Phase Condition for the Grover Algorithm

Quantum algorithms standardly use two techniques: Fourier transforms [1] and amplitude amplification. The Grover search algorithm is based on the latter. The problem addressed by the Grover algorithm is to find a marked term (or target term in [2]) in an unsorted database of size N . To accomplish this, a quantum computer needs O (√ N ) queries using the Grover algorithm [3]. In Grover’s original version [3], the algorithm consists of a sequence of unitary operations on a pure state, i.e., the algorithm is Q = −I 0 WI τ W , where W is the Walsh–Hadamard transformation and I x = I − 2|x〉|〈x|, which inverts the amplitude in the state |x〉; here, I 0 and I τ invert the amplitudes in the respective initial and marked basis states |0〉 and |τ〉. To extend his original algorithm, Grover [2] replaced the Walsh–Hadamard transformation with any quantum mechanical operation and thus obtained the quantum search algorithm Q = −I γ U−1I τ U , where U is any unitary operation and U−1 is the adjoint (the complex conjugate of the transpose) of U . Boyer et al. gave analytic expressions for the amplitude of the states for the original Grover algorithm with the Walsh–Hadamard transformation and the inversion of the amplitudes and established tight bounds on quantum searching [4]. To generalize the Grover algorithm further, we must allow amplitudes to be rotated by arbitrary phases, not just be inverted. An example is the quantum algorithm Q = −I γ U−1I τ U , where θ and φ are the rotation angles of the amplitude phases in the respective initial basis state |γ〉 and marked basis state |τ〉. Recently, several authors have contributed to general quantum search algorithms with any unitary operations and arbitrary phase rotations [5]–[10]. For general quantum search algorithms, the following problems must be solved: