All optical XOR, CNOT gates with initial insight for quantum computation using linear optics

All optical XOR, CNOT gates with initial insightfor quantum computation using linear optics

Omar Shehab

Department of Computer Science and Electrical EngineeringUniversity of Maryland, Baltimore County

Baltimore, Maryland 21250

[email protected]

April 25, 2012

Basic ideas

New design of an all-optical XOR gate.

Splits the input beams and let them cancel or strengthen eachother selectively or flip the encoded information based on theirpolarization properties.

The information is encoded in terms of polarization of thebeam.

Based on a similar idea, the design of an all optical CNOTgate is proposed.

Requires no additional power supply, extra input beam orancilla photon to operate.

Doesn’t require the expensive and complex single photonsource and detector.

Only narrowband laser sources are required to operate thesegates.

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, andDallAra [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


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].


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].


The XOR gate

|x〉 , |y〉 −→ |x ⊕ y〉

Input 1 Input 2 Output

0 0 00 1 11 0 11 1 0


The CNOT gate

|x〉 , |y〉 −→ |x〉 , |x ⊕ y〉

Control Target Control Output

0 0 0 00 1 0 11 0 1 11 1 1 0


Encoding

Definition:

Logic 0 = H.

Logic 1 = V.

Phase shift doesn’t loose the information. So,

-H = Logic 0.

-V = Logic 1.


The optical XOR logic


H H HH V VV H VV V H


The optical CNOT logic


H H H HH V H VV H V VV V V H


Schematic of the XOR gate


Schematic of the CNOT gate


The XOR gate

Operational regions


Five operational regions


Building the truth table

Table: Blank truth table


H H ?H V ?V H ?V V ?


Input: H, H


Input: H, V


Input: V, H


Input: V, V


The truth table

Table: Table complete


H H HH V VV H VV V H


Linearity of XOR operation


H 0 HV 0 V -H0 H H0 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 ).


Truth table for optical CNOT logic


H H H HH V H -VV H V VV V V -H


Ignoring the phase shift


H H H HH V H VV H V VV V V H


Universal quantum gates with linear optics

According to the Solovay-Kitaev theorem (Kitaev et al.[2002]), the Hadamard, rotation and CNOT gates comprisethe set of universal quantum gates.

It is well known that a beam splitter behaves like a Hadamardgate (Ramakrishnan and Talabatulla [2009]).

Recently, Kieling demonstrated that phase rotation gate ispossible to be implemented with beam splitter and wave plateusing linear optics (Kieling [2008]).

So, linear optical beam may be used to implement the completeset of universal quantum gates.


Implementations

The author recommends to investigate the application ofphotonic crystals in realizing the above mentioned gates.

It has been shown that linear optical components like waveplates (Zhang et al. [2009]), beam splitters (Ramakrishnanand Talabatulla [2009], Lin et al. [2002]), beam combiners (T.and Gu [2002]) and phase shifters (Dai et al. [2010]) can befabricated from photonic crystals.

So, there is a possibility of building linear optical quantumlogic gates from photonic crystals based on the ideaspresented in this paper.

Moreover, as the polarization property of coherent bulkphotons has been used, the decoherence problem is not goingto prohibit the system to be scalable and sustainable.


Recent developments I

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


Recent developments II

We have found following truth values so far. For simplicity, thenormalization factors are omitted. Here, C. C. I. = CNOT ControlInput 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)


Acknowledgments

The author thanks his supervisor Professor Samuel J Lomonaco Jr.for encouraging with his insights. He is also grateful to ProfessorJames D Franson, Dr. Vincenzo Tamma, Sumeetkumar Bagde,Tanvir Mahmood and Asif M Adnan for their suggestions.


Bibliography I

J.U. Ahmed and A.A.S. Awwal. General purpose computing using polarization-encoded optical shadow casting. InAerospace and Electronics Conference, 1992. NAECON 1992., Proceedings of the IEEE 1992 National, pages1146–1151, 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 bandgap. 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-dopednonlinear 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. PhDthesis, 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 gateusing 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 inMathematics. 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.


Bibliography II

S. Y. Lin, E. Chow, J. Bur, S. G. Johnson, and J. D. Joannopoulos. Low-loss, wide-angle y splitter at approximately1.6- 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 differencefrequency generation and implementation of fast binary adders using frequency encoding and nonlineardielectric 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 quantumcontrolled-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 PhotonicFiber and Crystal Devices: Advances in Materials and Innovations in Device Applications III, San Diego, CA,2009.

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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 fornrz based on toad. In Progress In Electromagnetics Research Symposium, PIERS Proceedings, pages1286–1290, 2009.

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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.


Bibliography III

Wenfu Zhang, Jihong Liu, Wei-Ping Huang, and Wei Zhao. Self-collimating photonic-crystal wave plates. OpticsLetters, 34:2676–2678, 2009.

Shufen Zhou, Shuqin Guo, Jianfen Wang, Pan Zhuang, and Limiao Zhu. All-optical logic xor gate exploiting xpmand polarization rotation in single highly nonlinear fiber. In Intelligent Computation Technology andAutomation (ICICTA), 2011 International Conference on, pages 401–403, 2011.


Questions?


THANK YOU!


Extra slides

Extra truth values for the XOR gate


Input: H, 0


Input: V, 0


Input: 0, H


Input: 0, V


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All optical XOR, CNOT gates with initial insight for quantum computation using linear optics