The Phase Transition of Task-Solvability: Quantum Evasion of a Classical Computational Limit

The success of autonomous agents in complex environments is often contingent on resource availability. Here, we demonstrate that for clas- sical agents operating under local perception and movement rules, the probability of successfully completing a resource-gathering task exhibits a sharp phase transition. This transition is not merely a gradual increase in success but a critical phenomenon analogous to percolation in statisti- cal physics. Using numerical simulations and finite-size scaling analysis, we pinpoint the critical resource density, pc,agent, at which the environ- ment shifts from a collection of isolated resource-patches to a globally connected network, enabling task-solvability. We find this critical point, pc,agent ≈ 0.30, aligns remarkably well with the theoretical site percolation threshold for a 2D triangular lattice. We identify the microscopic mecha- nism of failure in the sub-critical regime as the ”Local Resource Prison” (LRP), where an agent exhausts its local resource neighborhood and be- comes trapped. We propose that this classical limitation can be overcome by quantum agents. By exploiting non-local quantum effects such as tun- neling or superposition, a quantum agent could ”evade” the criticality, bypassing the physical barriers of the LRP and achieving high success rates even in sub-critical environments. This work frames task-solvability as a critical phenomenon and suggests a new class of problems where quantum computation offers a fundamental advantage by transcending the classical limits imposed by environmental topology.