量子安全直接通訊與量子電路驗證

Quantum Secure Direct Communication and Quantum Circuit Verification

王秀安(Shiou-An Wang)

指導教授 ： 郭斯彥

國立臺灣大學/電機資訊學院/電機工程學研究所/博士(2008年)

https://doi.org/10.6342/NTU.2008.00050

摘要

近年來，一些研究已將量子力學應用在通訊領域如量子密鑰傳遞、量子安全直接通訊等。在本論文中，我們提出一個利用EPR量子對的量子安全直接通訊協定。以往的量子安全直接通訊協定為了要傳遞一個量子位元，通常需要消耗一對EPR量子。假如使用者甲想要傳遞一個n位元的訊息給使用者乙，他需要使用至少n/2對EPR量子。且這是當他使用高密度編碼的方式才做得到。在我們所提出的協定中，如果使用者甲、乙和一個可信賴的伺服器事先分享2c+1對EPR量子，在這裡c是一個常數。使用者甲便可以傳遞任意多位元的訊息給使用者乙。而且在傳遞秘密訊息之前，不需要先傳遞EPR量子對。我們的協定可以防止任何竊聽者得知或破壞這個秘密訊息。這個方法可以降低量子通道的使用量以及風險。除此之外，可以透過伺服器驗證使用者的身分以避免任何的冒充者。
量子電路的設計是建造一部量子電腦的基礎。驗證設計好的量子電路是否能執行正確的功能是很重要的。我們提出了一個可以驗證不同量子電路的演算法，而這些量子電路可以是由不同的量子閘所組成的。這個演算法使用一個以二元決策圖（BDD）為基礎的資料結構，這個新的資料結構稱為XQDD（X-分解量子決策圖）。在這個方法中，量子運算可以用圖形的方式表示。藉由比較二個圖形，我們就可以驗證二個量子電路是否具有相同的功能。如果二個量子電路轉換成相同的XQDD，則表示這二個量子電路具有相同的功能。我們也提出了一個可以驗證可逆電路的演算法。即使這些可逆電路含有不同數量的垃圾量子位元，這個演算法也可以用來驗證它們。在大多數的情況下，XQDD所需要的節點數比其他的方法（如QuIDD）來的少。所以這個方法在時間和空間上的使用上都是比較有效率的，甚至可以在多項式時間驗證許多的量子電路。
XQDD可以表示量子運算也可以用來執行矩陣運算，我們用它來驗證量子電路和可逆電路，不論在時間上或空間上都是相當有效率的。所以接下來，我們將XQDD擴充到多值量子邏輯上。我們用XQDD來表示一個多值量子運算及執行矩陣運算。即使多值量子電路或可逆電路是用不同的設計方法設計出來的，XQDD也可以用來檢查這些電路是否執行正確的功能。根據我們的實驗結果也顯示XQDD所需的空間少於其他的表示方法。

關鍵字

量子計算 ； 量子通訊 ； 量子電路

並列摘要

Quantum mechanics has been applied in the field of communication in the last couple of decades. Quantum Key Distribution (QKD) and Quantum Secure Direction Communication (QSDC) protocols have been presented. In this dissertation, we propose a quantum secure direct communication protocol based on Einstein-Podolsky-Rosen (EPR) pairs. Previous QSDC protocols usually consume one EPR pair to transmit a single qubit. If Alice wants to transmit an n-bit message, she needs at least n/2 EPR pairs when dense coding scheme is used. In our protocol, if both Alice and Bob pre-share 2c+1 EPR pairs with the trusted server where c is a constant, Alice can transmit arbitrary number of qubits to Bob. It is not necessary to transmit EPR pairs before transmitting the secret message. No eavesdropper can steal or change the secret message without being detected. It reduces both the utilization of the quantum channel and the risk. In addition, the server can authenticate the users to avoid any pretender.
Synthesis of quantum circuits is essential for building quantum computers. It is important to verify that the designed circuits perform the correct functions. We propose an algorithm which can be used to verify the quantum circuits consisting of different quantum gates. The proposed data structure is based on BDD (Binary Decision Diagram) and is called X-decomposition Quantum Decision Diagram (XQDD). In this method, quantum operations are modeled using a graphic method and the verification process is based on comparing these graphic diagrams. If the XQDD representations of two quantum circuits are identical, they perform the same function. We also develop an algorithm to verify reversible circuits even if they have a different number of garbage qubits. In most cases, the number of nodes used in XQDD is less than that in other representations such as QuIDD. In general, the proposed method is more efficient in terms of space and time and can be used to verify many quantum circuits in polynomial time.
X-decomposition Quantum Decision Diagram (XQDD) can represent a quantum operation and perform matrix operations. It can be used to verify quantum and reversible circuits even if the reversible circuits have different number of garbage qubits. It is efficient in terms of space and time. In this dissertation, we extend the original XQDD to multiple-valued quantum logic. The extended XQDD can represent a multiple-valued quantum operation and perform matrix operations. It can be used to check the equivalence of two multiple-valued quantum or reversible circuits which are synthesized by different approaches. We show that the number of nodes used in multiple-valued XQDD is less than other representations.

並列關鍵字

quantum computing ； quantum communication ； quantum circuit

參考文獻

[1] P. W. Shor, "Algorithms for Quantum Computation: Discrete Logarithms and Factoring", Proc. 35th Annu. IEEE Symp. Foundations of Computer Science, pp. 124-134, 1994.

[2] L. Grover, "A Fast Quantum Mechanical Algorithm for Database Search", Proc. 28th Annu. ACM Symp. Theory of Computing, pp. 212-219, 1996.

[3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters, "Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels," Phys. Rev. Lett. 70, pp. 1895-1899, 1993.

[4] C. Bennett and S.J. Wiesner. "Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states," Phys. Rev. Lett., 69, 2881, 1992.

[5] C. H. Bennett and G. Brassard, "Quantum Cryptography: Public Key Distribution and Coin Tossing," Proc. Int'l Conf. Computers, Systems & Signal Processing, pp. 175-179, 1984.