The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries

A. Papageorgiou, H. Woźniakowski
2006 Quantum Information Processing  
We show how a number of NP-complete as well as NP-hard problems can be reduced to the Sturm-Liouville eigenvalue problem in the quantum setting with queries. We consider power queries which are derived from the propagator of a system evolving with a Hamiltonian obtained from the discretization of the Sturm-Liouville operator. We use results of our earlier paper concering the complexity of the Sturm-Liouville eigenvalue problem. We show that the number of power queries as well the number of
more » ... s needed to solve the problems studied in this paper is a low degree polynomial. The implementation of power queries by a polynomial number of elementary quantum gates is an open issue. If this problem is solved positively for the power queries used for the Sturm-Liouville eigenvalue problem then a quantum computer would be a very powerful computation device allowing us to solve NP-complete problems in polynomial time. Power queries are used in the well-known phase estimation algorithm, see [17] , which plays a central role in Shor's factorization algorithm [21] . In a recent paper [20] we dealt with power queries in the study of the quantum complexity of the Sturm-Liouville eigenvalue problem. In this paper, we show how to reduce NP-complete problems to the Sturm-Liouville eigenvalue problem whose complexity in the classical and quantum settings has been studied in [20] . Obviously, it would be enough to show this reduction for one NP-complete problem. We choose to present this reduction for several problems to show how the number of power queries and qubits depends on the particular NP-complete problem. In particular, that is why we consider satisfiability and the traveling salesman problem, as well as their NP-hard versions. The reductions presented in this paper can be summarized in the following diagram. Here, SAT stands for the satisfiability problem, TSP for the traveling salesman problem, MIN for the minimization problem of choosing the smallest number out of N real numbers, GRO for Grover's problem, BOOL for the Boolean mean problem, INT for the integration problem, and finally SLE for the Sturm-Liouville eigenvalue problem. These reductions mean, in particular, that the satisfiability problem is reduced to the Boolean mean problem for a specific Boolean function which is reduced to the integration problem for a specific integrand, which in turn is reduced to the Sturm-Liouville eigenvalue problem for a specific function, and finally the last problem is solved by the quantum algorithm using power queries. The Sturm-Liouville problem is defined in the next section. For the moment we mention that we want to approximate the smallest eigenvalue of a specific differential operator, and this smallest eigenvalue is given in a variational form as the minimum of specific integrals. We use a formula relating the Sturm-Liouville eigenvalue problem to a weighted integration problem, see [20] . Many computational problems including the discrete problems mentioned above can be recasted as this weighted integration problem. Thus, we can solve them using the algorithms of [20] for solving the Sturm-Liouville eigenvalue problem. These algorithms use of order ε −1/3 bit queries or log ε −1 power queries and compute an ε-approximation of the smallest eigenvalue with probability 3 4 . The bounds on bit and power queries are sharp up to multiplicative constants, see [5, 20] . Hence, exponentially fewer power queries than bit queries are needed to solve the Sturm-Liouville eigenvalue problem. As we shall see, the same is true for the problems studied in this paper. In the quantum setting with bit queries, we do not obtain surprising results. The polynomial number of bit queries, ε −1/3 , implies that the solution of NP-complete problems by modifications of the algorithm for the Sturm-Liouville eigenvalue problem will require exponentially many queries in terms of the NP problem size. The situation is quite different if we consider power queries. The logarithmic number of power queries, log ε −1 , implies that NP-complete problems can be solved by modifications of the algorithm for the Sturm-Liouville eigenvalue problem and the number of power queries is polynomial in the problem size.
doi:10.1007/s11128-006-0043-0 fatcat:ipbcw6iq2vhotn6ij3lrrjl3se