design and realization of a quantum controlled not gate ......whole design methodology was simulated...

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

mailto:[email protected]

http://en.wikipedia.org/wiki/Quantum_field_theory

http://en.wikipedia.org/wiki/Quantum_field_theory

http://en.wikipedia.org/wiki/Quantum_state

http://en.wikipedia.org/wiki/Energy

http://en.wikipedia.org/wiki/Controlled_NOT_gate



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

http://en.wikipedia.org/wiki/Mirror

http://www.rp-photonics.com/waveguides.html

http://www.rp-photonics.com/photonic_integrated_circuits.html

http://www.rp-photonics.com/photonic_integrated_circuits.html

http://www.rp-photonics.com/beam_combining.html

http://www.rp-photonics.com/interference.html



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



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.



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 ) .




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 ) .



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 .



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 .

design and realization of a quantum controlled not gate ......whole design methodology was simulated...

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