Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer

Abstract

The fundamental security aspect of the classical crypto-system depends on integer factorization and discrete logarithm problems. The quantum factorization problem is a crucial problem in quantum computing as it has significant implications for cryptography and security. Shos’s paper on “Polynomial-time algorithms for discrete logarithms and factoring on a quantum compute” has become a threat to the classical crypto-system, influencing many researchers to work on factorization problems using quantum computing. Quantum Computers (QC) can be essential in running different factorization algorithms in polynomial time. However, practical implementation on larger numbers is still a major challenge due to the requirement for error correction and massive quantum devices. Although the quantum factorization issue puts traditional cryptographic systems at risk, it also opens up new possibilities for quantum communication and encryption. The challenge of factorization opens up an entirely new field for research into quantum communication protocol security, including quantum key distribution and the development of quantum-resistant cryptographic systems. This paper surveys the various quantum algorithms for factoring a number into prime integers. We present a simulation study of the Quantum factorization algorithm to determine the period of a process. Turning the factoring problem into the challenge of determining a function’s period is the essential strategy of practical implementation using the quantum circuit.

Full text

Noname manuscript No.
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Study on Implementation of Shor’s Factorization
Algorithm on Quantum Computer
Mandeep Kumar ·Bhaskar Mondal*
Received: date / Accepted: date
Abstract The fundamental security aspect of the classical crypto-system depends on
integer factorization and discrete logarithm problems. The quantum factorization prob-
lem is a crucial problem in quantum computing as it has significant implications for
cryptography and security. Shos’s paper on "Polynomial-time algorithms for discrete
logarithms and factoring on a quantum compute" has become a threat to the classical
crypto-system, influencing many researchers to work on factorization problems using
quantum computing. Quantum Computers (QC) can be essential in running differ-
ent factorization algorithms in polynomial time. However, practical implementation on
larger numbers is still a major challenge due to the requirement for error correction and
massive quantum devices. Although the quantum factorization issue puts traditional
cryptographic systems at risk, it also opens up new possibilities for quantum communi-
cation and encryption. The challenge of factorization opens up an entirely new field for
research into quantum communication protocol security, including quantum key distri-
bution and the development of quantum-resistant cryptographic systems. This paper
surveys the various quantum algorithms for factoring a number into prime integers.
We present a simulation study of the Quantum factorization algorithm to determine
the period of a process. Turning the factoring problem into the challenge of determin-
ing a function’s period is the essential strategy of practical implementation using the
quantum circuit.
Keywords Shor’s algorithm ·Factorization ·Quantum Period finding ·Quantum
Phase estimation ·Quantum Fourier Transformation
M. Kumar
Department of Computer Science and Engineering
National Institute of Technology Patna
India 800005 E-mail: mandeepk.ph21.cs@nitp.ac.in *Corresponding author
B. Mondal
Department of Computer Science and Engineering
National Institute of Technology Patna
India 800005 E-mail: bhaskar.cs@nitp.ac.in *Corresponding author

2 Mandeep Kumar, Bhaskar Mondal*
1 Introduction
Midway through the 1970s, Robert Solovay and Volker Strassen presented a randomised
algorithm that could identify a composite and probably prime number. David Deutsch
used the rules of physics in 1985 to provide a more robust solution to computing prob-
lems inspired by the Solovay-Strassen algorithm. This gave rise to the QC. Culminating,
Peter Shor introduced two important issues in 1994: the discrete logarithm problem
and finding an integer’s factor [27].
Today’s classical cryptography relies heavily on discrete logarithm and integer fac-
torization as its core components, which support the entire communication system.
Furthermore, the classical public key algorithm’s security strength comes from the
fact that it is challenging to invert encryption stages with just the public key, depen-
dent on discrete logarithmic and integer factorization problems. Integer factorization
poses a significantly greater computational challenge compared to integer multiplica-
tion. One fundamental assumption in the field of quantum computing is that it can help
solve integer factorization problems more quickly. Notably, factorization and discrete
logarithmic issues are the foundations of both RSA and ECC, which are commonly
used in online transactions to guarantee security. The security aspect of these cryp-
tographic methods is dependent on the intricacy of integer factorization; nevertheless,
this presents a vulnerability to prospective quantum assaults.
The conventional method, such as RSA, which uses a public key Nthat is the prod-
uct of two enormously big prime numbers, makes it more difficult to factor a fixed huge
number into two integers. RSA can be cracked in a polynomial amount of time using
a QFA, a typical quantum algorithm. A variation of QF technique has been used to
attack even more public key crypto-systems. A very well-known QFA algorithm named
Shor’s algorithm, which factors a number Nin O((logN )3) [24] time and O(logN )
space, is an important quantum algorithm. A fundamental Shor’s algorithm strategy
is to use quantum computing to convert the factoring problem into the problem of cal-
culating the period of a function. The period-finding problem can only be solved using
quantum computing. In quantum computing, one uses unitary maps acting on complex
vector spaces rather than Boolean operations acting on binary strings, quantum states
rather than bit values, and qubits in place of bits. A qubit is a quantum bit and a
quantum state is a vector characterising a physical system in quantum Physics. Any
quantum computing algorithm runs on a quantum circuit [33].
Furthermore, the utilization of quantum circuits can be optimized through simula-
tion on classical systems before execution on actual QCs. The IBM Qiskit, Amazone
Braket, and other tools simulate and execute a quantum algorithm on a QC. Once QCs
are practical, they can be used for many tasks, including artificial intelligence, machine
learning, communication, control, and security.
This paper provides a simulation-based study of Shor’s factorization problem and
discusses different approaches to solving the integer factorization problem. Section 2
provides basic requirements to be used in integer factorization problems. Section 3
literature review followed by section 4, which provide insight to Shor’s algorithm, and
also provides a simulation-based study on the quantum network. Section 5 provide the
paper’s outcome. Section 6 and section 7, provide a brief discussion about the problem
and the research question and conclude the summary of the paper, respectively.

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 3
2 Prerequisites
2.1 Quantum Fourier Transformation (QFT)
In mathematics and computer science, the most effective method for solving a problem
is to transform it into the sub-problems for which a solution is already known. Some
of these transformations can be computed on a QC much quicker than on a classical
computer, allowing for the development of a fast quantum algorithm for example,
discrete Fourier transforms (DFT) [17]. The DFT in mathematical notation requires a
vector of complex numbers x0,··· , xL−1as input, where L is the length of the vector
and a fixed parameter. The transformed data is output as a vector of complex numbers
y1,··· , yL−1,. The vector of complex number (yk)is defined as follows:
yk=1
√L
L−1
X
j=0
xje2πijk/L.(1)
The DFT over amplitudes of a wave function is implemented in quantum mechanics
as the QFT [8] [31]. In particular, Shor’s algorithm and quantum phase estimation use
it. Similar to the equation (1), the QFT acts on an orthonormal basis (|0⟩,··· ,|L−1⟩)
and is defined to be the linear operation on the basis state with the following action
|j⟩ −→ 1
√L
L−1
X
j=0
xje2πijk/L |k⟩.(2)
where yjand xjare amplitudes and yjare the DFT of xj. The the unitary (UQF T )
matrix is defined as :
UQF T =1
√L
L−1
X
j=0
L−1
X
k=0
ωjk
L|k⟩⟨j|(3)
where ωjk
L=e2πi
Lis Lth unity root.
2.1.1 Intuition
The QFT transform the computational Z-basis to the Fourier basis [30]. The binary
numbers are stored in the states |0⟩and |1⟩in the computational bases. The Hadamard
gate is an example of the single qubit QFT, which transforms between the Z-basis states
|0⟩and |1⟩to the X-basis states |+⟩and |−⟩. Similarly, all multi-qubit states in the
computational basis have corresponding states in the Fourier basis. The QFT is simply
a function that transforms between these bases.
|State in Computational Basis⟩QFT
−−−→ |State in Fourier Basis⟩
The QFT over a finite abelian group can be expressed as a direct product of cyclic
group [30]. Thus the derivation of transformation for L= 2n,QFTLacting on the
state |x⟩=|x1. . . xn⟩where x1is the most significant bit.
QF TL|x⟩=1
√L
L−1
X
y=0
ωxy
L|y⟩(4)
=1
√N|0⟩+e
2πi
2x|1⟩⊗|0⟩+e
2πi
22x|1⟩⊗. . .⊗|0⟩+e
2πi
2n−1x|1⟩⊗|0⟩+e
2πi
2nx|1⟩
(5)

4 Mandeep Kumar, Bhaskar Mondal*
2.2 The QFT Circuit
In order to achieve the Fourier transform (FT), the circuit includes quantum gates
(single-qubit Hadamard gates Hand UROT gates). A two-qubit quantum gate called
controlled rotation (CROTa) gate use U ROTagate, which is represented as:
CROTa=I0
0UROTa, U ROTa="1 0
0e(2πi
2a)#(6)
and the control and target qubit are represented as CROTa|0xk⟩=|0xk⟩and
CROTa|1xk⟩=e(2πi
2axk)|1xk⟩. Provided two gates, the figure 1 shows a QFT circuit
for n-numbers of qubits.
|x1⟩· · ·
|x2⟩· · ·
.
.
.· · ·
|xn−1⟩
|xn⟩
HURO T2UR OTn−1U ROTn
HURO Tn−2UR OTn−1
HURO T2
1 2 3 4
Fig. 1 The quantum Fourier circuit with two-qubit UROT controlled rotation
With n-qubit input state |x1x2. . . xn⟩, the circuit takes input state, Hadamard
gate and UROT2operates on qubit 1, gives the following transform value:
UROT2.H1|x1x2. . . xn⟩=1
√2h|0⟩+e(2πi
2x1)|1⟩i⊗ |x2x3. . . xn⟩(7)
similarly, applying the same sequence of operation on 2to n-qubit, the final trans-
form state is:
1
√2h|0⟩+e(2πi
2nx1)|1⟩i⊗1
√2|0⟩+e2πi
2n−1x2|1⟩⊗. . .⊗1
√2h|0⟩+e(2πi
21xn)|1⟩i(8)
The QFT output state appear in the same sequence as provided input states.
2.3 Quantum Phase Estimation
Quantum phase estimation is a central building block of almost every quantum al-
gorithm. It gives the eigenvalue for an eigenvector unitary operator U. For a given
U,e2πiθ (eigenvalue) is the estimated phase in state U|ψ⟩=e2πiθ |ψ⟩, where |ψ⟩is an
eigenvector. A phase kickback technique is used to write the phase value of Uin the
counting register. The Fourier basis (U) is converted to the computational basis after
applying the inverse QFT.
The circuit (shown in the figure 2) estimate the phase (θ) of U, with the operation
as follow:

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 5
Setup The state |ψ⟩is in one set of qubit registers. An additional set of nqubits form
the counting register on which we will store the value 2nθ:
|ψ0⟩=|0⟩⊗n|ψ⟩(9)
Superposition A n-qubit Hadamard gate operation [5] is efficiently applied on the
counting register gives the following value:
|ψ1⟩=1
2n
2
(|0⟩+|1⟩)⊗n|ψ⟩(10)
Controlled Unitary Operations (CU ) The CU is applied on the Uon the target bit
register if the controlled bit is |1⟩. The eigenvector of the unitary operator is |ψ⟩
such that U|ψ⟩=e2πiθ |ψ⟩means:
U2j|ψ⟩=U2j−1U|ψ⟩=U2j−1e2πiθ |ψ⟩=··· =e2πi2jθ|ψ⟩(11)
After the ncontrolled Operation (CU ) using the relation |0⟩⊗|ψ⟩+|1⟩⊗e2πiθ |ψ⟩=
|0⟩+e2πiθ |1⟩⊗ |ψ⟩the value of the step is:
|ψ2⟩=1
2n
2
2n−1
X
k=0
e2πiθk |k⟩ ⊗ |ψ⟩(12)
where kis the n-bit binary numbers.
Inverse FT The eq. (12) is the result of the FT over the n-qubit input state. Applying
the inverse QFT on the eq. (12) gives
|ψ3⟩=1
2n
2
2n−1
X
k=0
e2πiθk |k⟩ ⊗ |ψ⟩QF T −1
n
−−−−−→ 1
2n
2n−1
X
x=0
2n−1
X
k=0
e−2πik
2n(x−2nθ)|x⟩ ⊗ |ψ⟩
(13)
Measurement The expression in eq. (13) peaks near x= 2nθ. Measuring in the com-
putational basis for the case when 2nθis an integer, give the phase with the high
probability:
|ψ4⟩=2nθ⊗ |ψ⟩(14)
If 2nθis not an integer, the eq. (14) comes up with with probability better than
4/π2≈40%.
The figure 1 takes n(n−1)
2rotation gates with nHadamard gates haing the complex-
ity (n2) where the n=⌈logN ⌉[30]. The QFT circuit over n-qubit can be approximate
with O(nlogn)[22].
3 Progress in Quantum Factorization
Peter Shor [27] addressed a critical issue in the field of cryptography. The challenge
was to find an efficient algorithm for solving the problems of discrete logarithms and
factorization, which are crucial for the security of many cryptographic systems. These
problems have traditionally been considered hard and the existing algorithms for solv-
ing them were inefficient and time-consuming.

6 Mandeep Kumar, Bhaskar Mondal*
Shor used a QC to solve these problems and showed that a QC could solve the
problems of discrete logarithms and factorization in polynomial time, which was a
major breakthrough in the field. This had significant implications for the security of
cryptographic systems, as it revealed that many of the commonly used systems were
vulnerable to attack by a QC. The findings of Shor’s paper have since had a ma jor
impact on the design of secure communication systems and have led to the development
of new and more secure cryptographic systems that are resistant to attack by a QCr.
The paper has also led to further research in the field of quantum algorithms and their
applications in cryptography.
The issue of the speed of Shor’s algorithm for factoring huge numbers has been
addressed by Burke et al. [9]. Finding a method to make the technique more effective
and to cut down on factorization time was a difficulty. Their solution to the issue
offered the r-algorithm, a modification of Shor’s technique that they showed could
greatly speed up factorization for big numbers. They discovered that the r-algorithm
factorised numbers up to 1000 bits with reasonable ease and was substantially faster
than earlier techniques. The r-algorithm is a vast advancement over existing techniques
and has the potential to dramatically increase communication system security.
In the context of noise and decoherence, Shepherd et al. [26] have discussed the
problem of quantum computation. Finding a method to carry out quantum compu-
tation that is resistant to environmental noise and decoherence, which can greatly
decrease the precision of the computation and the security of the system, was the
problem. The idea of temporally unstructured quantum computation, which includes
introducing random gates into the system, was first proposed by them. A major ad-
vancement over earlier approaches, temporally unstructured quantum computation has
the potential to greatly increase the security of quantum communication networks.
The subject of linear optics’ complexity and its potential effects on quantum com-
puting has been addressed by Aronson et al. [1] Understanding the computational
complexity of linear optics and its capacity to resolve some issues that are challenging
for conventional computers was a difficult task. They suggested a fresh strategy for
solving the issue and developed the idea of boson sampling, a method that applies
linear optics to carry out a particular kind of quantum processing. According to their
research, Boson Sampling has the ability to resolve some issues that conventional com-
puters are unable to handle. One of the most important aspects of the security of many
quantum communication systems is the capacity to do quantum computation with lin-
ear optics. A prominent area of research in the realm of quantum computing is the use
of linear optics and boson sampling to factorization difficulties, with the potential to
offer fresh perspectives and solutions to this significant issue.
Hamdi et al. [15] contrasted the General Number Field Sieve (GNFS) and Shor’s
quantum factoring algorithm while factoring huge integers. While GNFS is a classical
algorithm and is currently the most well-known way to factor huge numbers, Shor’s
algorithm is a quantum algorithm that can effectively factor enormous numbers on a
QC.
Ziyuan et al. [16] in the year 2018 proposed a scheme called location-based services
(LBS) used to preserve the location against quantum attacks. They used homomor-
phic encryption and quantum-resistant hash functions to protect the privacy of user’s
location information in LBS. Though the paper was theoretically secure against the
classical and quantum attacks, it has computational overhead associated with the ho-
momorphic encryption, due to which the scalability of the system gets compromised.

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 7
X. Fu et al. [13] in 2019 introduced an executable quantum instruction set archi-
tecture (eQASM) that offers a high-level programming interface for QC. The eQASM
aims to provide a standardised set of instructions that can be used across various kinds
of quantum hardware, making it simpler for researchers to programme QC and create
new quantum algorithms. They have used a case study of quantum factorization, a sig-
nificant issue for quantum computing, to show the effectiveness of eQASM. They use
eQASM to implement the Shor’s algorithm and execute it on a superconducting quan-
tum processor to factorise a 15-bit number. In comparison to the conventional method
of directly programming the quantum processor, the findings demonstrate that eQASM
offers a significant reduction in programming effort and a 20% increase in execution
time.
A.B. de Avila et al. [4] in year 2020 provided an overview on advanced quantum com-
puting simulators with an emphasis on the DGM simulator’s features, improvements,
and optimisations. They used a case study of quantum factorization, a significant issue
for quantum computing, to show the DGM simulator’s efficacy. The DGM simulator is
an alternative to work on quantum factorization and other quantum computing prob-
lems. In order to solve the quantum factorization problem, Lai et al. [19] proposed
suggests two different kinds of dynamic quantum state secret-sharing protocols based
on tensor network states. Tensor network states, entanglement, local operations, and
classical communication (LOCC) protocols are some of the protocols used. It offers ad-
vancements in quantum computing, which give participants a safe and effective method
to share a secret quantum state and dynamically update it throughout the factorization
process.
Mousavi et al. [21] proposed a method for lowering the distributed quantum fac-
torization circuit’s price using controlled-NOT gate (CNOT gate), a particular kind
of quantum gate that is a key component of quantum computations. In order to re-
duce the number of gates needed, the suggested approach optimises the placement
of CNOT gates in the distributed quantum factorization circuit. They demonstrate
that their method can greatly reduce the number of gates needed for the circuit when
compared to existing approaches by using a genetic algorithm to look for the ideal
gate placement. They use simulation experiments to evaluate the effectiveness of their
method, and the results demonstrate that it significantly reduces the number of gates
needed for the circuit.
Hui-Chao et al. [29] in 2022 proposed a method to optimize the Grover algorithm
circuit for integer factorization and analyzes the effect of noise on the circuit perfor-
mance. The technique was based on using gate merging and parallelization methods to
cut down on the overall number of gates needed for the circuit. In comparison to cur-
rent circuits for integer factorization using the Grover algorithm, the optimised circuit
has a lower gate count and a smaller depth. Due to the decreased amount of gates,
which are sources of errors in quantum computations, the execution time and error
rate are speed up.
Manish [18] in year 2022 suggested a new hybrid scheme that combines the Nieder-
reiter and McEliece cryptosystems in an effort to get around both systems’ drawbacks
and produce a more effective and safe post-quantum cryptographic scheme. Amaricci
et al. [3] in 2022 presented a new software package for exact diagonalization of quantum
impurity models. The exact diagonalization algorithm developed by Lanczos forms the
foundation of the Edipack software package’s architecture and execution. The pack-
age is written in C++ and contains several parallelization methods, including message
passing interface (MPI) and domain decomposition, to enhance performance on con-

8 Mandeep Kumar, Bhaskar Mondal*
temporary parallel computing architectures. Edipack is a useful tool for analysing the
behaviour of highly correlated electron systems, which could help to advance quantum
computing techniques like quantm factorization.
Pan et al. [35] 2022, suggested a novel method for solving optimisation issues that
is motivated by the idea of quantum coupling. This method to address the issue of
shift planning at airports. It entails graph representation of the optimisation issue,
with nodes denoting individual shifts and edges denoting dependencies among shifts.
The ground state of this quantum state is then discovered by converting the graph
into a quantum state using the methods of quantum adiabatic evolution and quan-
tum annealer. Quantum entanglement-inspired hard constraint handling, a technique
used in the optimisation process, includes adding a penalty term to the optimisation
function. The authors demonstrate that their method finds high-quality solutions at a
significantly lower computational cost than the conventional method.
Mohammed et al. [34] in the year 2022 suggested algorithm includes encoding
the input expressions into a quantum state using a quantum circuit, then performing
logical operations on the quantum state using a series of quantum gates. The solution
to the logical equality puzzle is then determined by measuring the circuit’s output. The
logical equivalence problem is closely linked to the factorization problem. Factorization
involves determining a number’s prime factorization, which is analogous to a logical
expression that involves multiplication and division.
Fei et al. [11] proposed a quantum algorithm based on the quantum singular value
transformation (QSVT) for state estimation in microgrids. A simulation of a microgrid
system made up of numerous distributed energy resources (DERs), including solar
panels and wind turbines, was used to verify the algorithm. The simulation results
demonstrate that, in terms of accuracy and speed, the suggested quantum algorithm
performs better than the conventional state estimation algorithms.
Dey et .al. [10] proposed a new post-quantum signcryption scheme, the isogeny-
based cryptography to establish secure communication channels. The suggested scheme
creates a shared secret key between the sender and receiver using the supersingular
isogeny Diffie-Hellman (SIDH) protocol. The communication is then encrypted with
the help of this key, which also creates a digital signature. Based on the difficulty of the
computational problem of finding isogenies between elliptic curves, the suggested sign-
cryption scheme is shown to be resistant to quantum attacks. Post-quantum schemes
like post-quantum signcryption scheme may be required to protect the security of sensi-
tive data, including factorization-related issues, if QC become strong enough to defeat
conventional cryptographic protocols like RSA and ECC. The estimation of value func-
tions, a crucial step in many reinforcement learning methods, was addressed by Yi-Pei
et al. [20] in year 2022 using a quantum reinforcement learning (QRL) approach. Com-
pared to traditional algorithms, the technique can estimate the value function exponen-
tially faster. In addition, the writers offer a quantum-enhanced Q-learning algorithm
that combines the QRL technique and the Q-learning algorithm.

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 9
Table 1: A survey on Quantum factorization
Author &
Year
Algorithm Method Advancement Limitation List of Test Threat to
Shor [27]
1994
Shor’s Algo-
rithm
Integer Fac-
torization
Polynomial-time algo-
rithms for prime fac-
torization and discrete
logarithms on a QC
Difficult Implementa-
tion for bigger numbers
−RSA
Burke et al.
[9] 2008
Shor’ Algo-
rithm
Iterative
Shor’s
Proving linear conver-
gence in 2-dimensional
case
Available number of
qubits
Exact Line
Search & Convex
quadratic
Factorisation
problem based
Cryptographic
methods
Shepherd
et al. [26]
2009
IQP Quantum
Computation
Temporally unstruc-
tured quantum compu-
tation
Difficult to verify QUATH [2], P-
hard [7]
−
Aaronson
et al. [1]
2011
Bason Sam-
pling
PH infinite
or approx.
counting
The computational
complexity of linear
optics
Difficult to verify PH infinite count-
ing
Breaks as-
sumption of
PH infinite
Hamdi et
al. [15]
2014
General Num-
ber Field Sieve
General
Number Field
Sieve
Most efficient algo-
rithms, cannot factor
integers which are 1000
decimal digits
Calculation speed of
classical computers
Comparsion with
Shor’s Algorithm
RSA
Ziyuan et
al. [16]
2018
Privacy-
preserving
location-based
services query
scheme against
quantum at-
tacks
LWE-
based key-
homomorphic
pseudo ran-
dom function
−Protection of location
based service data from
cloud server
−Location based
service
X. Fu et al.
[13] 2019
eQASM QC Proposes eQASM, that
is translatable from
QASM, and supports
comprehensive quan-
tum program flow
control
Compile-time optimi-
sations limited
Validated by
performing sev-
eral experiments
on a two-qubit
quantum pro-
cessor, eg. Rabi
oscillation
−

10 Mandeep Kumar, Bhaskar Mondal*
Table 1: A survey on Quantum factorization
Author &
Year
Algorithm Method Advancement Limitation List of Test Threat to
A.B. de
Avila et al.
[4] 2020
Extension to
Distributed
Geometric Ma-
chine (DGM)
LIQUi∥’s sim-
ulator
Six quantum sim-
ulators are studied
and speedups with
LIQUi∥’s simulator
Provides a novel in-
sight of discrete con-
strained optimization
Speeds of Shor’s
Algorithms
−
Lai et al.
[19] 2021
Affleck
Kennedy
Lieb Tasaki
(AKLT) model
Affleck
Kennedy
Lieb Tasaki
(AKLT)
model, AKLT
model
Proposed two (n, n)
threshold secret shar-
ing schemes for quan-
tum states
Mostly theoretical − −
Mousavi
et al. [21]
2021
Unnamed
(Shor’ Algo-
rithm Circuit)
CNOT and
single qubit
gates
Discusses distributed
quantum computing
as a system composed
of a number of small-
capacity quantum
circuits
Requires that one
qubit be teleported to
the other partition,
which is a costly pro-
tocol
Factorisation of
15
Factorisation
Problem
Hui-Chao
et al. [29]
2022
Efficient
Grover
IBM Q quan-
tum cloud
platform
Influence of different
noises on the circuit
under different itera-
tions was explored
Noise in quantum cir-
cuits
No Noise Exper-
iment, Experi-
ment with Noise
−
Kumar,
Manish
[18] 2022
Post-Quantum
Cryptography
Algorithms
Quantum
Circuits
Discussion on Quan-
tum algorithms and fu-
ture scope
Recommends millions
of qubits, not practical
currently
−RSA
Amaricci et
al. [3] 2022
EDIpack Look-up
method
Exact diagonaliza-
tion package to solve
generic quantum im-
purity problems
Exponential growth of
the Hilbert space is the
bottleneck of ED calcu-
lations
Lanczos or P-
Arpack algorithm
−
Pan et al.
[35] 2022
QEI-GA Quantum
Computation
Leverage the hard
constraint handling
strength of quantum
entanglement for opti-
mization
It cannot handle in-
equality constraints
Robustness Test −

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 11
Table 1: A survey on Quantum factorization
Author &
Year
Algorithm Method Advancement Limitation List of Test Threat to
Mohammed
et al. [34]
2022
Quantum Logi-
cal Equivalence
Verification
IBM Q Ex-
perience Sim-
ulator
Combination of Quan-
tum Walk and Grover’s
Algorithm
Number of qubit avail-
able
Logical equiva-
lent verification
test
−
Fei et al.
[11] 2022
GQSE
and Pre-
conditioned
quantum linear
solver
Quantum
Computation
Resolving the ill-
conditioned matrix
issue in GQSE
Number of qubit Correctness of
GQSE, PQLS
and EQSE
−
Dey et .al.
[10] 2022
Isogeny based
signcryption
scheme
SimS [12] and
SeaSign
Post-Quantum Cryp-
tographic Algorithm
Not practical currently IND −CCA and
EUF −CMA se-
curity checks
−
Yi-Pei et
al. [20]
2022
Quantum Re-
inforcement Al-
gorithm
Quantum
Computation
Combination of Quan-
tum Walk and Grover’s
Algorithm
Number of qubit avail-
able can increase the
accuracy
− −
Runge et
al. [25]
2022
Deutsch–Jozsa
algorithm
Deutsch-like
algorithm
Two quantum-like al-
gorithms which exploit
classical entanglement
(i.e., nonseparability)
of elastic waves
− − −

12 Mandeep Kumar, Bhaskar Mondal*
Runge et al. [25] 2022 used conventional acoustic qubit analogues to implement a
two-bit controlled-NOT (CNOT) gate. They suggested a brand-new category of "qubit-
analogue" conventional acoustic qubit that can mimic the behaviour of a quantum bit
(qubit) in specific circumstances. They demonstrated through numerical models that,
in the presence of specific kinds of noise, the behaviour of their qubit-analogues closely
resembles that of a genuine quantum bit. The table 1 provide an overview of different
methods related to factorization along with their limitation and threat.
Willsch et al. [32] have shown in their study that Shor’s method performs well
while working with numbers that exceed the processing capacity of both existing and
upcoming quantum hardware when using large GPU-based supercomputers. The first
step of the inquiry is to examine Shor’s original factoring algorithm. The actual re-
sults show that the average success probability exceeds 50%, even though theoretical
constraints indicate success rates in the range of 3–4%. This divergence from theoret-
ical predictions is due to a large proportion of "lucky" events, reflecting instances of
successful factorization even in the lack of met necessary criteria.
Additionally, Skosana et al. [28] present a proof-of-concept demonstration showcas-
ing a quantum order-finding algorithm’s application in factoring the integer 21. The
implementation involves the utilization of a compiled version of the quantum phase
estimation routine, thereby extending and enhancing the capabilities established in a
preceding demonstration.
4 Factorization problem in Quantum Computation
The challenges for which classical computer has had difficulty finding a solution are
better solved by quantum computing. The two algorithms that best demonstrate the
application of quantum computing are Shor’s [27] and Grover’s [14]. Shor’s algorithm
is a powerful tool for factorizing large numbers, and has significant implications for
cryptography, as many popular encryption methods rely on the difficulty of factorizing
large numbers. The key advantage of Shor’s algorithm is that it can efficiently solve the
period-finding problem using quantum parallelism, which allows it to find the period
much faster than classical algorithms. A crucial aspect of Shor’s algorithm, a quantum
method for effectively tackling the factorization of huge composite numbers, is the
period-finding problem.
4.1 Period Finding
The goal of the period finding is to determine the smallest positive number rsuch that
the function repeats its values after riterations. The function in question is a modular
exponential function, which converts a large composite number Nto its power modulo
another integer a(where a∈ {0,1,...,N −1}), and is used in Shor’s algorithm. The
period function is set to be a mode function shown as eq. (15):
f(x) = axmod N(15)
where, x∈(0,1,...,2L)and N2≤2L>2N2.
The lowest positive even integer such that the group of the number ais finite is
called the number rwhich is the smallest finite order that the number amay have.

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 13
If xis set to 0, this relationship still exists for the same number r. Thus we have the
relation as:
armod N≡a0mod N(16)
armod N≡1 mod N(17)
Hence,
ar−1 = (ar
2+ 1)(ar
2−1) ≡0 mod N(18)
Applying r= 0 in eq. (18), satisfies the eq. (18) but 0is not the expected result.
Therefore another non-trivial value of ris required to satisfy the eq. (18), can be
obtained from the eq. eq. (19), and eq. (20):
N1= gcd(ar
2+ 1, N )(19)
N2= gcd(ar
2+ 1, N )(20)
Hence we have the following result:
N=N1∗N2(21)
4.2 Algorithm
Shor’s algorithm is a quantum method for effectively factoring large composite numbers
into their prime components has significant ramifications for cryptography and the
creation of safe encryption techniques. The steps of Shor’s algorithm are as follows:
1 Input: A composite number nand integer pwhere n2≤2p<2n2.
2 Check: For prime, even, or integer powers of prime numbers, Shor’s algorithm is not
applicable. This conduction can be checked using classical computers.
3 Create quantum registers: Create quantum registers Rand apply the two equal par-
tition (R1, R2), the state of the register is |R1, R2⟩.
–To store the value p−1, register R1must have an adequate number of qubits.
–To store the value n−1, register R2must have an adequate number of qubits.
4 Load: A superposition of all numbers from 0to (p−1) with equal weights is loaded
into R1, and R2with all 0. And the state |R1, R2⟩after the load is:
1
√p
p−1
X
a=0 |a, 0⟩(22)
5 QFT: Select a random number ibetween 1and n−1co-prime to n, and on quantum
state of iand apply QFT:
–Transform iamod nfor each value stored in R1and store the result in R2.
–QC calculate i|a⟩mod n, the state |a⟩is superposition state created in step (4
Load).
–State after QFT is:
1
√p
p−1
X
a=0 a, iamod n(23)

14 Mandeep Kumar, Bhaskar Mondal*
6 Measurement: Measuring R2gives the value of kand also collapse the R1into the
superposition state for each value 0to (p−1) such that:
iamod n=k(24)
, and hence the state of Ris:
1
p||A|| X
a′=a′∈Aa′, k(25)
where Ais the set of a’s and ||A|| is number of elements. Applying QFT on state
|a⟩gives:
|a⟩=1
√p
p−1
X
c=0 |c⟩ ∗ e
2πiac
p(26)
The final state of register Rafter the QFT is given by:
1
p||A|| X
a′∈A
1
√p
p−1
X
c=0 |c, k⟩ ∗ e
2πia′c
p(27)
7 Order Finding: The state of the register R1provide the value mon measurement,
where mhas very high probability of being multiple of p
r. This mis used to calculate
the desired period r.
8 Classical Computation: On the classical computer post-processing is required to cal-
culate the period rusing mand p. The following steps are performed based on the
knowledge we have:
–The value mhas a higher probability of being in the =y×p
r, where yis an
integer
–Calculate the rational approximation using the relation m
pby taking floating
point division, where the denominator should be less than equal to p.
–If the selected denominator is the candidate of rand it is odd, if either doubling
the candidate, it leads to a value less than or equal to p.
9 Output: Once the ris obtained, the prime factors of Nwill be obtained as gcd(ir
2+
1, n) and gcd(ir
2−1, n) as shown in the section 4.1. If it fails to give the result, go
to step 3and repeat all steps.
If Shor’s algorithm fails to provide factors of the Nthere are a few reasons for it. The
measurement result of QFT in step 6 could be 0; hence step 8becomes impossible to
solve. Sometimes the result of the factorization algorithm will be 1and Nis not useful
either; hence in the failed case, the algorithm jumps to step 3.
5 Results and Analysis
The Quantum phase estimation refers to section 2.3 is used to get the solution of Shor’s
algorithm. This phased estimation is applied on the unitary operator Ugives:
U|j⟩ ≡ |aj mod N⟩(28)
Let’s start with state |1⟩; after each subsequent application of U, the state in the register
increases by amod N. Finally, after rsubsequent applications returns the state |1⟩.

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 15
Taking a= 3 and N= 35,the following result has been observed after applying U,r
times on state |1⟩:
U|1⟩=|3⟩
U2|1⟩=|9⟩
U3|1⟩=|27⟩
.
.
.
U(r−1) |1⟩=|12⟩
Ur|1⟩=|1⟩
Similarly a superposition of state |v0⟩in this order would be the eigenstate of
unitary operator Ugiven as:
|v0⟩=1
√r
r−1
X
k=0 akmod NE(29)
Using the result of unitary operations shown above, assign the values of aand Ninto
the eq. 29, the result will be:
|v0⟩=1
√12 (|1⟩+|3⟩+|9⟩ ·· · +|4⟩+|12⟩)
U|v0⟩=1
√12 (U|1⟩+U|3⟩+U|9⟩ ·· · +U|4⟩+|12⟩)
=1
√12 (|3⟩+|9⟩+|27⟩ ·· · +|12⟩+|1⟩)
=|1⟩
Similarly, the eigenstate with different phases for all the computational basis for
the kth state is as follows:
|v1⟩=1
√r
r−1
X
k=0
e−2πik
rakmod NE(30)
U|v1⟩=e2πi
r|v1⟩(31)
Let’s replace the value of aand Nin the eq. 30 and apply Uwe get the similar result
as in the eq. 31. The result is:
|v1⟩=1
√12 (|1⟩+e−t2πi
12 |3⟩+e−4πi
12 |9⟩ ·· · +e−20πi
12 |4⟩+e−22πi
12 |12⟩
U|v1⟩=1
√12 (|3⟩+e−2πi
12 |9⟩+e−4πi
12 |27⟩ ·· · +e−20πi
12 |12⟩+e−22πi
12 |1⟩)
U|v1⟩=e2πi
12 ·1
√12 (e−2πi
12 |3⟩+e−4πi
12 |9⟩+e−6πi
12 |27⟩ ·· · +e−22πi
12 |12⟩+e−24πi
12 |1⟩)

16 Mandeep Kumar, Bhaskar Mondal*
U|v1⟩=e2πi
12 |v1⟩(32)
As a result, r= 12 appears in the denominator of the desired phase in eq. (32).
To guarantee that the phase differences between the r computational basis states are
equivalent, the rmust be included. To further generalise this, add an integer, z, to the
phase difference, and the resulting eigenvalue is:
|vz⟩=1
√r
r−1
X
k=0
e−2πizk
rakmod NE
U|vz⟩=e2πiz
r|vz⟩
(33)
Given that the computational basis state |1⟩is a superposition of these eigenstates,
perform Quantum Phase Estimation (QPE)on Uusing the state |1⟩, provide the phase:
ϕ=s
r(34)
Where the random number zis between 0and r−1. In order to get r, apply the
continuing fractions technique to ϕ. Figure 2 is the circuit schematic, which makes use
of the Qiskit qubit ordering convention:
|0⟩H⊗nQF T †e2πiz
r|Vz⟩
|1⟩U20U21U2(n−1) U2(n−2)
Fig. 2 Quantum Phase Estimation (QPE) Circuit
The factor of 35 is found using the equation (35) that is 5and 7, and the table 2
provides the register reading and the corresponding phase value of each round to find
the factor. When the phase value becomes one the desired answer will be achieved.
Here the register value for phase 1is 12, and we will stop the process and the final
result will be the desired factor.
gcd(ar
2±1, N )(35)

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 17
Table 2 Phase reading for each register reading
Round Register Reading Phase
1 0 0.00
2 12 0.86
3 24 1.00
6 Research discussion
Shor’s algorithm is a quantum algorithm first introduced by mathematician Peter Shor
in 1994. It is a quantum-based algorithm that is used to solve the prime factorization
problem. The prime factorization problem is determining the prime numbers multiplied
to obtain a given composite number. In 1994, Peter Shor showed that a QC could solve
this problem exponentially faster than any classical algorithm.
The prime factorization problem is important for many cryptographic applications,
such as RSA, which is used for secure communication on the Internet. Therefore it has
the potential to break many of the encryption schemes that are used to protect sensitive
information.
The heart of Shor’s algorithm is quantum phase estimation, a quantum-mechanical
technique used to estimate the phase of a unitary operation. This phase is then used to
obtain the period of a certain function, which can then be used to factorize the given
number.
Quantum phase estimation is performed using the quantum state of a system of
qubits. The quantum state of a qubit is described by a complex vector in a two-
dimensional Hilbert space. This vector can be represented as a superposition of two
basis states, 0 and 1. The phase of a unitary operation is then estimated by transforming
the initial state of the qubits into an eigenstate of the unitary operation.
The results of the quantum phase estimation are then used to determine the period
of a certain function. The period of this function is used to factorize the given number.
The number is first transformed into a quantum state, and then used to perform the
quantum phase estimation.
In order to perform the factorization, the number is first transformed into a quan-
tum state using the QFT. The QFT is a quantum algorithm that transforms the
quantum state of a system of qubits into the Fourier space. The period of the function
is then estimated using the quantum phase estimation.
The results of Shor’s algorithm have been verified in experimental studies using
small-scale QC. These experiments have demonstrated the exponential speedup pos-
sible with a QC compared to classical algorithms. This has theoretical and practical
implications for research and technology.
The emergence of new, quantum-resistant encryption algorithms has been prompted
by quantum factorization, and post-quantum cryptography is a major area of ongoing
study. However, the consequences of quantum factorization go beyond cryptography
and could impact mathematical theory, scientific research, and computational complex-
ity theory. Shor’s algorithm and the development of quantum computing have opened
up new perspectives on the limitations of conventional computing and helped identify
new classes of problems like (Integer factorization, Optimization problems, Machine
learning, Database search, etc.) that QC can solve.
It is founded on a number-theoretic result known as the RSA assumption that
factoring large numbers is a computationally challenging issue. The RSA assumption

18 Mandeep Kumar, Bhaskar Mondal*
would be challenged, and there would be repercussions for the more significant subject
of number theory if a practical QC could factor large numbers using Shor’s algorithm.
Shor’s algorithm offers a potent factorization tool with significant uses in chemistry
and number theory. It could be used, for instance, to factor large numbers that come
up when studying prime numbers or to model the behaviour of molecules and other
substances. The potential of Shor’s algorithm has sparked study and development work
on building usable QC, which have the potential to revolutionise many fields of science
and technology. Although creating practical QC is still difficult, the promise of these
devices has spurred substantial investment in the field and led to the development of
novel quantum computation tools and algorithms. In conclusion, Shor’s algorithm has
possible effects on security and cryptography.
Qubits are used by Shor’s algorithm to carry out the factorization procedure. Qubits
are used to conduct multiple calculations at once because they can exist in a super-
position of states, in contrast to classical bits, which can only be in a state of either
0 or 1. These calculations are accomplished by employing a quantum circuit that car-
ries out a number of processes, such as the application of a QFT, which permits the
superposition of various states. In order to determine the phase of the function, the
circuit also has a modular exponentiation operation that is applied to a superposition
of states.
Shor’s algorithm requires a large number of qubits, which are difficult to maintain
due to their tendency for decoherence. This is one of the major difficulties in implement-
ing Shor’s algorithm. As a result, creating useful QC has become extremely difficult.
Shor’s algorithm can now be put into practice more easily due to new developments in
quantum technology, such as the development of error correction methods.
7 Conclusion
Shor’s algorithm, a quantum-based approach, uses quantum phase estimation and pe-
riod finding on a system of qubits to solve the prime factorization problem much faster
than classical techniques. The algorithm can evaluate functions in parallel thanks to
this technique, which makes factoring big numbers extremely effective. The 6-qubit
quantum system under consideration in the simulation study was able to factor the
composite number 35 with a registered count of 24 for phase value 1, which resulted
in factors 5and 7. Shor’s algorithm’s exponential speedup in experiments presents a
serious threat to classical cryptosystems, especially when it comes to RSA key security
being compromised as the number of qubits in quantum computers rises. As demon-
strated by developing post-quantum cryptography studies [6] and [23], safeguarding
sensitive data today depends on the creation of new cryptographic algorithms immune
to quantum technology. Moreover, the potential benefits of quantum factorization go
beyond encryption and promise more effective resource allocation in scheduling and
route planning, among other optimization problems. The urgent need to transition to
post-quantum cryptographic algorithms is made clear by the fact that factorization-
based encryption techniques are now in danger of being compromised by quantum
developments.

Study on Implementation of Shor’s Factorization Algorithm on Quantum Computer 19
Declarations
Ethical Approval No human and/ or animal studies have been presented in the manuscript.
Hence, no ethical approval is needed.
Competing interests The authors do not have any financial or personal conflict of
interest to disclose related to this manuscript.
Authors’ contributions Mandeep Kumar drafted the main manuscript text, Dr. Bhaskar
Mondal prepared all the figures and did the proofreading and arranging the systematic
review throughout. All authors equally contributed to the scientific work and reviewed
the manuscript.
Funding The authors hereby declare that there was no full or partial financial support
from any organization.
Availability of data and materials No data was generated during the research to dis-
close. No Code is available to share.
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