Matched-multiphase Grover algorithm for a small number of marked states

Recently, we proposed a multiphase-matching method for the Grover algorithm with a matching rule for multiple phases αj and βj, j=1,…,k, where k is the number of Grover operations. The phases are matched such that αj=−βk−j+1 globally over a sequence of k Grover operations. The success probability P6(λ) for k=6 was found to be almost constant and unity over a wide range of λ, i.e., 0.10777⩽λ⩽1, where λ is the fraction of marked items in a database state. For λ<0.10777, however, P6(λ) decreases rapidly to zero as λ decreases and the efficiency of the method deteriorates. In this Brief Report we show that the difficulty with small values of λ mentioned above can be alleviated by increasing the number of operations k. With k=20, for example, we find a value of P20(λ) that is almost constant and unity in the region of small λ where the matching with six Grover operations is not effective. The matching with k=6 and the one with k=20 complement each other so that the entire range of 0≲λ<1 can be well covered.