arXiv:1808.02222 · Entropy

Coherence Depletion in Quantum Algorithms

Ye-Chao Liu, Jiangwei Shang, Xiangdong Zhang

Published Mar 7, 2019• 22 citations

Abstract

Besides the superior efficiency compared to their classical counterparts, quantum algorithms known so far are basically task-dependent, and scarcely any common features are shared between them. In this work, however, we show that the depletion of quantum coherence turns out to be a common phenomenon in these algorithms. For all the quantum algorithms that we investigated, including Grover's algorithm, Deutsch-Jozsa algorithm, and Shor's algorithm, quantum coherence of the system states reduces to the minimum along with the successful execution of the respective processes. Notably, a similar conclusion cannot be drawn using other quantitative measures such as quantum entanglement. Thus, we expect that coherence depletion as a common feature can be useful for devising new quantum algorithms in the future.

DOI: 10.3390/e21030260

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arXiv:1808.02222 · Entropy

Coherence Depletion in Quantum Algorithms

Ye-Chao Liu, Jiangwei Shang, Xiangdong Zhang

Published Mar 7, 2019• 22 citations

arXiv HTML PDF

Abstract

DOI: 10.3390/e21030260

Related papers

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G

, which can be decomposed into $G={H}^{\ensuremath{\bigotimes}n}P{H}^{\ensuremath{\bigotimes}n}O$, where

H

is Hadamard operator,

P

is the condition phase-shift operator, and

O

is the oracle operator. First, we show that

O

and

P

are incoherent operators while ${H}^{\ensuremath{\bigotimes}n}$ is coherent. Second, we prove that the amount of the operator coherence of the first ${H}^{\ensuremath{\bigotimes}n}$ and the operator coherence produced or depleted by ${H}^{\ensuremath{\bigotimes}n}$ depends not only on the size of the database and the success probability, but also on target states. Moreover, the amount of operator coherence is larger when the superposition state of targets is entangled rather than product. Third, we show that the two ${H}^{\ensuremath{\bigotimes}n}$ have different effects on coherence that one produces coherence and the other depletes coherence, and the depletion plays a major role. Therefore, the coherence is vibrating during the search process and the overall effect is that coherence is in depletion.

{\mathcal{C}}_{r}

), we show that the success probability depends on the QCD. The same phenomenon is also found by using the

{l}_{1}

norm of coherence measure (

{\mathcal{C}}_{{l}_{1}}

). In the limit case, the cost performance is defined to characterize the behavior about QCD in enhancing the success probability of GA, which is only related to the number of searcher items and the scale of the database, regardless of using

{\mathcal{C}}_{r}

{\mathcal{C}}_{{l}_{1}}

. In the generalized Grover search algorithm (GGA), the QCD for a class of states increases with the required optimal measurement time. In comparison, the quantification of other quantum correlations in GA, such as pairwise entanglement, multipartite entanglement, pairwise discord, and genuine multipartite discord, cannot be directly related to the success probability or the optimal measurement time. Additionally, we do not detect pairwise nonlocality or genuine tripartite nonlocality in GA since Clauser-Horne-Shimony-Holt inequality and Svetlichny's inequality are not violated.