The design for an all optical XOR gate is proposed. The basic idea is to split 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. Polarization of a light beam is well understood hence the design should be feasible to implement. The truth table of the optical circuit is worked out and compared with the expected truth table. Then it is demonstrated that the design complies with the linear behavior of the XOR function. In the next section, based on a similar idea, the design of an all optical CNOT gate is proposed. The truth table for the gate is verified. Then, it is discussed how this approach can be used for Linear Optics Quantum Computation (LOQC). It is shown that with a Hadamard gate and a rotation gate, the CNOT gate makes up a universal set of quantum gates based on linear optics. This novel approach requires no additional power supply, extra input beam or ancilla photon to operate. It also doesn\'t require the expensive and complex single photon source and detector. Only narrowband laser sources are required to operate these gates.

1. 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

2. 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

3. 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

4. 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

5. 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

6. The XOR gate
|x , |y −→ |x ⊕ y
Input 1 Input 2 Output
0 0 0
0 1 1
1 0 1
1 1 0
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7. 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
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8. 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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9. 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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10. 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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11. Schematic of the XOR gate
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12. Schematic of the CNOT gate
Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 12 / 38

13. The XOR gate
Operational regions
Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 13 / 38

14. Five operational regions
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15. 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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16. Input: H, H
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17. Input: H, V
Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 17 / 38

18. Input: V, H
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19. Input: V, V
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20. 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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21. 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

22. 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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23. 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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24. 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

25. 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

26. 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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27. 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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28. 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

29. Bibliography I
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Photonics Technology Letters, IEEE, 22:1096–1098, 2010.
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30. Bibliography II
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31. Bibliography III
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32. Questions?
Omar Shehab (UMBC) LOQC with All Optical CNOT April 25, 2012 32 / 38

33. THANK YOU!
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34. Extra slides
Extra truth values for the XOR gate
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35. Input: H, 0
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36. Input: V, 0
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37. Input: 0, H
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38. Input: 0, V
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