Out-of-equilibrium molecular computing for efficient solutions to NP-hard graph problems

Molecular computing holds great potential for solving NP-hard problems through massive parallelism. However, traditional approaches, such as DNA computing -- first demonstrated by Aldeman in 1994 to tackle the Hamiltonian Path Problem (HPP) -- are significantly limited by high error rate, which exponentially diminishes yields as problem size increases. We propose an out-of-equilibrium molecular computing framework inspired by living systems to solve scalable HPP, with patchy particles serving as individual computational units. Our approach introduces dynamic temperature control and energy-driven state change mechanisms to stabilize the computation process and enhance error correction. Specifically, we implement a series of reaction cycles that maintain the system in a controlled, out-of-equilibrium statem allowing for more reliable and scalable computations. Programmable, enzyme-like interactions between patchy particles provide precise control over state transitions, facilitating the efficient exploration of potential solutions. This framework offers significant improvement in practicality and scalability of molecular computing, advancing the field towards real-world applications for solving computationally complex problems.