Quantum Searching amidst large uncertainties
D. Li, Xiao-Ming Li, Hongtao Huang, Xinxin Li · 2006
The quantum search algorithm is a way for finding an item with desired properties in an unsorted database of size N in only$\\sqrt{N}$ steps by means of a sequence of selective inversion and quantum diffusion operations. However, like most quantum algorithms, this requires the algorithm to execute the precise sequence of operations, any further probing into the database only worsens the result. Grover recently showed that by replacing the selective inversion by selective phase shifts of $\\pi /3$, the algorithm converges to the desired item monotonically. This paper builds on that result and considers the problem of retrieving a marked item from a database containing an unknown fraction of marked items, say $\\epsilon $. We derive optimal algorithms for different ranges of $\\epsilon .$ These are similar to Grover's recent algorithm but the phase shift turns out to vary with $\\epsilon $ - this is generally quite different from $\\pi /3$.