Complementarity between Success Probability and Coherence in Grover Search Algorithm

TL;DR

This paper reveals a surprising complementary relationship between coherence and success probability in Grover's search algorithm, valid in both ideal and noisy environments, offering new insights for quantum algorithm design.

Contribution

It introduces a normalization coherence measure and demonstrates a universal complementary relation with success probability in GSA, applicable in noisy quantum settings.

Findings

Success probability plus normalized coherence approximately equals one.

The complementary relation holds under various noise conditions.

Numerical experiments confirm the relation across different database sizes.

Abstract

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.