All optical XOR, CNOT gates with initial insight for quantum computation using linear optics Omar Shehab Department of Computer Science and Electrical Engineering University of Maryland, Baltimore County Baltimore, Maryland 21250 shehab1@umbc.edu April 25, 2012
Basic ideas New design of an all-optical XOR gate. Splits the input beams and let them cancel or strengthen each other selectively or flip the encoded information based on their polarization properties. The information is encoded in terms of polarization of the beam. Based on a similar idea, the design of an all optical CNOT gate is proposed. Requires no additional power supply, extra input beam or ancilla photon to operate. Doesn’t require the expensive and complex single photon source and detector. Only narrowband laser sources are required to operate these gates. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 2 / 38
Related works on optical XOR gate I Semiconductor optic Kim, Jhon, Byun, Lee, Woo, and Kim [2002]. Soto, Erasme, and Guekos [2001]. Bintjas, Kalyvas, Theophilopoulos, Stathopoulos, Avramopoulos, Occhi, Schares, Guekos, Hansmann, and DallAra [2000]. Fjelde, Wolfson, Kloch, Dagens, Coquelin, Guillemot, Gaborit, Poingt, and Renaud [2000]. Terahertz optical asymmatric demultiplexer Wang, Wu, Shi, Yang, and Wang [2009]. Optical feedback Fok, Trappe, and Prucnal [2010]. Four wave mixing Yeh, Gu, Zhou, and Campbell [1993]. Fok and Prucnal [2010]. Polarization encoded optical shadow casting technique Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 3 / 38
Related works on optical XOR gate II Ahmed and Awwal [1992]. Highly nonlinear fiber Yu, Christen, Luo, Wang, Pan, Yan, and Willner [2005]. Zhou, Guo, Wang, Zhuang, and Zhu [2011]. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 4 / 38
Related works on LOQC I Discovery Knill, Laflamme, and Milburn [2000]. Optical CNOT gate O’Brien, Pryde, White, Ralph, and Branning [2003]. Nemoto and Munro [2004]. Mukherjee and Ghosh [2010]. Qureshi, Sen, Andrews, and Sen [2009]. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 5 / 38
The XOR gate | x � , | y � − → | x ⊕ y � Input 1 Input 2 Output 0 0 0 0 1 1 1 0 1 1 1 0 Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 6 / 38
The CNOT gate | x � , | y � − → | x � , | x ⊕ y � Control Target Control Output 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 0 Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 7 / 38
Encoding Definition: Logic 0 = H . Logic 1 = V . Phase shift doesn’t loose the information. So, -H = Logic 0 . -V = Logic 1 . Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 8 / 38
The optical XOR logic Input 1 Input 2 Output H H H H V V V H V V V H Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 9 / 38
The optical CNOT logic Control Target Control Output H H H H H V H V V H V V V V V H Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 10 / 38
Schematic of the XOR gate Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 11 / 38
Schematic of the CNOT gate Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 12 / 38
The XOR gate Operational regions Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 13 / 38
Five operational regions Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 14 / 38
Building the truth table Table: Blank truth table Input 1 Input 2 Output H H ? H V ? V H ? V V ? Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 15 / 38
Input: H, H Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 16 / 38
Input: H, V Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 17 / 38
Input: V, H Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 18 / 38
Input: V, V Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 19 / 38
The truth table Table: Table complete Input 1 Input 2 Output H H H H V V V H V V V H Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 20 / 38
Linearity of XOR operation Input 1 Input 2 Output H 0 H V 0 V - H 0 H H 0 V -( V + H ) XOR( H , 0 )+XOR( 0 , H ) ⇒ H + H ⇒ H ⇒ XOR( H , H ). XOR( H , 0 )+XOR( 0 V ) ⇒ XOR( H , V ). XOR( V , 0 )+XOR( 0 H ) ⇒ XOR( V , H ). XOR( V , 0 )+XOR( 0 , V ) ⇒ XOR ( V , V ). Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 21 / 38
Truth table for optical CNOT logic Control Target Control Output H H H H H V H -V V H V V V V V -H Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 22 / 38
Ignoring the phase shift Control Target Control Output H H H H H V H V V H V V V V V H Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 23 / 38
Universal quantum gates with linear optics According to the Solovay-Kitaev theorem (Kitaev et al. [2002]), the Hadamard, rotation and CNOT gates comprise the set of universal quantum gates. It is well known that a beam splitter behaves like a Hadamard gate (Ramakrishnan and Talabatulla [2009]). Recently, Kieling demonstrated that phase rotation gate is possible to be implemented with beam splitter and wave plate using linear optics (Kieling [2008]). So, linear optical beam may be used to implement the complete set of universal quantum gates. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 24 / 38
Implementations The author recommends to investigate the application of photonic crystals in realizing the above mentioned gates. It has been shown that linear optical components like wave plates (Zhang et al. [2009]), beam splitters (Ramakrishnan and Talabatulla [2009], Lin et al. [2002]), beam combiners (T. and Gu [2002]) and phase shifters (Dai et al. [2010]) can be fabricated from photonic crystals. So, there is a possibility of building linear optical quantum logic gates from photonic crystals based on the ideas presented in this paper. Moreover, as the polarization property of coherent bulk photons has been used, the decoherence problem is not going to prohibit the system to be scalable and sustainable. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 25 / 38
Recent developments I If a Hadamard gate is connected to the CNOT gate it is expected to generate the Bell states. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 26 / 38
Recent developments II We have found following truth values so far. For simplicity, the normalization factors are omitted. Here, C. C. I. = C NOT C ontrol I nput and C. T. I. = C NOT T arget I nput. Input 1 Input 2 C. C. I. C. T. I. Output 1 Output 2 H H H + V H H + V (H) + (V) H V H + V V H + V (-V) + (-H) V H H - V H H - V (H) + (H - V) V V H - V V H - V (-V) + (-V) Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 27 / 38
Acknowledgments The author thanks his supervisor Professor Samuel J Lomonaco Jr. for encouraging with his insights. He is also grateful to Professor James D Franson, Dr. Vincenzo Tamma, Sumeetkumar Bagde, Tanvir Mahmood and Asif M Adnan for their suggestions. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 28 / 38
Bibliography I J.U. Ahmed and A.A.S. Awwal. General purpose computing using polarization-encoded optical shadow casting. In Aerospace and Electronics Conference, 1992. NAECON 1992., Proceedings of the IEEE 1992 National , pages 1146–1151, 1992. C. Bintjas, M. Kalyvas, G. Theophilopoulos, T. Stathopoulos, H. Avramopoulos, L. Occhi, L. Schares, G. Guekos, S. Hansmann, and R. DallAra. 20 gb/s all-optical xor with uni gate. Photonics Technology Letters, IEEE , 12: 834–836, 2000. Qiao-Feng Dai, Sheng Lan, Li-Jun Wu, and He-Zhou Wang. Phase properties of reflected light in photonic band gap. Journal of Applied Physics , 107, 2010. T. Fjelde, D. Wolfson, A. Kloch, B. Dagens, A. Coquelin, I. Guillemot, F. Gaborit, F. Poingt, and M. Renaud. Demonstration of pogbitls all-optical logic xor in integrated soa-based interferometric wavelength converter. Electronics Letters , 36:1863–1864, 2000. Mable P. Fok and Paul R. Prucnal. Polarization effect on optical xor performance based on four-wave mixing. Photonics Technology Letters, IEEE , 22:1096–1098, 2010. Mable P. Fok, Wade Trappe, and Paul R. Prucnal. All-optical xor gate with feedback using highly ge-doped nonlinear fiber. In Optical Fiber Communication (OFC), collocated National Fiber Optic Engineers Conference, 2010 Conference on (OFC/NFOEC) , pages 1–3, 2010. Konrad Kieling. Linear optics quantum computing construction of small networks and asymptotic scaling . PhD thesis, Imperial College, London, 2008. Jae Hun Kim, Young Min Jhon, Young Tae Byun, Seok Lee, Deok Ha Woo, and Sun Ho Kim. All-optical xor gate using semiconductor optical amplifiers without additional input beam. Photonics Technology Letters, IEEE , 14: 1436–1438, 2002. A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. Classical and Quantum Computation . Graduate Studies in Mathematics. American Mathematical Society, Providence, RI,USA, July 2002. E. Knill, R. Laflamme, and G. Milburn. A scheme for efficient quantum computation with linear optics. Nature , 409:46–52, 2000. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 29 / 38
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