Quantum
NOT gate is implemented by phase encoding technique of light waves in our
previous work 26. The realization of phase encoded quantum NOT gate is
briefly described in the following section.

Quantum
NOT gate matrix is expressed by Pauli-x matrix 27

 

The
quantum states and  can be represented by the column vector
representation as,

 

   =    and  = .

 

The
block diagram to implement quantum NOT gate by phase encoding technique is
shown in fig.1. 

Fig.1 Schematic diagram of implementation of NOT gate by phase
encoding. Here ‘PS’, ‘BS’ and ‘M’ are ? phase shifter, beam splitter and mirror
respectively.

 

Here
the main light source is divided into three parts. The first one is considered
as the reference beam with the initial phase ?, and the other two parts serve
as the input signal beams. I1 and I2 respectively, which represent
two input light beams and O1 and O2 are the corresponding
output light beams. EOM1
and EOM2 are two Lithium niobate (LiNbO3) based electro-optic
modulators, operated simultaneously by the biasing voltage VNOT,
which causes an additional phase difference of ? to the input light beams.

Now,
at first, the quantum state  is
taken, then the phases of input light at I1 and I2 are
(?+?) and ? with respect to the reference beam (with initial phase angle ?). The
phase shifter (PS) is to be used in any of the two inputs (I1 or I2)
to satisfy the desired input state. So the portion in fig. 1 with dotted area
serves as NOT logic gate.

Then
after passing through the modulators, the phase of the output light beams O1
and O2 becomes (?+2?) and (?+?). Therefore in other words it can be
said that the first output O1 is in same phase with the reference
beam and the second one is ? out of phase with the reference beam. So the
output logic state () is the inverted version of the input
logic state (). Therefore a quantum NOT operation is
done by the system. Table. 1 shows the phase encoded truth table for
representing the phases of the input and output signal beams with respect to
the reference beam.

 

Table 1: Input and Output phase
differences with respect to the reference beam.

 

Phase difference of
Input signal beams  with respect to
reference beam

Phase difference of
Output signal beams  with respect to
reference beam

I1

I2

O1

O2

?

0

2? (0)

?

0

?

?

2 ? 
(0)