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Coherence plays a very important role in Grover search algorithm (GSA). Inthis paper, we define the normalization coherence N(C), where C is a coherencemeasurement. In virtue of the constraint of large N and Shannon's maximumentropy principle, a surprising complementary relationship between thecoherence and the success probability of GSA is obtained. Namely,P_s(t)+N(C(t))\simeq 1, where C is in terms of the relative entropy ofcoherence and l_1 norm of coherence, t is the number of the search iterationsin GSA. Moreover, the equation holds no matter in ideal or noisy environments.Considering the number of qubits is limited in the recent noisyintermediate-scale quantum (NISQ) era, some exact numerical calculationexperiments are presented for different database sizes N with different typesof noises. The results show that the complementary between the successprobability and the coherence almost always hold. This work provides a newperspective to improve the success probability by manipulating itscomplementary coherence, and vice versa. It has an excellent potential forhelping quantum algorithms design in the NISQ era.

Complementarity between Success Probability and Coherence in Grover Search Algorithm