Grover’s Search Algorithm and Quantum Lower Bounds

Searching an item in an unsorted database with size N costs a classical computer O(N) running time. Can a quantum computer search a needle in a haystack much more efficient than its classical counterpart? Grover, in 1996, affirmatively answered this question by proposing a search algorithm [4], which consults the database only O( N ) times. In contrast to algorithms based on the quantum Fourier transform, with exponential speedups, the search algorithm only provides a quadratic improvement. However, the algorithm is quite important because it has broad applications and the same technique can be used to improve solutions of NP-complete problems. One might think of having better improvements over the search algorithm. Nevertheless, it turns out that Grover’s search algorithm is optimal. At least Ω( N ) queries are needed to solve the problem [1,2,5]. This note details the quantum search algorithm and its lower bound in Section 1 and Section 2 respectively.