Fundamental Scaling Constraints for Equilibrium Molecular Computing

Molecular computing promises massive parallelization to explore solution spaces, but so far practical implementations remain limited due to off-target binding and exponential proliferation of competing structures. Here, we investigate the theoretical limits of equilibrium self-assembly systems for solving computing problems, focusing on the directed Hamiltonian Path Problem (HPP) as a benchmark for NP-complete problems. The HPP is encoded via particles with directional lock-key patches, where self-assembled chains form candidate solution paths. We determine constraints on the required energy gap between on-target and off-target binding for the HPP to be encoded and solved. We simultaneously examine whether components with the required energy gap can be designed. Combining these results yields a phase diagram identifying regions where HPP instances are both encodable and solvable. These results establish fundamental upper bounds on equilibrium molecular computation and highlight the necessity of non-equilibrium approaches for scalable molecular computing architectures.