Zermelo's navigation problem through the lens of quantum annealing: How the Landau-Zener approximation leads to an efficient classical solution

The river-crossing problem, also known as Zermelo's navigation problem, is a classic example of an optimization problem with practical relevance and a scalable degree of complexity. It asks for the optimal trajectory of a vessel moving through a water flow field and provides a setting in which physics, variational methods, and optimization are naturally intertwined. We state a version of Zermelo's problem and then solve it as formulate it as an adiabatic quantum-computing problem using quantum trits, or qutrits for short. The construction includes a penalty term that enforces the prescribed boundary conditions and an exploration term that allows the system to move through intermediate configurations toward the optimal feasible path. In the adiabatic description, the evolution proceeds through a sequence of avoided crossings, so that the resulting fidelity can be estimated using the Landau-Zener formula. Remarkably, the regime in which this approximation is valid also provides a deterministic way to identify the correct solution with computational effort that scales only quadratically with the problem size. Thus, a quantum formulation initially motivated by the apparent exponential complexity of the problem reveals an underlying classical structure that can be exploited efficiently. Our approach also provides a pedagogical illustration of how a real-world optimization problem can be cast into a quantum-annealing framework and then analyzed using the Schrödinger equation, avoided crossings, and Landau-Zener theory.