International Journal of Advancements in Research & Technology, Volume 1, Issue1, June-2012
ISSN 2278-7763
1
Design and realization of a quantum Controlled NOT
gate using optical implementation
K. K. Biswas, Shihan Sajeed
1(
Department of Applied Physics, Electronics and Communication Engineering (APECE), University of Dhaka, Bangladesh.(Phone: =
+8801724119834; e-mail: shorbiswas.377@gmail.com).
2
(Department of Applied Physics, Electronics and Communication Engineering (APECE), University of Dhaka, Bangladesh. (Phone: =
+8801817094868; e-mail: Shihan.sajeed@gmail.com)
ABSTRACT
In this work an optical implementation technique of a Controlled-NOT (CNOT) gate has been designed, realized and
simulated. The polarization state of a photon is used as qubit. The interaction required between two qubits for realizing the
CNOT operation was achieved by converting the qubits from polarization encoding to spatial encoding with the help of a
0
Polarizing Beam Splitter (PBS) and half wave plate (HWP) oriented at 45 . After the nonlinear interference was achieved the
spatially encoded qubits were converted back into polarization encoding and thus the CNOT operation was realized. The
whole design methodology was simulated using the simulation software OptiSystem and the results were verified using the
built-in instruments polarization analyzer, polarization meter, optical spectrum analyzer, power meters etc.
Keywords - Qubit, Quantum gate, Polarizing Beam Splitter, Half Wave Plate, Beam Splitter, Dual rail technique.
1 INTRODUCTION
2 Theory
Q
2.1 Representation of Quantum CNOT gate
Let us consider the computational basis states defined as
uantum computation (QC) was first proposed by Benioff
[1] and Feynman [2] and further developed by a number
of scientists, e.g. Deutsch [3,4], Grover [5,6], Lloyd [7,8] and
others [9-11]. QC is the study of information processing tasks
that can be accomplished using quantum mechanical systems
[12]. It is based on sequences of unitary operations on the input
qubits using quantum gates [13, 14].The quantum gates have
been successfully demonstrated in the past using photonic
interference process [15,16].In photon-based optical QC
processes has been instructed to photonic interference
phenomena [17,18].This photonic interference or interaction
phenomena has been executed using different types of optical
device. There has been also works interesting the use of Beam
Splitter to perform the photonic interference or interaction [1922]. In optical approach the principle of photonic interference
or interaction operation is obtained by changing reflectivity of
Beam Splitter.
In this work, a set of beam splitter (BS) structures are
proposed and used to demonstrate the operation of a quantum
CNOT gate. The difficulty in optical quantum computing has
been in achieving the two photon interactions required for a
two qubit gate. This two photon interaction provide non-linear
operation which occurred in non-linear phase shift. To require
this non-linearity through two photon interaction is
accomplished using extra ―ancilla‖ photons. Vacuum state
input modes provide extra ―ancilla‖ photons. In quantum field
theory, the vacuum state is the quantum state with the lowest
possible energy. Generally, it contains no physical particles.
The design of CNOT gate has been simulated by OptiSystem
software where the directional coupler has been used as BS and
obtained that the BS based structure can be worked as a
quantum CNOT gate.
Copyright © 2012 SciResPub.
1 
0  
0 
0 
1  
1 
(0.1)
(0.2)
The CNOT gate acts on 2 qubits, one control qubit and another
target qubit. It performs the NOT operation on the target qubit
only when the control qubit is
1 , and leaves the target qubit
unchanged otherwise. Operation of CNOT gate is as follows:
00  00
01  01
10  11
11  10
The action of the CNOT gate expression will be written as [23,
24]
CNOT  00 00  01 01  10 11  11 10
International Journal of Advancements in Research & Technology, Volume 1, Issue1, June-2012
ISSN 2278-7763
1
0

0

0
1
0

0

0
0 0 0 0
0 0 0 0

0 0 0  0
 
0 0 0  0
0 0 0 0
1 0 0 0

0 0 0  0
 
0 0 0  0
0 0 0  0
1 0 0  0

0 0 0  0
 
0 0 0  0
0 0 0 0
0 0 0 0

0 0 1  0
 
0 0 0  0
2
0 0 0
0 0 0
0 0 0

0 1 0
0 0 0
0 0 0 
0 0 1

0 1 0
(0.3)
1 0  1 0   0 0   0 1 





 0 0   0 1   0 1  1 0 
Fig. 1: A schematic of the Polarization of single photon mode. (A)
Horizontally Polarization, (B) Vertically Polarization, (c) Left Circular
Polarization, (d) Right Circular Polarization.
1 
0
   1 0  I    0 1  X
0
1 
 0 0 I  1 1 X
From equation (1.3) we can write the CNOT operator in matrix
form as:
U
CNOT
1
0

0

0
When input is
U CNOT
1
0
10  
0

0
0 0 0
1 0 0 
0 0 1

0 1 0
(0.4)
10 . So
0 0 0  0  0  0  0 
1 0 0 0 0  0  1 0 0
 
   1  1  11
0 0 1 1 0 1 0  1  1
     
0 1 0   0  1   1  1 
Other operation is also performed this same way. So
U
CNOT
matrix acts as a CNOT operator.
2.2 Polarization state of Single photon as qubit
Let, Horizontal Polarization state of single photon be
defined as qubit
0 and Vertical Polarization state of single
photon is defined as qubit
1 . Superposition of these two
states can also form new polarization states such as Left
Circular Polarization (LCP), Right Circular Polarization
(RCP), etc.
Some polarization states are shown in fig. 1.
Copyright © 2012 SciResPub.
2.3 Splitter
A beam splitter is an optical device which can split an
incident light beam into two beams, which may or may not
have the same optical power. Half-silvered mirror, Nicol
prism, Wollaston prisms etc are also used to make beam
splitter. Waveguide beam splitters are used in photonic
integrated circuits. Any beam splitter may in principle also be
used for combining beams to a single beam. The output power
is then not necessarily the sum of input powers, and may
strongly depend on details like tiny path length differences,
since interference occurs. Such effects can of course not occur
e.g. when the different beams have different wavelengths or
polarization. The beam reflected from above does not require a
phase change while a beam reflected from below acquires a
phase change of  and 50% beam splitter performs Hadamard
operation [25].
2.4 Half Wave Plate
A retardation plate that introduces a relative phase difference
of  radians or 180 between the o- and e-waves is known as
a half-wave plate or half-wave retarded. It will invert the
handedness of circular or elliptical light, changing right to left
and vice versa. If relative phase difference between e- and owaves has changed then the state of polarization of the wave
must be changed [26]. When angle between optical axis and
0
incident light axis or light propagation axis is
450 then HWP is
acted as NOT operator [27].
2.5 Polarizing Beam Splitter
Polarizing Beam Splitter (PBS) divides the incident beam
0
into two orthogonally polarized beams at 90 to each other.
Output of the PBS, one is parallel to the incident beam and
other one is perpendicular to the incident beam. This two
divided beams obtain at two different faces of PBS. Actually
PBS is made by adding one transmittance and one reflectance
International Journal of Advancements in Research & Technology, Volume 1, Issue1, June-2012
ISSN 2278-7763
3
type polarizer. Sometime is called addition of a horizontal and
a vertical polarizer we got PBS means that PBS is a
combination of special type of a horizontal and a vertical
polarizer [28].
2.6 Dual rail technique
With dual rail the photon number is the same for all logical
states. The logical 0 is represented by a single photon
occupation of one mode with the other in the vacuum state
[29]. The logical 1 is the opposite of the logical 0 state with a
single photon in the other mode. This means that the logical
0 L equals the state 10 , while the logical one, 1
zero,
equals the state
Fig. 3: Conversion from polarization to spatial encoding.
L
The reverse process converts the spatial encoding back to
polarization encoding. To convert from polarization, to spatial
encoding and back, while preserving the quantum information,
it is required that the phase relationship between the two basis
components be preserved throughout: the path lengths must be
sub-wavelength stable (interferometric stability)[19].
01 .
Fig. 2: A simple principle sketch of a polarization beam splitter.
Dual rail logic is often implemented using the horizontal and
vertical polarization modes of a single spatial mode. In dual
rail logic encoding strategy the qubit is encoded in a pair of
complementary optical modes. If we want to represent n
qubits with this representation we will need N  2n ports and
n photons where in case of single rail logic N  n ports and
n photons are needed [29]. Means that the logical
state
3.2 DESIGN METHODOLOGY of CNOT operation
The two paths used to encode the target qubit are mixed at a
50% reflecting beam splitter (BS) in fig.4 that performs the
Hadamard operation [21]. If the phase shift is not applied, the
second beam splitter (Hadamard) undoes the first, returning the
target qubit exactly the same state it started in (example of
classical interference).
00 L equals the number state 1010 and the logical state
11 L equals the number state 0101 [29].
Fig. 4: A possible realization of an optical quantum CNOT gate.
3 Design methodology
3.1Conversion from polarization to spatial encoding
It is most practical to prepare single photon qubits where the
quantum information is encoded in the polarization state.
 H   V    0   1  —polarization encoding,
where
H and V are the horizontal and vertical
To convert from polarization to spatial encoding a
polarizing beam splitter (PBS) and half wave plate (HWP) is
used. In fig. 3, the HWP rotates the polarization of the lower
0
beam to 90 when its optical axis is apt 45 to the beam.
After this rotation all components of the spatial qubits have the
same polarization state and they can interfere both classically
and non-classically.
Copyright © 2012 SciResPub.
i.e.
0  1 and 1  0 . When control qubit is 0 , then
phase shift is not applied and when control qubit is
1 , then
phase (π) shift is applied. So this phase shifting operation is
non-linear phase shift. A CNOT gate must implement this
phase shift when the control photon is in the ―
1 ‖ path,
otherwise not [21].
polarization states [19].
0
If π phase shift is applied i.e. non-classical interference is
occurred and target qubit is flipped, the NOT operation occurs,
3.3 Implementation of CNOT gate
In fig. 5 is shown a conceptual realization of CNOT gate
where the beam splitter reflectivity and asymmetric phase
shifts is indicated. Actually this gate is accurately given the
Non-linear operation by using two photon interactions and
performs CNOT gate operation [19]. From fig. 5 B1, B2, B3,
B4 and B5 are beam splitters (BS) and
vC and vT are vacuum
inputs. B1, B2, B3, B4 and B5 are assumed asymmetric in
phase. B1, B2 and B5 beam splitters have equal reflectivity of
International Journal of Advancements in Research & Technology, Volume 1, Issue1, June-2012
ISSN 2278-7763
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one third ( 1 3 ). B3 and B4 beam splitters have equal
reflectivity of 50:50 ( 1 2 ) [20].
Fig.6: Simulation model for polarization to spatial encoding.
Fig. 5: A conceptual realization of the CNOT gate.
A sign change ( phase shift) is occurred upon reflection off
the green side of the BSs. A spatially encoded single photon
qubit is prepared by separating the polarization components of
a polarization qubit on a polarizing beam splitter (not shown).
In this dual rail logic notation
C0,in  0 and C1,in  1 are
4.2 Result for simulation of polarization to spatial
encoding
From fig. 10(a) which is shown the result of upper arm
polarization state after the 1st PBS and it is seen from position
1(fig. 6) and it is horizontally polarized. Fig. 10(b) is also
shown the result of lower arm polarization state after the 1 st
PBS and it is also seen from position 2(fig. 6) and it is
vertically polarized. Finally fig. 10(c) is shown the result of
output polarization state and it is seen from position 3(fig. 6)
and arbitrary polarized state is the final output.
the two bosonic mode operators for the control qubit, and
T0,in  0
and
T1,in  1
those
for
the
target.
Transformation between this dual rail logic and polarization
encoding are achieved with a half wave plate and a polarizing
beam splitter [20].The two target modes are mixed and
recombined on two 50% ( 1 2 ) reflective beam splitters to form
4.3 Simulation of the CNOT gate
In fig. 7 is shown the simulation model of CNOT gate where
B1, B2, B3, B4 and B5 beam splitters are replaced by
directional coupler and their reflectivity value is also set up
according to their given values. CW laser is used to give the
CNOT gate input.
an interferometer which also included a 33% ( 1 3 ) reflective
beam splitter in each arm. One can understand the operation of
this gate in the computational basis by considering the case
where the
C0,in mode is occupied. The target interferometer is
balanced and the target qubit exit in the same mode as it enters
(if it is not lost through a beam splitter). Conversely, if the
C1,in mode is occupied, then a non-classical interference is
occurred of the two qubit at the central beam splitter and when
a coincidence event is observed, the target mode is flipped. So
when control qubit is
0 , there is no interaction between
control and target qubit and the target qubit exists in the same
state that it entered, when control qubit is
1 , the control and
target qubit interact non-classically and interaction causes the
target mode flipped[19].
Fig.7: Simulation model for the CNOT gate.
4.4 Result for input
In fig. 8 is shown the simulation circuit arrangement when
input
4 SIMULATION AND RESULT
4.1Simulation of polarization to spatial encoding
In fig. 6 is shown a simulation model of fig. 3 or simulation
of polarization to spatial encoding which is prepared by
OptiSystem built in instrument and result also verified in is
built in optical visualizer.
Copyright © 2012 SciResPub.
C0,in  0 and T1,in  1
C0,in  0 and T1,in  1 .When those input are applied,
the resultant output is shown in fig. 11.
International Journal of Advancements in Research & Technology, Volume 1, Issue1, June-2012
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( 00  00 , 11  10 ) was also demonstrated.
REFERENCES
[1]
[2]
[3]
C0,in  0
Fig.8: Simulation circuit arrangement when input
T1,in  1
and
.
[4]
[5]
Fig. 11(a) and fig. 11(c) is shown the control input
C0,in  0 and target input T1,in  1 . It is mentioned that
[6]
horizontally polarized single photon state is known as
[7]
0 ( 0  H ) and vertically polarized single photon is
[8]
1 ( 1  V ) .After that fig. 11(b) and fig.
[9]
qubit
known as qubit
11(d) is shown the control output
C0,out  0 and target
 1 . In this way one of the CNOT gate
output T1,out
01  01 .
[11]
C1,in  1 and T0,in  0
[12]
operations is demonstrated
4.5 Result for input
In fig. 8 is shown the simulation circuit arrangement when
input
[10]
C1,in  1 and T0,in  0 .When those input are applied,
[13]
[14]
the resultant output is shown in fig. 12.
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[19]
Fig.9: Simulation circuit arrangement
T0,in  0
when input
C1,in  1
and
.
Fig. 12(a) and fig. 12(c) is shown the control input
[20]
C1,in  1 and target input T0,in  0 . After that fig. 12(b)
and fig. 12(d) is shown the control output
target output T1,out
other
[21]
 1 . In this way another CNOT gate
operations is demonstrated
way
C1,out  1 and
two
10  11 . According to same
CNOT
Copyright © 2012 SciResPub.
gate
operation
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Fig.10 (a): Upper (position 1) arm polarization state after the 1 st
PBS (
Copyright © 2012 SciResPub.
0 ).
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Fig.10 (b): Lower (position 2) arm polarization state after the 1 st
PBS ( 
Fig.10(c):
PBS (
Final
(position
3)
polarization
0  1 ).
Copyright © 2012 SciResPub.
state
after
the
2 nd
1 ).
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ISSN 2278-7763
Fig.11 (a): Control input C0,in
Fig.11(c): Target input T1,in
 0  H
1 V
Copyright © 2012 SciResPub.
.
.
8
Fig.11 (b): Control output C0,out
Fig.11 (d): Target output T1,out
 0  H
1 V
.
.
International Journal of Advancements in Research & Technology, Volume 1, Issue1, June-2012
ISSN 2278-7763
Fig.12 (a): Control input C1,in
Fig.12(c): Target input T0,in
1 V
 0  H
Copyright © 2012 SciResPub.
.
.
9
Fig.12 (b): Control output C1,out
Fig.12 (d): Target output T1,out
1 V
1 V
.
.