Chapter 10: Quantum Search Algorithm
The quantum search algorithm was discovered by L. K. Grover in 1997, and is called Grover’s algorithm after his name. Let us consider how to find elements marked by a certain symbol out of a database consisting of N elements (say, multi-million elements). This problem is similar to how to find the name of the person from a telephone book knowing only his/her telephone number. This search takes by far more time than finding the phone number from the name of the person. The difference comes from the structure of the telephone book. In the telephone book, the names are arranged in the alphabetical order and constitute the structured database, while phone numbers do not form any structure in the telephone book. In classical computers, when we search one target element from the database with N elements, we need N/2 trials on average. Grover showed using the quantum search algorithm that √N trials will be enough on average to find the target element from the database with N elements. Shor’s quantum computation algorithm was an epoch-making algorithm which made possible to solve the factorization problem in polynomial time, while the classical computers need exponentially increasing time to solve it. In comparison, Grover’s algorithm makes the data search possible by O(√N) computation time, while it takes O(N) computation time by the classical computers. Concerning the reduction of computing time, Grover’s algorithm may look less powerful than Shor’s algorithm. However, it has good potentiality of application in the wide area such as NP-complete problems and structured databases.
