All optical XOR, CNOT gates with initial insight for quantum - - PowerPoint PPT Presentation

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

• ther 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.

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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

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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].

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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].

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The XOR gate

|x , |y − → |x ⊕ y Input 1 Input 2 Output 1 1 1 1 1 1

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The CNOT gate

|x , |y − → |x , |x ⊕ y Control Target Control Output 1 1 1 1 1 1 1 1

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Encoding

Definition: Logic 0 = H. Logic 1 = V. Phase shift doesn’t loose the information. So,

• H = Logic 0.
• V = Logic 1.

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The optical XOR logic

Input 1 Input 2 Output H H H H V V V H V V V H

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The optical CNOT logic

Control Target Control Output H H H H H V H V V H V V V V V H

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Schematic of the XOR gate

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Schematic of the CNOT gate

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The XOR gate

Operational regions

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Five operational regions

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Building the truth table

Table: Blank truth table

Input 1 Input 2 Output H H ? H V ? V H ? V V ?

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Input: H, H

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Input: H, V

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Input: V, H

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Input: V, V

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The truth table

Table: Table complete

Input 1 Input 2 Output H H H H V V V H V V V H

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Linearity of XOR operation

Input 1 Input 2 Output H H V V -H H H 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 ).

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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

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Ignoring the phase shift

Control Target Control Output H H H H H V H V V H V V V V V H

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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.

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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.

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Recent developments I

If a Hadamard gate is connected to the CNOT gate it is expected to generate the Bell states.

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Recent developments II

We have found following truth values so far. For simplicity, the normalization factors are omitted. Here, C. C. I. = CNOT Control Input and C. T. I. = CNOT Target Input. 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)

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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.

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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,

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Bibliography II

• S. Y. Lin, E. Chow, J. Bur, S. G. Johnson, and J. D. Joannopoulos. Low-loss, wide-angle y splitter at approximately

1.6- mum wavelengths built with a two-dimensional photonic crystal. Optics Letters, 27:1400–1402, 2002. Kousik Mukherjee and Parimal Ghosh. A novel frequency encoded all optical cnot gate exploiting difference frequency generation and implementation of fast binary adders using frequency encoding and nonlinear dielectric films. Optik - International Journal for Light and Electron Optics, 121:2195–2197, 2010. Kae Nemoto and W. J. Munro. A near deterministic linear optical cnot gate. Physical Review Letters, 93, 2004.

• J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning. Demonstration of an all-optical quantum

controlled-not gate. Nature, 426:264–267, 2003. M.S. Qureshi, P. Sen, J.T. Andrews, and P.K. Sen. All optical quantum cnot gate in semiconductor quantum dots. IEEE Journal of Quantum Electronics, 45:59–65, 2009.

• Rohit. K. Ramakrishnan and Srinivas Talabatulla. Photonic crystal based quantum hadamard gate. In Photonic

Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications III, San Diego, CA, 2009.

• H. Soto, D. Erasme, and G. Guekos. 5-gb/s xor optical gate based on cross-polarization modulation in

semiconductor optical amplifiers. Photonics Technology Letters, IEEE, 13:335–337, 2001. Zhang X. T. and P. F. Gu. Design and fabrication of ir/mmw dichroic beam combiner. Jiguang Yu Hongwai, 32: 292294, 2002. Yaping Wang, Chongqing Wu, Xiaojun Shi, Shuangshou Yang, and Yongjun Wang. An all optical xor logic gate for nrz based on toad. In Progress In Electromagnetics Research Symposium, PIERS Proceedings, pages 1286–1290, 2009. Pochi Yeh, Claire Gu, Shaomin Zhou, and Scott Campbell. Photorefractive nonlinear optics for optical computing. In Lasers and Electro-Optics Society Annual Meeting, 1993. LEOS ’93 Conference Proceedings. IEEE, pages 317–318, 1993. Changyuan Yu, Louis Christen, Ting Luo, Yan Wang, Zhongqi Pan, Lian-Shan Yan, and Alan E. Willner. All-optical xor gate using polarization rotation in single highly nonlinear fiber. Photonics Technology Letters, IEEE, 17:1232–1234, 2005. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 30 / 38

Bibliography III

Wenfu Zhang, Jihong Liu, Wei-Ping Huang, and Wei Zhao. Self-collimating photonic-crystal wave plates. Optics Letters, 34:2676–2678, 2009. Shufen Zhou, Shuqin Guo, Jianfen Wang, Pan Zhuang, and Limiao Zhu. All-optical logic xor gate exploiting xpm and polarization rotation in single highly nonlinear fiber. In Intelligent Computation Technology and Automation (ICICTA), 2011 International Conference on, pages 401–403, 2011. Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 31 / 38

Questions?

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THANK YOU!

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Extra slides

Extra truth values for the XOR gate

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Input: H, 0

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Input: V, 0

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Input: 0, H

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Input: 0, V

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