NEW QUANTUM ALGORITHM FOR STUDYING NP-COMPLETE PROBLEMS

Masanori Ohya, Igor V. Volovich

Dhananjay P. Mehendale

Pramod S. Joag, Dhananjay P. Mehendale

Masanori Ohya, Igor Vasil'evich Volovich

Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete problems and argue that the problem, in principle, can be solved in polynomial time by using our new quantum algorithm. keywords: Quantum Algorithm, NP-complete problem, Chaotic Dynamics 1

Dhananjay P. Mehendale

We develop three new quantum algorithms for searching the desired target state in the unstructured database of size N. The first algorithm requires Log N iterative steps. It constructs two quantum bags of equal size in terms of two quantum states, out of which exactly one quantum state will have nonzero overlap with the target state. This determination of overlap is done by taking the inner product, in Log N time [2], of the implicitly known target state with any one of these two quantum states. The second algorithm requires just one single step which uses a new suitable operator and the choice of this operator is problem dependent, i.e. it depends upon the number of qubits required to be used to represent an element in the index set. The third algorithm again requires only a single step and this algorithm makes use of a fixed (same) operator. It is known that algorithm for unstructured database search can be easily adaptable for solving NP-Complete problems. However, the computational complexity of NP-Complete problems after the adaptations of both the classical as well as quantum [1] search algorithms remains of the exponential order as the exponent for quantum [1] algorithm changes only to one-half times the exponent for classical algorithm. But for our quantum algorithms the exponent falls substantially so that our new quantum algorithms for unstructured search are capable if reducing the computational complexity of NP-Complete problems to polynomial order!

Makoto Ohya

We review the quantum chaos algorithm solving the NP-complete problems in polynomial time. This work has been done in the series of papers with Professor Igor Volovich for nearly ten years.