This thesis presents and evaluates a variational approach designed to address 3-SAT problems at critical clause density, examining how parameter-space trajectories generated by deterministic time-dependent variational principle (TDVP) dynamics, seeded by an initial imaginary-time evolution, can steer the ansatz toward low-energy configurations and, ultimately, the ground states of the encoded instances. Each Boolean formula is mapped to a spin Hamiltonian, whose low-energy spectrum encodes the satisfying assignments. Within this framework, the algorithm employs an ansatz constructed as a superposition of uncorrelated Bloch-disk product states. The initialization is carried out via a short imaginary-time evolution, starting from a transverse-field-like state with equal overlap with all classical configurations.