  Coherence plays a very important role in Grover search algorithm (GSA). In
this paper, we define the normalization coherence N(C), where C is a coherence
measurement. In virtue of the constraint of large N and Shannon's maximum
entropy principle, a surprising complementary relationship between the
coherence 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 of
coherence and l_1 norm of coherence, t is the number of the search iterations
in GSA. Moreover, the equation holds no matter in ideal or noisy environments.
Considering the number of qubits is limited in the recent noisy
intermediate-scale quantum (NISQ) era, some exact numerical calculation
experiments are presented for different database sizes N with different types
of noises. The results show that the complementary between the success
probability and the coherence almost always hold. This work provides a new
perspective to improve the success probability by manipulating its
complementary coherence, and vice versa. It has an excellent potential for
helping quantum algorithms design in the NISQ era.
