Quantum Matrices Using Quantum Gates

Manu Mitra*
Member of Golden Key International Honor Society, Alumnus of University of Bridgeport, USA
*Corresponding author: Manu Mitra, IEEE Member, Member of Golden Key International Honor Society, Alumnus of University of Bridgeport,
Bridgeport, CT, USA, Email
Submission: July 17, 2018; Published: September 04, 2018
Quantum Matrices Using Quantum Gates
Introduction
In physics, the size of iotas (atoms) and electrons demonstrates
that how things move at every day at different sizes and speeds.
In classical mechanics, objects exist in a definite place at a definite
time. However, in quantum mechanics, objects rather exist in a haze
of probability; they have a specific possibility of being at point A,
another chance of being at point B. Certain properties like position,
speed and colour can sometimes occur in exact, set amounts, like
a dial that “clicks” from number to number. This challenged a
fundamental assumption of classical mechanics, which said that
such properties should exist on an even, continuous spectrum. To
illustrate the possibility that a few properties “clicked” like a dial
with definite settings, experts coined the word “quantized” [1].
Beam Splitter
An example was demonstrated by Prof. Marek Perkowski from
Portland State University using beam splitter and based on that
how quantum circuits could work. “Half of the photons leaving the
light source arrive at detector A; the other half arrive at detector B”
[2,3] (Figure 1 & 2).
Research Article
COJ Electronics & Communications
C CRIMSON PUBLISHERS
Wings to the Research
1/22Copyright © All rights are reserved by Manu Mitra.
Volume - 1 Issue - 2
Abstract
Quantum mechanics explains the behavior of matter and its movement with energy in the scale of atoms and subatomic particles. In quantum
circuits there are many gates such as Hadamard Gate, Pauli Gates, many more gates for half turns, quarter turns, eighth turns, sixteenth turns and so
on, rest for spinning, parametrized etc. Linear operators in general and quantum mechanics can be represented in the form of vectors and in turn can
be viewed as matrices format because linear operators are totally equivalent with the matrices view point. This paper discloses creation of various
quantum matrices using Quantum bits. Square, Identity and Transposition of matrices are performed considering whole process in entanglement.
Angle, phase, coordinates, magnitude, complex numbers and amplitude has been noted and documented in this paper for further research.
Keywords: Quantum; Quantum mechanics; Quantum gates; Q bits; Quantum digital circuits; Logic circuits; Quantum matrix; Matrices
Figure 1: Demonstrates the light beam splitting to two directions [3].

COJ Elec Communicat Copyright © Manu Mitra
2/22How to cite this article: Manu M. Quantum Matrices Using Quantum Gates . COJ Elec Communicat. 1(2). COJEC.000506.2018.
Figure 2: Demonstrates simplest circuit of light beam splitter in Quantum Circuit [1].
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0
1
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Linear Algebra in Quantum Mechanics
The most convenient way to understand linear operators is in
terms of their matrix representations. In fact, the linear operator
and matrix viewpoints turn out to be entirely equivalent. For more
information on linear operator representation can refer Reference
No 7
Vectors can be written as:
For readability format it can be written as (1,2) and for two
level quantum systems used as qubits, we can identify the state |0”
with the vector (1, 0), and similarly |1” with (0, 1) [4-6].
Square Matrices using Quantum Gates
A simulation was performed on Hadamard Gates to generate
square matrices considering entanglements of quantum bits [7].
A. Square matrix using hadamard gate using two
quantum circuits (Figure 3) (Table 1)
Table 1: Measured output results of two quantum circuits.
Value Binary Value Magnitude
0 |00> 0.25
1 |01> 0.25
2 |10> 0.25
3 |11> 0.25
Figure 3: Demonstrates Square Matrix using two quantum
circuits and total usage of 4 quantum bits.
Notes:
Chance of being ON if measured: 50.00000%
Bloch sphere representation of local state
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Amplitude of |00>
Val: +0.50000+0.00000i
mag2
: 25.0000%, phase: 0.00o
Amplitude of |01>
Val: 0.50000+0.00000i
mag2
: 25.0000%, phase: 0.00o
Amplitude of |10>
Val: +0.50000+0.00000i
mag2
: 25.0000%, phase: 0.00o
Amplitude of |01>
Val: 0.50000+0.00000i
mag2
: 25.0000%, phase: 0.00o
B. Square matrix using hadamard gate using four
quantum circuits (Figures 4-6) (Table 2)
Table 2: Measured output results of four quantum circuits.
0 |0000> 0.0625
1 |0001> 0.0625
2 |0010> 0.0625
3 |0011> 0.0625
4 |0100> 0.0625
5 |0101> 0.0625
6 |0110> 0.0625
7 |0111> 0.0625
8 |1000> 0.0625
9 |1001> 0.0625
10 |1010> 0.0625
11 |1011> 0.0625
12 |1100> 0.0625
13 |1101> 0.0625
14 |1110> 0.0625
15 |1111> 0.0625

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How to cite this article: Manu M. Quantum Matrices Using Quantum Gates . COJ Elec Communicat. 1(2). COJEC.000506.2018.
Figure 4: Demonstrates the graph of two quantum circuits
square Matrix in QCAD (Quantum Computer Aided
Design) [5,6].
Figure 5: Demonstrates Square Matrix using four quantum
Figure 6: Demonstrates the graph of four quantum circuits
square Matrix in QCAD (Quantum Computer Aided Design).
Notes:
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Amplitude of |0000>
Val: +0.25000+0.00000i
mag2
: 6.25%, phase: 0.00o
C. Square matrix using hadamard gate using six quantum circuits (Figures 7 & 8) (Table 3)
Table 3: Measured output results of six quantum circuits.
0 |000000> 0.015625
1 |000001> 0.015625
2 |000010> 0.015625
3 |000011> 0.015625
4 |000100> 0.015625
5 |000101> 0.015625
6 |000110> 0.015625
7 |000111> 0.015625
8 |001000> 0.015625
9 |001001> 0.015625
10 |001010> 0.015625
11 |001011> 0.015625
12 |001100> 0.015625
13 |001101> 0.015625
14 |001110> 0.015625
15 |001111> 0.015625
16 |010000> 0.015625
17 |010001> 0.015625
18 |010010> 0.015625
19 |010011> 0.015625
20 |010100> 0.015625
21 |010101> 0.015625
22 |010110> 0.015625
23 |010111> 0.015625

24 |011000> 0.015625
25 |011001> 0.015625
26 |011010> 0.015625
27 |011011> 0.015625
28 |011100> 0.015625
29 |011101> 0.015625
30 |011110> 0.015625
31 |011111> 0.015625
32 |100000> 0.015625
33 |100001> 0.015625
34 |100010> 0.015625
35 |100011> 0.015625
36 |100100> 0.015625
37 |100101> 0.015625
38 |100110> 0.015625
39 |100111> 0.015625
40 |101000> 0.015625
41 |101001> 0.015625
42 |101010> 0.015625
43 |101011> 0.015625
44 |101100> 0.015625
45 |101101> 0.015625
46 |101110> 0.015625
47 |101111> 0.015625
48 |110000> 0.015625
49 |110001> 0.015625
50 |110010> 0.015625
51 |110011> 0.015625
52 |110100> 0.015625
53 |110101> 0.015625
54 |110110> 0.015625
55 |110111> 0.015625
56 |111000> 0.015625
57 |111001> 0.015625
58 |111010> 0.015625
59 |111011> 0.015625
60 |111100> 0.015625
61 |111101> 0.015625
62 |111110> 0.015625
63 |111111> 0.015625

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Figure 7: Demonstrates Square Matrix using six quantum
Figure 8: Demonstrates the graph of six quantum circuits
Notes:
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Amplitude of |000000>
Val: +0.12500+0.00000i
mag2
: 1.5625%, phase: 0.00o
D. Square matrix using hadamard gate using eight
quantum circuits (Figures 9 & 10)
Figure 9: Demonstrates Square Matrix using eight quantum
Figure 10: Demonstrates the graph of eight quantum
circuits square Matrix in QCAD (Quantum Computer Aided
Design).
Notes:
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Val: +0.06250+0.00000i
mag2
: 0.3906%, phase: 0.00o
E. Square matrix using hadamard gate using ten
quantum circuits (Figures 11 & 12)
Figure 11: Demonstrates Square Matrix using ten quantum
Figure 12: Demonstrates the graph of ten quantum circuits
Notes:
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Val: +0.03125+0.00000i
mag2
: 0.0977%, phase: 0.00o
F. Square matrix using hadamard gate using twelve
quantum circuits (Figure 13)

Figure 13: Demonstrates Square Matrix using twelve
quantum circuits and total usage of 4096 quantum bits
Notes:
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Val: +0.01562+0.00000i
mag2
: 0.0244%, phase: 0.00o
G. Square matrix using hadamard gate using fourteen
Figure 14: Demonstrates Square Matrix using fourteen
quantum circuits and total usage of 16384 quantum bits.
Notes:
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Amplitude of |00000000000000>
Val: +0.00781+0.00000i
mag2
: 0.0061%, phase: 0.00o
H. Square Matrix using hadamard gate using sixteen
Figure 15: Demonstrates Square Matrix using sixteen (max)
Notes:
r: +1.0000, φ+0.00o
, θ: +90o
x: +1.0000, y: 0.0000, z: +0.0000
Amplitude of |0000000000000000>
Val: + 0.00391+0.00000i
mag2
: 0.0014%, phase: 0.00o
Observations from square matrix
1. Successfully performed square matrix utilizing sixteen
(max) quantum circuits considering entanglement process
using only Hadamard gate
2. Real value of each quantum bit is equal for all quantum
bits and for every addition of two quantum circuit there is a
decrease of real value by half
3. Magnitude of each quantum bit decreases by half for every
addition of quantum circuit (In this case one fourth because
addition of two circuits)
4. Study of Hadamard gate has been performed
5. Graphs of twelve, fourteen and sixteen quantum circuits
are not used due to complexity of the circuit and limitation of
the tool to process large quantum circuits
6. Tables of eight, ten, twelve, fourteen and sixteen quantum
circuits are avoided due to results are similar as previous data
except for magnitudes.
Identity Matrices using Quantum Gates
A simulation was performed on various gates of Hadamard,
Pauli Y Gates and NOT gates to generate identity matrices
considering entanglements

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A. Identity matrix using hadamard, Pauli Y and NOT gate
using two quantum circuits (Figure 16 & 17) (Table 4)
0 |00> 0.5
1 |01> 0
2 |10> 0.5
3 |11> 0
Figure 16: Demonstrates Identity Matrix using Pauli Y,
Hadamard Gate and NOT gate utilizing two quantum
Figure 17: Demonstrates the graph of two quantum
circuits identity matrix in QCAD (Quantum Computer
Aided Design).
Notes:
r: +0.0000, φ+0.00o
, θ: +90o
x: +0.0000, y: 0.0000, z: +0.0000
Amplitude of |00>
Val: +0.00000+0.70711i
mag2
: 50.0000%, phase: +90.00o
Amplitude of |01>
Val: +0.00000+0.00000i
mag2
: 00.0000%, phase: +0.00o
Amplitude of |10>
Val: +0.00000+0.70711i
mag2
: 50.0000%, phase: +90.00o
Amplitude of |11>
Val: +0.00000+0.00000i
mag2
: 00.0000%, phase: +0.00o
B. Identity matrix using hadamard, Pauli Y and NOT gate
using four quantum circuits (Figure 18-20) (Table 5)
0 |0000> 0.25
1 |0001> 0
2 |0010> 0
3 |0011> 0
4 |0100> 0
5 |0101> 0.25
6 |0110> 0
7 |0111> 0
8 |1000> 0
9 |1001> 0
10 |1010> 0.25
11 |1011> 0
12 |1100> 0
13 |1101> 0
14 |1110> 0
15 |1111> 0.25
Figure 18: Demonstrates Identity Matrix using Pauli
Y, Hadamard Gate and NOT gate utilizing four quantum
Figure 19: Demonstrates four quantum circuit Identity
Matrix using QCAD.

Figure 20: Demonstrates the graph of four quantum
Aided Design)
Notes:
r: +0.0000, φ+0.00o
, θ: +90o
x: +0.0000, y: 0.0000, z: +0.0000
Amplitude of |0000>
Val: -0.50000+0.00000i
mag2
: 25.0000%, phase: +180.00o
Amplitude of |0001>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
C. Identity matrix using hadamard, Pauli Y and NOT gate
using six quantum circuits (Figure 21-23) (Table 6)
0 |0000> 0.25
1 |0001> 0
2 |0010> 0
3 |0011> 0
4 |0100> 0
5 |0101> 0.25
6 |0110> 0
7 |0111> 0
8 |1000> 0
9 |1001> 0
10 |1010> 0.25
11 |1011> 0
12 |1100> 0
13 |1101> 0
14 |1110> 0
15 |1111> 0.25
Hadamard Gate and NOT gate utilizing six quantum circuits
and total usage of 8 quantum bits.
Figure 22: Demonstrates six quantum circuit Identity
Matrix using QCAD.
identity matrix in QCAD (Quantum Computer Aided
Design).
Notes:
r: +0.0000, φ+0.00o
, θ: +90.00o
x: +0.0000, y: +0.0000, z: +0.0000
Val: +0.00000+0.35355i
mag2
: 12.5000%, phase: -90.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o

9/22
D. Identity matrix using hadamard, Pauli Y and NOT gate
using eight quantum circuits (Figure 24-26)
Hadamard Gate and NOT gate utilizing eight quantum
Figure 25: Demonstrates eight quantum circuit Identity
Matrix using QCAD.
Aided Design).
r: +0.0000, φ+0.00o
, θ: +90.00o
x: +0.0000, y: +0.0000, z: +0.0000
Val: +0.25000+0.00000i
mag2
: 6.2500%, phase: +00.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
E. Identity matrix using hadamard, Pauli Y and NOT gate
using ten quantum circuits (Figure 27-29)
Y, Hadamard Gate and NOT gate utilizing ten quantum
Figure 28: Demonstrates ten quantum circuit Identity
Matrix using QCAD
identity matrix in QCAD (Quantum Computer Aided Design).
Notes:
r: +0.0000, φ+0.00o
, θ: +90.00o
x: +0.0000, y: +0.0000, z: +0.0000
Val: +0.00000+0.17678i

mag2
: 3.1250%, phase: +90.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
F. Identity matrix using hadamard, Pauli Y and NOT gate
using twelve quantum circuits (Figure 30)
Hadamard Gate and NOT gate utilizing twelve quantum
circuits and total usage of 64 quantum bits
Notes:
r: +0.0000, φ+0.00o
, θ: +90.00o
x: +0.0000, y: +0.0000, z: +0.0000
Val: -0.12500+0.0000i
mag2
: 1.5625%, phase: +180.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
Hadamard Gate and NOT gate utilizing fourteen quantum
G. Identity matrix using hadamard, Pauli Y and NOT gate
using fourteen quantum circuits (Figure 31)
Notes:
r: +0.0000, φ+0.00o
, θ: +90.00o
x: +0.0000, y: +0.0000, z: +0.0000
Amplitude of |00000000000000>
Val: +0.12500+0.08839i
mag2
: 0.7812%, phase: -90.00o
Amplitude of |00000000000001>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
H. Identity matrix using hadamard, Pauli Y and NOT gate
using sixteen (max) quantum circuits (Figure 32)
Y, Hadamard Gate and NOT gate utilizing sixteen (max)
Notes:
r: +0.0000, φ +0.00o
, θ: +90.00o
x: +0.0000, y: +0.0000, z: +0.0000
Amplitude of |0000000000000000>
Val: +0.06250+0.00000i
mag2
: 0.3906%, phase: +00.00o
Amplitude of |0000000000000001>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
Observations from identity matrix
1. Successfully performed identity matrix utilizing sixteen
(max) quantum circuits considering entanglement process
using Hadamard, Pauli X and NOT gates

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2. Real and complex values of each quantum bit is equal for
all quantum bits
3. Real values decrease by half for addition of fourth, eighth,
twelve and sixteenth quantum circuit
4. Complex value decreases by half for addition of second,
sixth, tenth and fourteenth quantum circuit
5. Magnitude of each quantum bit decreases by half for
every addition of two quantum circuit
circuits are avoided due to results are similar as previous data
except for magnitudes.
Transposing of Matrices using quantum Gates
A simulation was performed for transposition of Matrix
A. Transposition of matrix using hadamard, Pauli Y, NOT,
and swap gate using two quantum circuits (Figure 33)
Figure 33: Demonstrates before transposition of Matrix
using Pauli Y, Hadamard, NOT and swap gates utilizing two
Notes:
Chance of being ON for Y Pauli Gate if measured: 50.00000%
Bloch sphere representation of local state for first quantum
circuit
r: +1.0000, φ+0.00o
, θ : +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for second quantum
circuit
r: +1.0000, +0.00o
, : +90.00o
x: +0.0000, y: +0.0000, z: +1.0000
Amplitude of |00>
Val: +0.00000+0.70711i
mag2
: 50.0000%, phase: +90.00o
Amplitude of |10>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figure 34) (Table 7) (Figure 35)
0 |00> 0.5
1 |01> 0.5
2 |10> 0
3 |11> 0
Figure 34: Demonstrates the graph of two quantum
circuits before transposition of matrix in QCAD (Quantum
Computer Aided Design)
Figure 35: Demonstrates after transposition of Matrix
using Pauli Y, Hadamard, NOT and swap gates utilizing two
Notes:
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for second quantum
circuit
r: +1.0000, +0.00o
, : +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Amplitude of |00>
Val: +0.00000+0.70711i
mag2
: 50.0000%, phase: +90.00o
Amplitude of |01>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figure 36) (Table 8)

0 |00> 0.5
1 |01> 0
2 |10> 0.5
3 |11> 0
Figure 36: Demonstrates the graph of two quantum circuits
after transposition of matrix in QCAD (Quantum Computer
Aided Design).
B. Transposition of Matrix using Hadamard, Pauli Y, NOT,
and swap gate using four quantum circuits (Figure 37)
using Pauli Y, Hadamard, NOT and swap gates utilizing four
Notes:
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for third quantum
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Amplitude of |0000>
Val: +0.50000+0.00000i
mag2
: 25.0000%, phase: +180.00o
Amplitude of |0100>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figure 38) (Table 9) (Figure 39)
Value Binary value Magnitude
0 |0000> 0.25
1 |0001> 0.25
2 |0010> 0.25
3 |0011> 0.25
4 |0100> 0
5 |0101> 0
6 |0110> 0
7 |0111> 0
8 |1000> 0
9 |1001> 0
10 |1010> 0
11 |1011> 0
12 |1100> 0
13 |1101> 0
14 |1110> 0
15 |1111> 0
using Pauli Y, Hadamard, NOT and swap gates utilizing four
Notes:
Chance of being ON for Hadamard Gate if measured: 50.00000%
circuit
r: +1.0000, φ+0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for third quantum
circuit

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r: +1.0000, φ +0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Amplitude of |0000>
Val: +0.50000+0.00000i
mag2
: 25.0000%, phase: +180.00o
Amplitude of |0001>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
circuits after transposition of matrix in QCAD (Quantum
Table 10: Measured output results of four quantum
circuits.
0 |0000> 0.25
1 |0001> 0
2 |0010> 0
3 |0011> 0
4 |0100> 0.25
5 |0101> 0
6 |0110> 0
7 |0111> 0
8 |1000> 0.25
9 |1001> 0
10 |1010> 0
11 |1011> 0
12 |1100> 0.25
13 |1101> 0
14 |1110> 0
15 |1111> 0
C. Transposition of Matrix using Hadamard, Pauli Y, NOT,
and swap gate using six quantum circuits (Figure 41)
using Pauli Y, Hadamard, NOT and swap gates utilizing six
Notes:
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for fourth quantum
circuit
r: +1.0000, φ+0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Val: +0.00000-0.35355i
mag2
: 12.5000%, phase: -90.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figure 42) (Table 11) (Figure
43)
0 |000000> 0.125
1 |000001> 0.125
2 |000010> 0.125
3 |000011> 0.125
4 |000100> 0.125
5 |000101> 0.125
6 |000110> 0.125

7 |000111> 0.125
8 |001000> 0
9 |001001> 0
10 |001010> 0
11 |001011> 0
12 |001100> 0
13 |001101> 0
14 |001110> 0
15 |001111> 0
16 |010000> 0
17 |010001> 0
18 |010010> 0
19 |010011> 0
20 |010100> 0
21 |010101> 0
22 |010110> 0
23 |010111> 0
24 |011000> 0
25 |011001> 0
26 |011010> 0
27 |011011> 0
28 |011100> 0
29 |011101> 0
30 |011110> 0
31 |011111> 0
32 |100000> 0
33 |100001> 0
34 |100010> 0
35 |100011> 0
36 |100100> 0
37 |100101> 0
38 |100110> 0
39 |100111> 0
40 |101000> 0
41 |101001> 0
42 |101010> 0
43 |101011> 0
44 |101100> 0
45 |101101> 0
46 |101110> 0
47 |101111> 0
48 |110000> 0
49 |110001> 0
50 |110010> 0
51 |110011> 0

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52 |110100> 0
53 |110101> 0
54 |110110> 0
55 |110111> 0
56 |111000> 0
57 |111001> 0
58 |111010> 0
59 |111011> 0
60 |111100> 0
61 |111101> 0
62 |111110> 0
63 |111111> 0
Figure 42: Demonstrates the graph of six quantum
using Pauli Y, Hadamard, NOT and swap gates utilizing six
Notes:
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for fourth quantum
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Val: +0.00000-0.35355i
mag2
: 25.0000%, phase: +180.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
Table 12: Measured output results of six quantum circuits
0 |000000> 0.125
1 |000001> 0
2 |000010> 0
3 |000011> 0
4 |000100> 0
5 |000101> 0
6 |000110> 0
7 |000111> 0
8 |001000> 0.125
9 |001001> 0
10 |001010> 0

11 |001011> 0
12 |001100> 0
13 |001101> 0
14 |001110> 0
15 |001111> 0
16 |010000> 0.125
17 |010001> 0
18 |010010> 0
19 |010011> 0
20 |010100> 0
21 |010101> 0
22 |010110> 0
23 |010111> 0
24 |011000> 0.125
25 |011001> 0
26 |011010> 0
27 |011011> 0
28 |011100> 0
29 |011101> 0
30 |011110> 0
31 |011111> 0
32 |100000> 0.125
33 |100001> 0
34 |100010> 0
35 |100011> 0
36 |100100> 0
37 |100101> 0
38 |100110> 0
39 |100111> 0
40 |101000> 0.125
41 |101001> 0
42 |101010> 0
43 |101011> 0
44 |101100> 0
45 |101101> 0
46 |101110> 0
47 |101111> 0
48 |110000> 0.125
49 |110001> 0
50 |110010> 0
51 |110011> 0
52 |110100> 0
53 |110101> 0
54 |110110> 0
55 |110111> 0

17/22
56 |111000> 0.125
57 |111001> 0
58 |111010> 0
59 |111011> 0
60 |111100> 0
61 |111101> 0
62 |111110> 0
63 |111111> 0
Aided Design)
D. Transposition of matrix using hadamard, Pauli Y, NOT,
and swap gate using eight quantum circuits (Figure 45)
using Pauli Y, Hadamard, NOT and swap gates utilizing
eight quantum circuits and total usage of 16 quantum bits
Notes:
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for fifth quantum
circuit
r: +1.0000, φ+0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Val: +0.25000-0.00000i
mag2
: 6.2500%, phase: +0.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figure 46 & 47)
Computer Aided Design).
eight quantum circuits and total usage of 16 quantum bits.
Notes:
circuit
r: +1.0000, φ+0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for fifth quantum
circuit
r: +1.0000, φ +0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000

Val: +0.25000+0.00000i
mag2
: 6.2500%, phase: +0.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figure 48)
circuits after transposition of matrix in QCAD (Quantum
E. Transposition of matrix using hadamard, Pauli Y, NOT,
and swap gate using ten quantum circuits (Figure 49)
using Pauli Y, Hadamard, NOT and swap gates utilizing ten
Notes:
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for sixth quantum
circuit
r: +1.0000, φ+0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Val: +0.00000-0.17678i
mag2
: 3.1250%, phase: +90.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figures 50 & 51)
Figure 50: Demonstrates the graph of ten quantum
using Pauli Y, Hadamard, NOT and swap gates utilizing ten
Notes:
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for sixth quantum
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Val: +0.00000+0.17678i
mag2
: 3.1250%, phase: +90.00o
Val: +0.00000+0.00000i

19/22
mag2
: 0.0000%, phase: +0.00o
(Figure 52)
Aided Design).
F. Transposition of matrix using hadamard, Pauli Y, NOT,
and swap gate using twelve quantum circuits (Figure 53)
twelve quantum circuits and total usage of 64 quantum
bits.
Notes:
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for seventh quantum
circuit
r: +1.0000, φ+0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Val: -0.12500-0.00000i
mag2
: 1.5625%, phase: +180.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
(Figure 54)
Figure 54: Demonstrates after transposition of Matrix using
Pauli Y, Hadamard, NOT and swap gates utilizing twelve
Notes:
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for seventh quantum
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Val: -0.12500+0.00000i
mag2
: 1.5625%, phase: +180.00o
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
G. Transposition of matrix using hadamard, Pauli Y, NOT,
and swap gate using fourteen quantum circuits (Figure 55)
fourteen quantum circuits and total usage of 128 quantum
bits.

Notes:
circuit
r: +1.0000, φ +0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for eighth quantum
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Amplitude of |00000000000000>
Val: +0.00000-0.08839i
mag2
: 1.5625%, phase: +180.00o
Amplitude of |00000010000000>
Val: +0.00000+0.0000i
mag2
: 0.0000%, phase: +0.00o
(Figure 56)
Pauli Y, Hadamard, NOT and swap gates utilizing fourteen
Notes:
circuit
r: +1.0000, φ+0.00o
, θ: +0.00o
x: +1.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for eighth quantum
circuit
r: +1.0000, φ +0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Amplitude of |00000000000000>
Val: +0.00000-0.08839i
mag2
: 0.7812%, phase: -90.00o
Amplitude of |00000000000001>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
H. Transposition of matrix using hadamard, Pauli Y, NOT,
and swap gate using sixteen quantum circuits (Figure 57)
sixteen quantum circuits and total usage of 256 quantum
bits.
Notes:
circuit
r: +1.0000, φ +0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Bloch sphere representation of local state for ninth quantum
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Amplitude of |0000000000000000>
Val: +0.06250-0.00000i
mag2
: 0.3906%, phase: +0.00o
Amplitude of |0000000100000000>
Val: +0.00000+0.0000i
Pauli Y, Hadamard, NOT and swap gates utilizing sixteen

21/22
mag2
: 0.0000%, phase: +0.00o
(Figure 58)
Notes:
circuit
r: +1.0000, φ +0.00o
, θ: +0.00o
x: +0.0000, y: +0.0000, z: +1.0000
Bloch sphere representation of local state for ninth quantum
circuit
r: +1.0000, φ+0.00o
, θ: +90.00o
x: +1.0000, y: +0.0000, z: +0.0000
Amplitude of |0000000000000000>
Val: +0.06250+0.00000i
mag2
: 0.3906%, phase: +0.00o
Amplitude of |0000000000000001>
Val: +0.00000+0.00000i
mag2
: 0.0000%, phase: +0.00o
Observations from transposition of matrix
1. Successfully performed transposition of matrix utilizing
sixteen (max) quantum circuits considering entanglement
process using Hadamard, Pauli X and NOT gates
2. Magnitude and phase don’t change in transposition of
Matrix
3. Magnitude of each quantum bit decreases by half for
every addition of two quantum circuit
circuits are avoided due to limitation of the tool to process large
quantum circuits.
Applications of Quantum Matrices
1. Quantum Matrices can be used to solve system
performance with a big RAM and/or a high clock rate for
Quantum Central Processing Unit (QCPU)
2. Studying Quantum Matrices and their properties is of
crucial importance in trying to improve linear solvers efficiency
in Quantum Processor/Quantum computer
3. If constructing a larger system in a quantum processor
with the same pattern (2 on the diagonal, -1 on the upper and
lower diagonal) and put a right-hand side with only positive
entries, we can still claim that the solution exists and it’s
positive and the number of operations needed to solve it, where
n is the size of the system [7].
Results
What claimed are
1. Successfullyprocessed65536quantumbitsofinformation
for 16 quantum circuits to achieve square matrix considering
whole process in entanglement
2. Square matrix for quantum bits is generated with respect
to phase, angle, complex values and magnitude of the results
3. Successfully processed 256 quantum bits of information
for identity matrix and transposition of matrix with respect to
phase, angle, real, complex values and magnitude.
4. Identity matrix for quantum bits is generated with respect
to phase, angle, complex values and magnitude of the results.
5. Transposition of matrix for quantum bits is generated
with respect to phase, angle, complex values and magnitude of
the results.
6. Study of Hadamard Gates are performed and observations
are noted.
7. Study of various phases, angles, complex values,
magnitudes and amplitudes of the quantum bits are performed
and is documented in this paper
8. Based on results and above data other matrices such as
row matrix, column matrix, rectangular matrix, diagonal matrix,
scalar matrix, triangular matrix, null matrix can be constructed
using quantum bits.
Conclusion
Based on the results Quantum Matrices can be constructed
and processed 65536 quantum bits have been processed. And any
quantum matrices can be constructed using this method which will
help in performance of the Quantum Computers.
Novelty
Novelty in this paper are:
Constructed square Matrix using quantum bits
Constructed Identity Matrix using quantum bits
Performed Transposition of Matrix using quantum bits
Processed 65536 (max) quantum bits of information for 16
quantum circuits considering whole process in entanglement for
square matrix
Processed 256 quantum bits of information for 16 quantum
circuits considering whole process in entanglement for identity
matrix
Processed 256 quantum bits of information for 16 quantum
circuits considering whole process in entanglement for
transposition of matrix.
Nomenclature
r: position

: represents wavefuntion to solve from the Schrödinger
equation (varies with situation and number of quantum bits)
: Angle in degrees
i: the imaginary unit
x: x component
y: y component
z: z component
Val: Value
mag2
: magnitude in square
Acknowledgment
Author would like to thank Prof. Navarun Gupta, Prof. Hassan
Bajwa, Prof. Linfeng Zhang and Prof. Hmurcik for their academic
support. Author also thanks anonymous reviewers for their
comments.
References
1. Coolman R (2014) What Is Quantum Mechanics? Live Science.
2. Perkows M (2002) Introductory Lecture on Quantum Computing.
3. Edmund Optics What are Beam splitters? | Edmund Optics, Barrington,
USA.
4. Quirk Quirk: Quantum Circuit Simulator.
5. QCAD QCAD - CAD for Quantum Computer Simulator.
6. Bartgol What is the usefulness of matrices?
7. Nielsen MA, Chuang IL (2016) Quantum computation and quantum
information. Cambridge University Press, Cambridge, UK.
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