Cons-training tensor networks

In this study, we introduce a novel family of tensor networks, termed constrained matrix product states (MPS), designed to incorporate exactly arbitrary linear constraints into sparse block structures. These tensor networks effectively bridge the gap between U(1) symmetric MPS and traditional, unconstrained MPS. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop canonical forms for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary cost functions subject to linear constraints. We use this to solve the quadratic knapsack problem and show a superior performance against a leading nonlinear integer programming solver, highlighting the potential of our method in tackling complex constrained combinatorial optimization problems