In this study, we introduce a novel family of tensor networks, termed \textit{constrained matrix product states} (MPS), designed to incorporate exactly arbitrary discrete linear constraints, including inequalities, into sparse block structures. These tensor networks are particularly tailored for modeling distributions with support strictly over the feasible space, offering benefits such as reducing the search space in optimization problems, alleviating overfitting, improving training efficiency, and decreasing model size. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in U(1) symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop a novel canonical form for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules and permit optimal truncation schemes. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary objective functions subject to discrete linear constraints. Our method's efficacy is demonstrated by solving the quadratic knapsack problem, achieving superior performance compared to a leading nonlinear integer programming solver. Additionally, we analyze the complexity and scalability of our approach, demonstrating its potential in addressing complex constrained combinatorial optimization problems.

翻译：本研究提出了一种新型张量网络家族，称为\textit{约束矩阵乘积态}（MPS），旨在将任意离散线性约束（包括不等式）精确地纳入稀疏块结构。这些张量网络专门为严格在可行空间上具有支撑的分布建模而设计，具有减少优化问题搜索空间、缓解过拟合、提高训练效率以及减小模型规模等优势。我们方法的核心是量子区域的概念，这是对传统用于U(1)对称张量网络的量子数概念的扩展，适用于捕捉任何线性约束，包括无约束情形。我们进一步为这些新型MPS开发了一种新颖的规范形式，允许根据量子区域融合规则合并和分解张量块，并支持最优截断方案。利用这种规范形式，我们采用无监督训练策略来优化受离散线性约束的任意目标函数。通过求解二次背包问题，我们的方法展示了其有效性，其性能优于领先的非线性整数规划求解器。此外，我们分析了该方法的复杂性和可扩展性，证明了其在解决复杂约束组合优化问题方面的潜力。