使用直接图公式，九个使用因子状态公式，其中六个通过二元线性可行性获得。对于每条路径，我们记录了维度、系数结构和恢复规则。我们分析了交互计数、系数范围、目标分离、扰动容差、支持恢复和解码开销。分离界量化了团大小、因子权重和偏移的影响。在完整的因子状态构造中，每个赋值（包括每个次优赋值）都是严格局部最小值。
英文摘要
The standard quadratic program (StQP) minimizes a quadratic form over nonnegative variables that sum to one. We compose classical graph reductions with regularized Motzkin--Straus clique formulations to express discrete optimization problems in this continuous domain. The graph matrix has diagonal entries $τ$, zeros on edges, and ones on nonedges. For $0<τ<1$, its minimum is $τ/ω(G)$, where $ω(G)$ is the clique number. Its strict local minimizers are precisely the uniform distributions on maximal cliques, and its global minimizers encode maximum cliques. At $τ=1/2$, integer scaling gives coefficients in $\{0,1,2\}$ and minimum $1/ω(G)$, yielding an NP-complete StQP threshold problem with a restricted coefficient alphabet. We give explicit formulations for satisfiability, coloring, Hamiltonian cycles, independent set, vertex cover, set packing, three-dimensional matching, and graph isomorphism. A regularized weighted clique formulation combined with local-state compatibility graphs gives an exact compiler for finite-domain factor models specified by complete local tables, including QUBO, with at most four simplex coordinates per binary pair factor. The catalog covers Karp's 21 problems: twelve use direct graph formulations, and nine use factor-state formulations, including six obtained through binary-linear feasibility. For each route we record dimensions, coefficient structure, and recovery rules. We analyze interaction count, coefficient range, objective separation, perturbation tolerance, support recovery, and decoding overhead. The separation bounds quantify the effects of clique size, factor weights, and offsets. In the complete factor-state construction, every assignment, including each suboptimal assignment, is a strict local minimum.
