  Constrained combinatorial optimization problems abound in industry, from
portfolio optimization to logistics. One of the major roadblocks in solving
these problems is the presence of non-trivial hard constraints which limit the
valid search space. In some heuristic solvers, these are typically addressed by
introducing certain Lagrange multipliers in the cost function, by relaxing them
in some way, or worse yet, by generating many samples and only keeping valid
ones, which leads to very expensive and inefficient searches. In this work, we
encode arbitrary integer-valued equality constraints of the form Ax=b, directly
into U(1) symmetric tensor networks (TNs) and leverage their applicability as
quantum-inspired generative models to assist in the search of solutions to
combinatorial optimization problems. This allows us to exploit the
generalization capabilities of TN generative models while constraining them so
that they only output valid samples. Our constrained TN generative model
efficiently captures the constraints by reducing number of parameters and
computational costs. We find that at tasks with constraints given by arbitrary
equalities, symmetric Matrix Product States outperform their standard
unconstrained counterparts at finding novel and better solutions to
combinatorial optimization problems.
