臺灣博碩士論文加值系統

隨著量子物理的快速發展，量子網路中能夠利用量子糾纏在兩個通信實體之間傳輸量子資訊的路由
受到了大量的關注。許多研究都在探討如何在量子網路中為單個或多個來源終端對尋找路徑以建立
遠距糾纏，其中糾纏所需的量子位元數量通常被假設不小於量子通道的數量。然而，實際上，每個
在量子網路中的節點會有不同數量的量子位元與量子通道，所以難以決定是否都存在有足夠的量子
位元可以透過量子通道來建立量子糾纏。此外，保真度，一種糾纏的品質，被用來表示量子資訊的資料
精準度。這促使我們在本論文中研究在考慮網路資源和滿足糾纏品質的前提下，使多個來源終端
對的遠距糾纏數量最大化的問題。此外，我們還進行了模擬實驗以展示提出的演算法之效能。

With the rapid development of quantum physics, the routing in
quantum networks that can use quantum entanglement to transmit
quantum information between two communicating entities receives a
great deal of attentions. Many research works study on how to find the
routing paths for single or multiple source-destination pairs in
quantum networks to establish remote entanglement, where the quantum
bits required for entanglement are often assumed to be not less than
the quantum channels. However, in reality, each node in a quantum
network has different numbers of quantum bits and quantum channels,
and therefore, it is hard to ensure that there always exist enough
quantum bits to establish entanglement via quantum channels. In
addition, the fidelity, a quality of entanglement, is used to represent
the data accuracy of the quantum information. This motivates us to study
the problem of maximizing the number of remote entanglements for multiple
source-destination pairs while considering the network resources and
satisfying the quality of entanglement in this thesis. For this problem,
the linear programming and heuristic algorithms are proposed. In addition,
the simulation is conducted to show the performances of the proposed
algorithms.

摘要.......................................................i
Abstract...................................................ii
致謝.......................................................iii
目錄.......................................................iv
表目錄.....................................................vi
圖目錄.....................................................vii
第一章 緒論.................................................1
第二章 網路模型與問題定義.....................................3
2.1 網路模型................................................3
2.2 問題定義................................................9
第三章 研究方法..............................................12
3.1 線性規劃................................................12
3.1.1 Qubit Assignment.....................................12
3.1.2 Entanglement Assignment..............................19
3.1.3 線性規劃鬆弛法.........................................20
3.1.4 隨機進為演算法.........................................21
3.2 啟發式演算法.............................................22
3.2.1 Qubit Assignment......................................22
3.2.2 Entanglement Assignment...............................23
第四章 模擬實驗與結果分析.....................................26
4.1 節點數量比較.............................................27
4.2 請求數量比較.............................................27
4.3 網路生成參數比較.........................................27
4.4 網路規模比較.............................................28
4.5 糾纏的初始保真度比較......................................28
4.6 糾纏交換的機率比較........................................28
4.7 量子位元數量比較..........................................31
4.8 量子通道數量比較..........................................31
第五章 結論..................................................33
5.1 研究結論.................................................33
5.2 未來展望.................................................33
參考文獻.....................................................34

[1] A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gherardini,
and G. Bianchi, “Quantum internet: Networking challenges in distributed
quantum computing”, IEEE Network, vol. 34, no. 1, pp. 137–143, 2020.
doi: 10.1109/MNET.001.1900092.
[2] M. Caleffi, A. S. Cacciapuoti, and G. Bianchi, “Quantum internet”, Pro-
ceedings of the 5th ACM International Conference on Nanoscale Computing
and Communication, Sep. 2018. doi: 10.1145/3233188.3233224. [Online].
Available: http://dx.doi.org/10.1145/3233188.3233224.
[3] H. J. Kimble, “The quantum internet”, Nature, vol. 453, no. 7198, pp. 1023–
1030, Jun. 2008, issn: 1476-4687. doi: 10.1038/nature07127. [Online].
Available: http://dx.doi.org/10.1038/nature07127.
[4] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for
the road ahead”, Science, vol. 362, eaam9288, Oct. 2018. doi: 10.1126/
science.aam9288.
[5] J. L. Park, “The concept of transition in quantum mechanics”, Foundations
of Physics, vol. 1, no. 1, pp. 23–33, Mar. 1970. doi: 10.1007/BF00708652.
[6] C. H. Bennett and G. Brassard, “Quantum cryptography: Public key distri-
bution and coin tossing”, p. 175, 1984.
[7] A. K. Ekert, “Quantum cryptography based on bell’s theorem”, Phys. Rev.
Lett., vol. 67, pp. 661–663, 6 Aug. 1991. doi: 10.1103/PhysRevLett.
67.661. [Online]. Available: https://link.aps.org/doi/10.1103/
PhysRevLett.67.661.
[8] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck,
D. Englund, T. Gehring, C. Lupo, C. Ottaviani, and et al., “Advances in
quantum cryptography”, Advances in Optics and Photonics, vol. 12, no. 4,
p. 1012, Dec. 2020, issn: 1943-8206. doi: 10.1364/aop.361502. [Online].
Available: http://dx.doi.org/10.1364/AOP.361502.
[9] Q. Jia, K. Xue, Z. Li, M. Zheng, D. Wei, and N. Yu, “An improved qkd pro-
tocol without public announcement basis using periodically derived basis”,
Quantum Information Processing, vol. 20, Feb. 2021. doi: 10.1007/s11128-
021-03000-8.
[10] M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S.
Miki, T. Yamashita, Z. Wang, A. Tanaka, and et al., “Field test of quan-
tum key distribution in the tokyo qkd network”, Optics Express, vol. 19,
no. 11, p. 10 387, May 2011, issn: 1094-4087. doi: 10.1364/oe.19.010387.
[Online]. Available: http://dx.doi.org/10.1364/OE.19.010387.
[11] J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y.
Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li,
Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X.
Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R.
Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement
distribution over 1200 kilometers”, 2017. arXiv: 1707.01339 [quant-ph].
[12] P. Kómár, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and
M. D. Lukin, “A quantum network of clocks”, Nature Physics, vol. 10, no. 8,
pp. 582–587, Jun. 2014, issn: 1745-2481. doi: 10.1038/nphys3000. [Online].
Available: http://dx.doi.org/10.1038/nphys3000.
[13] S. Olmschenk, D. N. Matsukevich, P. Maunz, D. Hayes, L.-M. Duan, and C.
Monroe, “Quantum teleportation between distant matter qubits”, Science,
vol. 323, no. 5913, pp. 486–489, Jan. 2009, issn: 1095-9203. doi: 10.1126/
science.1167209. [Online]. Available: http://dx.doi.org/10.1126/
science.1167209.
[14] J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental
entanglement swapping: Entangling photons that never interacted”, Phys.
Rev. Lett., vol. 80, pp. 3891–3894, 18 May 1998. doi: 10.1103/PhysRevLett.
80.3891. [Online]. Available: https://link.aps.org/doi/10.1103/
PhysRevLett.80.3891.
[15] H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo,
T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and et al.,
“Heralded entanglement between solid-state qubits separated by three me-
tres”, Nature, vol. 497, no. 7447, pp. 86–90, Apr. 2013, issn: 1476-4687. doi:
10.1038/nature12016. [Online]. Available: http://dx.doi.org/10.1038/
nature12016.
[16] M. Caleffi, “Optimal routing for quantum networks”, IEEE Access, vol. 5,
pp. 22 299–22 312, 2017. doi: 10.1109/ACCESS.2017.2763325.
[17] L. Gyongyosi and S. Imre, “Decentralized base-graph routing for the quan-
tum internet”, Physical Review A, vol. 98, no. 2, Aug. 2018, issn: 2469-9934.
doi: 10.1103/physreva.98.022310. [Online]. Available: http://dx.doi.
org/10.1103/PhysRevA.98.022310.
[18] M. Pant, H. Krovi, D. Towsley, L. Tassiulas, L. Jiang, P. Basu, D. Englund,
and S. Guha, “Routing entanglement in the quantum internet”, 2017. arXiv:
1708.07142 [quant-ph].
[19] C. Li, T. Li, Y.-X. Liu, and P. Cappellaro, “Effective routing design for
remote entanglement generation on quantum networks”, npj Quantum In-
formation, vol. 7, Jan. 2021. doi: 10.1038/s41534-020-00344-4.
[20] J. Li, M. Wang, Q. Jia, K. Xue, N. Yu, Q. Sun, and J. Lu, “Fidelity-guarantee
entanglement routing in quantum networks”, 2021. arXiv: 2111.07764
[quant-ph].
[21] S. Shi and C. Qian, “Concurrent entanglement routing for quantum net-
works: Model and designs”, SIGCOMM ’20, pp. 62–75, 2020. doi: 10.
1145/3387514.3405853. [Online]. Available: https://doi.org/10.1145/
3387514.3405853.
[22] Y. Zhao and C. Qiao, “Redundant entanglement provisioning and selection
for throughput maximization in quantum networks”, pp. 1–10, 2021. doi:
10.1109/INFOCOM42981.2021.9488850.
[23] P. C. Humphreys, N. Kalb, J. P. J. Morits, R. N. Schouten, R. F. L. Ver-
meulen, D. J. Twitchen, M. Markham, and R. Hanson, “Deterministic de-
livery of remote entanglement on a quantum network”, Nature, vol. 558,
no. 7709, pp. 268–273, Jun. 2018, issn: 1476-4687. doi: 10.1038/s41586-
018-0200-5. [Online]. Available: http://dx.doi.org/10.1038/s41586-
018-0200-5.
[24] N. K. Bernardes, “Long-distance quantum communication: Decoherence-
avoiding mechanisms”, Apr. 2013.
[25] X.-M. Hu, C.-X. Huang, Y.-B. Sheng, L. Zhou, B.-H. Liu, Y. Guo, C. Zhang,
W.-B. Xing, Y.-F. Huang, C.-F. Li, and et al., “Long-distance entanglement
purification for quantum communication”, Physical Review Letters, vol. 126,
no. 1, Jan. 2021, issn: 1079-7114. doi: 10.1103/physrevlett.126.010503.