量子搜尋與糾纏態分享於量子資訊科學之應用

量子資訊科學是一利用量子物理特性作為資訊計算處理的新興研究學門。由於其跨領域的特性，各種傳統計算機領域在量子系統中有了新的研究探討空間和可能解決方案。在本論文中，我們研究量子搜尋的電路設計和量子糾纏態分享方式，並提出其在計算與安全通訊的可能應用。首先，我們探討量子邏輯電路設計的細節以實現量子演算法及其兩種實際生活應用：電話簿搜尋和對稱式密碼系統破解。我們發現，量子搜尋演算法的效率雖然理論上可超越傳統演算法，但若要作為實際生活應用，還有許多需要考量的地方，例如其規模大小的影響等。為此我們定義適用的實際應用類型為某種類的單向函數，並為之設計出一種量子電路通用模型。其次，我們提出一個量子糾纏態的安全分享傳輸協定，並且說明如何應用在點對點的資訊傳輸和多點廣播通訊。在這些應用中，只有合法的使用者可以正確的解密，其安全性是由量子的海森堡不確定性原理所保障。最後，由於糾纏態分享協定中的量子通道在某些現實情境中可能無法取得，我們引用另一種糾纏態分享方式：糾纏態交換。我們提出兩種系統化的設計方式，在不需要傳送量子的情況下，可以直接且安全的分享應用所需的客制量子糾纏態。第一種方式是基於哈達馬矩陣，可以應用在認證的秘密資訊傳輸。第二種方式利用傅立葉轉換，可以更進一步的用在多量子多階層態的量子系統中及應用於秘密分享。不同於傳統密碼系統，這些量子通訊協定的安全性皆是根基於量子不可破的量子自然物理特性。

關鍵字

量子資訊科學；量子搜尋；量子糾纏態

並列摘要

Quantum information science is a fascinating field of research that studies how to process information based on quantum mechanisms. Due to its interdisciplinary nature, potential applications can be found almost in every field of computer science and electrical engineering. This thesis aims to discuss how to apply quantum mechanics to different applications in quantum computing and quantum cryptography, including quantum search and entanglement distribution. First, based on Grover's quantum search algorithm, we give the quantum circuits for two applications, searching a phone book and breaking a symmetric cryptosystem. Although these two applications have quite different types of search criteria, they are both one-way functions and the proposed circuits can actually be generalized to any such problems. In this perspective, we propose a template of quantum circuits that is capable of searching the solution of a certain class of one-way functions. Second, we propose a protocol to securely distribute entanglement states and show how to use it to perform encrypted communication and broadcasting. In these applications, only legitimate users can decrypt the messages, whereas eavesdroppers, if existing, will receive garbage. In comparison with classical cryptographic protocols, the security of our protocols is guaranteed by the Heisenberg uncertainty principle of quantum mechanics, instead of unproven mathematical hard problems. Third, since the quantum channel needed in our entanglement sharing protocol might not be available under some circumstances, we study a quantum phenomenon called entanglement swapping, which can be used to distribute entanglement states without transmitting qubits. We propose two systematic methodologies so application-specific entanglement states among participants can be generated and distributed directly. The first method is based on Hadamard matrix, and can be used to perform authenticated encryption. The other one is based on quantum Fourier transform, and can be applied on multi-particle multi-level quantum systems. We describe its circuit design and demonstrate that it can be applied to distributed applications such as secret sharing. In comparison with classical cryptographic protocols, its security is guaranteed by quantum physics, instead of any unproven mathematic conjecture.

並列關鍵字

quantum information science ； quantum search ； quantum entanglement

參考文獻

[1] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM Journal on Computing, vol. 26, pp. 1484–1509, 1997.

[2] L. Grover, “A fast quantum mechanical algorithm for database search,” in Proceedings of the 28th Annual ACM Symposium on the Theory of Computing, pp. 212–219, 1996.

[4] C.-M. Yu, I.-M. Tsai, Y.-H. Chou, and S.-Y. Kuo, “Improving the network flow problem using quantum search,” in Proceedings of the 2007 IEEE Conference on Nanotechnology, pp. 1126–1129, Aug. 2007.

[6] C. Zalka, “Grover’s quantum searching algorithm is optimal,” Physical Review A, vol. 60, pp. 2746–2751, 1999.

[7] G. Chen, S. A. Fulling, and J. Chen, “Generalization of grover’s algorithm to multiobject search in quantum computing, part I, continuous time and discrete time,” in Mathematics of Quantum Computation (R. K. Brylinski and G. Chen, eds.), pp. 135–

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量子搜尋與糾纏態分享於量子資訊科學之應用

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