design and realization of a quantum controlled not gate ......whole design methodology was simulated...
TRANSCRIPT
International Journal of Advancements in Research & Technology, Volume 1, Issue1, June-2012 1 ISSN 2278-7763
Copyright © 2012 SciResPub.
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: [email protected]). 2(Department of Applied Physics, Electronics and Communication Engineering (APECE), University of Dhaka, Bangladesh. (Phone: =
+8801817094868; e-mail: [email protected])
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
Polarizing Beam Splitter (PBS) and half wave plate (HWP) oriented at045 . 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
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 [19-
22]. 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.
2 Theory
2.1 Representation of Quantum CNOT gate
Let us consider the computational basis states defined as
10
0
(0.1)
01
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]
00 00 01 01 10 11 11 10CNOT
Q
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1 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 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 1 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
(0.3)
1 0 1 0 0 0 0 1
0 0 0 1 0 1 1 0
1 0
1 0 0 10 1
I X
0 0 1 1I X
From equation (1.3) we can write the CNOT operator in matrix
form as:
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
CNOTU
(0.4)
When input is 10 . So
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 1 0 010 1 1 11
0 0 0 1 1 0 1 0 1 1
0 0 1 0 0 1 1 1
CNOTU
Other operation is also performed this same way. So CNOTU
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.
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.
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 0180 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 o-
waves has changed then the state of polarization of the wave
must be changed [26]. When angle between optical axis and
incident light axis or light propagation axis is 045 then HWP is
acted as NOT operator [27].
2.5 Polarizing Beam Splitter
Polarizing Beam Splitter (PBS) divides the incident beam
into two orthogonally polarized beams at 090 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 3 ISSN 2278-7763
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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
zero, 0L
equals the state 10 , while the logical one, 1L
equals the state 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 2N n 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 00L
equals the number state 1010 and the logical state
11L
equals the number state 0101 [29].
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.
0 1H V —polarization encoding,
where H and V are the horizontal and vertical
polarization states [19].
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
beam to 090 when its optical axis is apt
045 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.
Fig. 3: Conversion from polarization to spatial encoding.
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].
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).
Fig. 4: A possible realization of an optical quantum CNOT gate.
If π phase shift is applied i.e. non-classical interference is
occurred and target qubit is flipped, the NOT operation occurs,
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].
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 Cv and Tv 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 4 ISSN 2278-7763
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one third ( 13 ). B3 and B4 beam splitters have equal
reflectivity of 50:50 ( 12 ) [20].
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 0, 0inC and 1, 1inC are
the two bosonic mode operators for the control qubit, and
0, 0inT and 1, 1inT 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% ( 12 ) reflective beam splitters to form
an interferometer which also included a 33% ( 13 ) reflective
beam splitter in each arm. One can understand the operation of
this gate in the computational basis by considering the case
where the 0,inC 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
1,inC 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].
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.
Fig.6: Simulation model for polarization to spatial encoding.
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 1st
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.
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.
Fig.7: Simulation model for the CNOT gate.
4.4 Result for input 0, 0inC and 1, 1inT
In fig. 8 is shown the simulation circuit arrangement when
input 0, 0inC and 1, 1inT .When those input are applied,
the resultant output is shown in fig. 11.
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Fig.8: Simulation circuit arrangement when input 0, 0inC and
1, 1inT .
Fig. 11(a) and fig. 11(c) is shown the control input
0, 0inC and target input 1, 1inT . It is mentioned that
horizontally polarized single photon state is known as
qubit 0 ( 0 )H and vertically polarized single photon is
known as qubit 1 ( 1 )V .After that fig. 11(b) and fig.
11(d) is shown the control output 0, 0outC and target
output 1, 1outT . In this way one of the CNOT gate
operations is demonstrated 01 01 .
4.5 Result for input 1, 1inC and 0, 0inT
In fig. 8 is shown the simulation circuit arrangement when
input 1, 1inC and 0, 0inT .When those input are applied,
the resultant output is shown in fig. 12.
Fig.9: Simulation circuit arrangement when input 1, 1inC and
0, 0inT .
Fig. 12(a) and fig. 12(c) is shown the control input
1, 1inC and target input 0, 0inT . After that fig. 12(b)
and fig. 12(d) is shown the control output 1, 1outC and
target output 1, 1outT . In this way another CNOT gate
operations is demonstrated 10 11 . According to same
way other two CNOT gate operation
( 00 00 , 11 10 ) was also demonstrated.
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Fig.10 (a): Upper (position 1) arm polarization state after the 1st
PBS ( 0 ) .
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Fig.10(c): Final (position 3) polarization state after the 2nd
PBS ( 0 1 ) .
Fig.10 (b): Lower (position 2) arm polarization state after the 1st
PBS ( 1 ) .
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Fig.11 (a): Control input 0, 0inC H .
Fig.11(c): Target input 1, 1inT V .
Fig.11 (b): Control output 0, 0outC H .
Fig.11 (d): Target output 1, 1outT V .
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Fig.12 (a): Control input 1, 1inC V .
Fig.12(c): Target input 0, 0inT H .
Fig.12 (b): Control output 1, 1outC V .
Fig.12 (d): Target output 1, 1outT V .