Solving NP - complete problem by simulations

The mathematical formalism of quantum resonance combined with tensor product decomposability of unitary evolutions is mapped onto a class of NP-complete combinatorial problems. It has been demonstrated that nature has polynomial resources for solving NP-complete problems and that may help to develop a new strategy for artificial intelligence, as well as to re-evaluate the role of natural selection in biological evolution. 1.Introduction. In this work an attempt is made to simulate combinatorial optimization that is the main obstacle to artificial intelligence. It is a well-established fact that nature exploits combinatorial optimization for natural selection. It is also known that even ants collectively solve combinatorial problems (such as the shortest path to food in a labyrinth) more efficiently than man-made artificial devices. That is why combinatorial problems are not only an obstacle, but is the greatest challenge to artificial intelligence. In this work, a new approach to simulation of NP-complete problems is introduced: combinatorial properties of tensor product decomposability of unitary evolution of many-particle quantum systems are mapped to solutions of NPcomplete problems, while the reinforcement and selection of a desired solution is executed by quantum resonance. 2. Quantum Resonance. Consider a quantum system characterized by a discrete spectrum of energy eigenstates subject to a small perturbing interaction, and let the perturbation be switched on at zero time. The Hamiltonian of the system can be presented as a sum of the time-independent and oscillating components ∫ + = ω ω ω ω ξ ε td H H H sin ) ( 1 0 0 (1) where 0 H and 1 H are constant Hermitian matrices, ω is the frequency of perturbations, and ) (ω ξ is the spectral density. The probability of transition from state k to q in the first approximation is proportional to the product, [1] ) ( ] ) ( 2 1 [sin | | 2 2 1 * ω ω φ φ − − ∝ qk qk q k kq a t a H P (2) Here i φ are the eigenstates of 0 H : . ,... 2 , 1 , 0 n j E H j j j = = φ φ (3) where i E are the energy eigenvalues, n q k E E a q k kq ,... 2 , 1 , , = − =  (4) and  is the Planck constant. The resonance, i.e., a time-proportional growth of the transition probability Pkq occurs when : qk a = ω t H P q k kq ) ( | | 2 1 * ω ξ φ φ ∝ (5) 3. Combinatorial problems. Combinatorial problems are among the hardest in the theory of computations. They include a special class of so called NP-complete problems which are considered to be intractable by most theoretical computer scientists. A typical representative of this class is a famous traveling-salesman problem (TSP) of