  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.
