Trade-offs in the quantum search algorithm
Lov K. Grover · Physical Review A · 2002
Quantum search has been proved to be the best possible algorithm for the exhaustive search problem in the sense that the number of queries it requires cannot be reduced. However, the number of nonquery operations, and thus the total number of operations, can be reduced. The number of nonquery unitary operations can be reduced by a factor of $\mathrm{log}N/\ensuremath{\alpha}\mathrm{log}(\mathrm{log}N)$ while increasing the number of queries by a factor of only $[1+(\mathrm{log}{N)}^{\ensuremath{-}\ensuremath{\alpha}}].$ For example, by choosing \ensuremath{\alpha} to be $O(\mathrm{log}N/\mathrm{log}(\mathrm{log}N)),$ the number of nonquery unitary operations can be reduced by 40% while increasing the number of queries by just two.