Программные продукты и системы, 2024, том 37, № 1

 ПРОГРАММНЫЕ ПРОДУКТЫ И СИСТЕМЫ
Научный журнал 

2024. Т. 37. № 1
DOI: 10.15827/0236-235X.142

Главный редактор 

Г.И. САВИН, академик РАН

Научные редакторы номера:

Н.А. СЕМЕНОВ, д.т.н., профессор

А.Н. СОТНИКОВ, д.ф.-м.н., профессор

Издатель НИИ «Центрпрограммсистем»

(г. Тверь, Россия)

Учредитель В.П. Куприянов

Журнал зарегистрирован в Роскомнадзоре

3 марта 2020 г.

Регистрационное свидетельство ПИ № ФС 77-77843

Подписной индекс в каталоге

Урал-Пресс 70799

ISSN 0236-235X (печатн.)
ISSN 2311-2735 (онлайн)

РЕДАКЦИОННАЯ КОЛЛЕГИЯ

Семенов Н.А. – заместитель главного редактора, д.т.н., профессор Тверского государственного технического 
университета (г. Тверь, Россия)
Сотников А.Н. – заместитель главного редактора, д.ф.-м.н., профессор, заместитель директора 
Межведомственного суперкомпьютерного центра РАН (г. Москва, Россия)
Афанасьев А.П. – д.ф.-м.н., профессор Московского физико-технического института (технического университета), 
заведующий Центром распределенных вычислений Института проблем передачи информации РАН (г. Москва, Россия)
Баламетов А.Б. – д.т.н., профессор Азербайджанского научно-исследовательского и проектно-изыскательского института 
энергетики (г. Баку, Азербайджан)
Борисов В.В. – д.т.н., профессор филиала Национального исследовательского университета «МЭИ» 
в г. Смоленске (г. Смоленск, Россия)
Голенков В.В. – д.т.н., профессор Белорусского государственного университета информатики и радиоэлектроники 
(г. Минск, Беларусь)
Елизаров А.М. – д.ф.-м.н., профессор Института математики и механики им. Н.И. Лобачевского Казанского федерального 
университета (г. Казань, Россия)
Еремеев А.П. – д.т.н., профессор Национального исследовательского университета «МЭИ» (г. Москва, Россия)
Кузнецов О.П. – д.т.н., профессор Института проблем управления РАН (г. Москва, Россия)
Мамросенко К.А. – к.т.н., доцент Московского авиационного института (национального исследовательского университета), 
руководитель Центра визуализации и спутниковых информационных технологий НИИСИ РАН (г. Москва, Россия)
Палюх Б.В. – д.т.н., профессор Тверского государственного технического университета (г. Тверь, Россия)
Сулейманов Д.Ш. – академик АН Республики Татарстан, д.т.н., профессор Казанского государственного технического 
университета (г. Казань, Республика Татарстан, Россия)
Татарникова Т.М. – д.т.н., доцент, профессор Санкт-Петербургского государственного 
электротехнического университета «ЛЭТИ» им. В.И. Ульянова (Ленина) (г. Санкт-Петербург, Россия)
Ульянов С.В. – д.ф.-м.н., профессор, ведущий научный сотрудник Объединенного института ядерных исследований 
(г. Дубна, Россия)
Хорошевский В.Ф. – д.т.н., профессор Московского физико-технического института (технического университета) 
(г. Москва, Россия)
Шабанов Б.М. – д.т.н., чл.-корр. РАН, директор Межведомственного суперкомпьютерного центра РАН (г. Москва, Россия)
Язенин А.В. – д.ф.-м.н., профессор Тверского государственного университета (г. Тверь, Россия)

АССОЦИИРОВАННЫЕ ЧЛЕНЫ РЕДАКЦИИ

Национальный исследовательский университет «МЭИ», г. Москва, Россия
Технологический институт Южного федерального университета, г. Таганрог, Россия
Тверской государственный технический университет, г. Тверь, Россия

АДРЕС ИЗДАТЕЛЯ И РЕДАКЦИИ 
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 SOFTWARE & SYSTEMS 
Research journal
2024, vol. 37, no. 1
DOI: 10.15827/0236-235X.142

Editor-in-chief 
G.I. SAVIN, Academician of RAS

Science editors of the issue:

N.A. Semenov, Dr.Sc. (Engineering), Professor
A.N. Sotnikov, Dr.Sc. (Physics and Mathematics), Professor

Publisher Research Institute 

CENTERPROGRAMSYSTEM (Tver, Russian Federation)

Founder V.P. Kupriyanov

The journal is registered with the Federal Service 

for Supervision of Communications, Information Technology 

and Mass Communications (Roskomnadzor) 

March 3rd, 2020

Registration certificate ПИ № ФС 77-77843

ISSN 0236-235X (print)
ISSN 2311-2735 (online)

EDITORIAL BOARD

Semenov N.A. – Deputy Editor-in-Chief, Dr.Sc. (Engineering), Professor of the Tver State Technical University
(Tver, Russian Federation)
Sotnikov A.N. – Deputy Editor-in-Chief, Dr.Sc. (Physics and Mathematics), Professor, Deputy Director
of the Joint Supercomputer Center of the Russian Academy of Sciences (Moscow, Russian Federation)
Afanasiev A.P. – Dr.Sc. (Physics and Mathematics), Professor of the Moscow Institute of Physics and Technology, 
Head of Centre for Distributed Computing of Institute for Information Transmission Problems 
(Moscow, Russian Federation)
Balametov A.B. – Dr.Sc. (Engineering), Professor of the Azerbaijan Scientific-Research & Design-Prospecting Power Engineering 
Institute (Baku, Azerbaijan)
Borisov V.V. – Dr.Sc. (Engineering), Professor of the MPEI Branch in Smolensk (Smolensk, Russian Federation)
Golenkov V.V. – Dr.Sc. (Engineering), Professor of the Belarusian State University of Informatics and Radioelectronics 
(Minsk, Republic of Belarus)
Elizarov A.M. – Dr.Sc. (Physics and Mathematics), Professor of the N.I. Lobachevsky Institute of Mathematics 
and Mechanics of the Kazan Federal University (Kazan, Russian Federation)
Eremeev A.P. – Dr.Sc. (Engineering), Professor of the National Research University Moscow Power Engineering 
Institute (Moscow, Russian Federation)
Kuznetsov O.P. – Dr.Sc. (Engineering), Professor of the Institute of Control Sciences of the Russian Academy 
of Sciences (Moscow, Russian Federation)
Mamrosenko K.A. – Ph.D. (Engineering), Associate Professor of the Moscow Aviation Institute (National Research
University), Head of the Center of Visualization and Satellite Information Technologies SRISA RAS 
(Moscow, Russian Federation)
Palyukh B.V. – Dr.Sc. (Engineering), Professor of the Tver State Technical University (Tver, Russian Federation)
Suleimanov D.Sh. – Academician of TAS, Dr.Sc. (Engineering), Professor of the Kazan State Technical University
(Kazan, Republic of Tatarstan, Russian Federation)
Tatarnikova T.M. – Dr.Sc. (Engineering), Associate Professor, Professor of the St. Petersburg Electrotechnical 
University LETI (St. Petersburg, Russian Federation)
Ulyanov S.V. – Dr.Sc. (Physics and Mathematics), Professor of the Dubna International University for Nature, 
Society and Man (Dubna, Russian Federation)
Khoroshevsky V.F. – Dr.Sc. (Engineering), Professor of the Moscow Institute of Physics and Technology
(Moscow, Russian Federation)
Shabanov B.M. – Dr.Sc. (Engineering), Corresponding Member of the RAS, Director of the Joint Supercomputer Center 
of the Russian Academy of Sciences (Moscow, Russian Federation)
Yazenin A.V. – Dr.Sc. (Physics and Mathematics), Professor of the Tver State University (Tver, Russian Federation)

ASSOCIATED EDITORIAL BOARD MEMBERS

National Research University Moscow Power Engineering Institute, Moscow, Russian Federation
Technology Institute at Southern Federal University, Taganrog, Russian Federation
Tver State Technical University, Tver, Russian Federation

EDITORIAL BOARD AND PUBLISHER OFFICE ADDRESS 
Nikolay Korytkov Ave, 3а, Tver, 170024, Russian Federation
Phone: (482-2) 39-91-49  Fax: (482-2) 39-91-00
E-mail: red@cps.tver.ru
Website: www.swsys.ru

Release date 16.03.2024

Printed in printing-office Faktor i K

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Published quarterly. 37th year of publication

Format 6084 1/8. Circulation 1000 copies

Prod. order № 4. Wordage 132 pages. Price 550,00 rub. 

Вниманию авторов

Журнал «Программные продукты и системы» публикует материалы научного и научно-практического 

характера по новым информационным технологиям, результаты академических и отраслевых исследований 
в области использования средств вычислительной техники. Практикуются выпуски тематических номеров 
по искусственному интеллекту, системам автоматизированного проектирования, по технологиям разработки 
программных средств и системам защиты, а также специализированные выпуски, посвященные научным 
исследованиям и разработкам отдельных вузов, НИИ, научных организаций. 

Журнал «Программные продукты и системы» внесен в Перечень ведущих рецензируемых научных жур
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форме Web of Science в виде базы данных RSCI.

Автор статьи отвечает за подбор, оригинальность и точность приводимого фактического материала. 

При перепечатке ссылка на журнал обязательна. Статьи публикуются бесплатно.

Условия публикации

К рассмотрению принимаются оригинальные материалы, отвечающие редакционным требованиям и со
ответствующие тематике журнала. Группы научных специальностей: 

1.2. Компьютерные науки и информатика 
1.2.1. Искусственный интеллект и машинное обучение (физико-математические науки). 
1.2.2. Математическое моделирование, численные методы и комплексы программ (физико-математиче
ские науки, технические науки)

2.3. Информационные технологии и телекоммуникации
2.3.1. Системный анализ, управление и обработка информации, статистика (технические науки, физико
математические науки). 

2.3.2. Вычислительные системы и их элементы (технические науки).
2.3.3. Автоматизация и управление технологическими процессами и производствами (технические 

науки). 

2.3.5. Математическое и программное обеспечение вычислительных систем, комплексов и компьютер
ных сетей (технические науки, физико-математические науки).

2.3.6. Методы и системы защиты информации (технические науки, физико-математические науки).
2.3.7. Компьютерное моделирование и автоматизация (технические науки, физико-математические науки).
2.3.8. Информатика и информационные процессы (технические науки).
Работа представляется в электронном виде в формате Word. Объем статьи вместе с иллюстрациями – не 

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должен включать не менее 10 пунктов.

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Порядок рецензирования

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нала, эксперты из числа крупных специалистов в области информатики и вычислительной техники ведущих 
вузов страны, а также ученые и специалисты НИИСИ РАН, МСЦ РАН (г. Москва) и НИИ «Центрпрограммсистем» (г. Тверь).

Редакция журнала «Программные продукты и системы» в своей работе руководствуется сводом правил 

Кодекса этики научных публикаций, разработанным и утвержденным Комитетом по этике научных публикаций (Committee on Publication Ethics – COPE).

Программные продукты и системы / Software & Systems
37(1), 2024

5

Software & Systems
doi: 10.15827/0236-235X.142.005-017
2024, 37(1), pp. 5–17

Software emulator of quantum algorithms for sophisticated simulation 

on a conventional computer

Sergey V. Ulyanov 1, 2, Viktor S. Ulyanov 3

1 Dubna State University – Institute of System Analysis 
and Management, Dubna, 141980, Russian Federation

2 Joint Institute for Nuclear Research –

Meshcheryakov Laboratory of Information Technologies, 

Dubna, 141980, Russian Federation

3 Moscow State University of Geodesy and Cartography (MIIGAiK), 

Moscow, 105064, Russian Federation

For citation
Ulyanov, S.V., Ulyanov, V.S. (2024) ‘Software emulator of quantum algorithms for sophisticated simulation on a conventional computer’, Software & Systems, 37(1), pp. 5–17 (in Russ.). doi: 10.15827/0236-235X.142.005-017
Article info
Received: 14.09.2023
After revision: 21.09.2023
Accepted: 05.10.2023

Abstract. A quantum software engineering platform includes quantum computing methods, a quantum algorithm theory 
and quantum programming. These areas develop according to a technological structure of nanotechnology development 
for hardware design of various configurations. In about 10 to 30 years we expect the appearing of an industrial quantum 
computer for real software engineering; this fact is due to overcoming a number of technological difficulties in implementing hardware, as well as the fundamental difficulty of eliminating decoherence physical phenomenon and correcting errors 
in quantum computers in near future. A key question in quantum computing is searching for quantum algorithms that 
potentially have a significant advantage and supremacy over classical algorithms for problems of practical interest. Therefore, currently, an approach is being developed to create quantum algorithm structures for quantum simulators with the 
possibility of effective implementation on classical architecture computers. This paper proposes an effective modelling 
method with information analysis of quantum search and decision-making algorithm structures in order to eliminate redundancy in practical implementation of a simulator on a classical structure computer. As an example, we demonstrate the 
method of modeling Grover's quantum search algorithm with stopping the search for a good solution based on the Shannon 
information entropy minimum principle. There are modeling examples to demonstrate the effectiveness of the developed 
approach in quantum software engineering and intelligent control robotics.
Keywords: quantum algorithm, quantum software engineering, quantum computing, quantum simulator, minimum of Shannon information entropy, termination criteria

Introduction. The history of quantum compu
ting starts around the 1980s when during the First 
Conference on the Physics of Computation Richard Feynman showed that it is not effective to simulate a quantum system evolution on a classical 
computer. An effective simulation of quantum system has a run-time in polynomial size, i.e. the computational time is smaller than a polynomial function of the problem size. Therefore, relevant simulations of quantum computers will always be larger 
in size than polynomial time. This leads to superpolynomial time simulations of quantum algorithms; these kinds of simulations have a long 
runtime for large problems. By separating the 
problems in smaller parts, we can avoid long runtime. For example, simulating Shor’s factoring algorithm on a classical computer takes super-polynomial time. The simulation of quantum algorithms is still constructive for parts of a larger 
problem and it gives us a basis for comparing experimental and theoretical results. The results from 
Shor’s algorithm might be verified by multiple factors from an algorithm outcome and hence it is 

simple to check the results from Shor’s factoring 
algorithm implemented on a quantum computer. 
It might be more complicated to check the outputs 
from future algorithms.

However, it is possible to show that Shor’s al
gorithm gives mathematically correct results. But
how can we verify that implementing Shor’s algorithm on a quantum computer coincides with its 
mathematical model? A simulation of a quantum 
algorithm on a classical computer allows comparing a quantum computer outcome with an output 
form a physically more stable classical computer. 
When developing quantum algorithms, it is interesting to check new algorithms on a classical computer. This study examines quantum algorithm 
simulation on a classical computer. The program 
code implemented on a classical computer will be 
a straight connection between the mathematical 
formulation of quantum mechanics and computational methods. A computational language includes such terms as a quantum state, a superposition and other quantum operators.

Программные продукты и системы / Software & Systems
37(1), 2024

6

Quantum algorithm general structure

The problem solved by a quantum algorithm 

(QA) can be stated in the symbolic form: 

Input
Function f: {0, 1}n → {0, 1}m.

Problem Find a certain property of function f.
A given function f is a map of one logical state 

into another, QA estimates qualitative properties 
of function f. Fig. 1 demonstrates a general circuit 
description of QA.

Three main quantum operators (as the superpo
sition, the entanglement (quantum oracle) and the 
interference) are a background of QA structure design for implementing quantum massive parallel 
computing. Therefore, they include the matrix design form of three quantum operators: superposition (Sup), entanglement (UF) and interference (Int)
(see, below Fig. 2). 

The structure of a quantum algorithmic gate 

(QAG) in Fig. 1 in a general form can be defined 
as follows:

(
)

1
,

h
n
n
m

F
QAG
Int
I
U
H
S

+




=







(1)

where I is an identity operator; symbol  denotes 
a tensor product; S is equal to I or H and depends 
on a problem description. The type of operator UF
physically describes the qualitative properties of 
function f. 

Figure 2 shows QA steps including described 

qualitative peculiarities of function f and physical 
interpretation of applied quantum operators.

The quantum circuit (Fig. 2) is a high-level de
scription of a method for composing smaller matrices using tensor and dot products in order to generate a finite QAG.

For example, Fig. 3 represents a general ap
proach to Grover’ QAG design [1].

The presented HW performs all functional 

steps of a Grover’s QSA. A termination condition
criterion is a minimum entropy-based method that 
is implemented in a digital part together with display output [2]. 

There are fast algorithms to simulate most of 

known QAs on classical computers [1] and in computational intelligence toolkit: 1) Matrix-based 
approach; 2) Model representations of quantum 
operators in fast QAs; 3) Algorithmic-based approach when matrix elements are calculated on demand; 4) Problem-oriented approach, where we 
succeeded to run Grover’s algorithm with up to 64 
and more qubits with Shannon entropy calculation
(up to 1024 without a termination condition); 
5) Quantum algorithms with a reduced number of 
operators (entanglement-free QA, and so on). 

In this article we briefly describe main blocks 

in Fig. 3: a) unified operators; b) problem-oriented 
operators; c) benchmarks of QA simulation on 
classical computers; d) quantum control algorithms based on quantum fuzzy inference (QFI) 

Fig. 1. QA general description

Input

n

m

0

0

H

H

...

x

x

S

S

...

UF

INT

Superposition
Entanglement
Interference
Output

h

h

h

h

bit

bit

bit

bit

...
...

M
E
A
S
U
R
E
M
E
N
T

...
...

Repeated k times

Fig. 2. QA general structure

Qualitative 
properties of 

function

Quantum Massive Parallel Computing

 fin
=

QC output
Classical input

Quantum Fourier  

transformation

Problem oriented operator 

Hadamard  

transformation

Coding of 
function 
properties

Qualitative 
properties of 

function

Quantum Oracle Black BOX

 initial
[ (Interference) (Quantum Oracle) ] (Superposition)

Answer
QAG design

Quantum knowledge 

base optimizer

Problem

Soft Computing 

optimizer

Программные продукты и системы / Software & Systems
37(1), 2024

7

and quantum genetic algorithm (QGA) as new 
QSA types.

Description of quantum operators: 

SW smart toolkit support

In terms of simulation, we consider the struc
ture of quantum operators as a superposition, entanglement and interference. In this case a superposition and an interference have a more complicated structure and differ from an algorithm to an 
algorithm [3–5]. We also focus on considering entanglement operators, since they have a similar 
structure for all QAs and differ only by an analyzed function [6–8].

QA superposition operators. In general form, a 

superposition operator is a combination of tensor 
products of Hadamard H operators with identity 
operator I: 

1
1
1
0
1
,
.
1
1
0
1
2

H
I




=
=




−





The superposition operator of most QAs (see 

Fig. 1) can be expressed as: 

1
1

n
m

i
i
Sp
H
S

=
=





= 
 









,

where n and m are the numbers of inputs and outputs respectively. Operator S may be Hadamard H
operator or identity operator I depending on the algorithm. Table 1 presents the number of outputs m, 
as well as the structures of corresponding superposition and interference operators for different QAs.

Table 1

Parameters of superposition and interference 

operators of main quantum algorithms

Algorithm
Superposition
m
Interference

Deutsch’s
H
I
1
H
H


Deutsch-Jozsa’s
nH
 H
1
nH
I


Grover’s
nH
 H
1
Dn
I


Simon’s
n
n
H
 I
n
n
n
H
I


Shor’s
n
n
H
 I
n

n

QFTn
I


Elements of the Walsh-Hadamard operator 

could be obtained as following:

(
)

*

/2
/2
,

1
  1,    if 
 is even,
1

1,   if 
 is odd,
2
2

i j

n

n
n
i j

i
j
H
i
j

−




=
=



−



(2)

where 
0,1, ..., 2 ,
0,1, ..., 2 .
n
n
i
j
=
=
Its elements

could be obtained by the simple replication according to the rule presented in Eq. (2). Thus, this approach greatly improves the performance of classical simulation of the Walsh–Hadamard operators.

Interference operators of main QAs. Interfer
ence operators must be selected individually for 
each algorithm according to the parameters presented in Table 1; for Grover’s algorithm they can 
be written as a block matrix:

/2
,

/2
/2

1

2

1
1
1
,
2
2

Grover
n

n
n
i j

n
n

i
j
i
j

Int
D
I
I
I

I
I

=






=

=
−

=











= − +


=









Fig. 3. Circuit and quantum gate representation of Grover’s QSA

Программные продукты и системы / Software & Systems
37(1), 2024

8

/2

,
,
1

,
,
2 n

I i
j

I i
j

−
=

=




(3)

where 
0, ..., 2
1,
0, ..., 2
1
n
n
i
j
=
−
=
− , Dn refers to

a diffusion operator: 


1
(
)

/2
,

( 1)

2

AND i
j

n
n
i j
D

=
= −
[9–12]. 

Note that with the increasing number of qubits, the 
gain coefficient becomes smaller. 

The matrix dimension increases according to 2n,

but each element can be extracted using Eq. (3) 
without allocating the entire operator matrix. In a 
certain form, the operator Dn (diffusion – inversion 
about average) in Grover’s algorithm is decomposed as follows: 

1
0
0
0

1
1
0
1
0
0
1
1
1

1
1
0
0
0
1
1
2

0
0
0
1

n

n
n

n
n
D






−










=








−
−










and can be implemented with 
( )
(log( ))
O n
O
N
=

quantum gates. In terms of efficient computation, 
it means that the form in Eq. (3) is more preferable.

Entanglement operators of main QAs. Entan
glement operators in a general form are a part 
of QA; the information about the function (being 
analyzed) is coded as an input-output relation. In the 
general approach to coding binary functions into 
corresponding entanglement gates, an arbitrary binary function is considered as: 




: 0,1
0,1
,

n
m
f
→

such that 
0
1
0
1
(
, ...,
)
(
, ...,
)
n
m
f x
x
y
y
−
−
=
. First, irre
versible function f transforms into reversible function F as following: 




: 0,1
0,1
,

m n
m n
F

+
+
→
and 

(
)
0
1
0
1

0
1
0
1
0
1

, ...,
,
, ...,

(
, ...,
,
(
, ...,
)
(
, ...,
)),

n
m

n
n
m

F x
x
y
y

x
x
f x
x
y
y

−
−

−
−
−

=

=


where  denotes addition by module 2. This transformation creates a unitary quantum operator and 
performs a similar transformation. It is possible design an entanglement operator matrix using reversible function F according to the following rule:



,
1 iff (
)
,

,
 0,..,0;1,..,1; .

B
B

B
B

F
i
j

n
m
n
m

U
F j
i

i j

+
+

=
=

 







B denotes binary coding.
A diagonal
block matrix
of the form: 

0

2
1

0

0
n

F

M

U

M
−






= 







is actually a resulted entan
glement operator. 

Each block
,
0, ..., 2
1
n

Mi
i =
−
can be ob
tained as follows:

1

0

,  iff ( , )
0,

, iff ( , )
1,

m

i
k

I
F i k
M
C
F i k

−

=

=

=  
=


(4)

and consists of m tensor products of I or C operators, where C stays for NOT operator.

Note that an entanglement operator is a sparse 

matrix and according to this property (4) it is possible to accelerate entanglement operation simulation.

Structure of QA simulation system 

in MatLab

Figure 4 shows a software system structure for 

QA simulation. 

The software system is divided into two general 

sections. The first section involves common functions. The second section involves algorithm-specific functions for implementing particular algorithms.

Common functions. The common functions in
clude:

•
superposition building blocks,

•
interference building blocks,

•
bra-ket functions,

•
measurement operators,

•
entropy calculation operators,

•
visualization functions,

•
state visualization functions,

•
operator visualization functions.

Algorithm specific functions. Algorithmic spe
cific functions include:

•
entanglement encoders,

•
problem transformers,

•
result interpreters,

•
algorithm execution scripts,

•
Deutsch algorithm execution script,

•
Deutsch Jozsa algorithm execution script,

•
Grover’s algorithm execution script,

•
Shor’s algorithm execution script,

•
quantum control algorithms as scripts.

Visualization functions. Visualization func
tions are functions that provide visualization display of quantum state vector amplitudes and quantum operator structure. 

Algorithmically-specific functions. Algorithmi
cally-specific functions provide a set of scripts for 
QA execution in a command line and tools for QA
simulation including quantum control algorithms. 
The functions of section 2 prepare the appropriate
operators of each algorithm and use common functions as operands. 

QA simulation by a command line. The exam
ple of the Grover’s algorithm script is presented at 
the link http://swsys.ru/uploaded/image/2024-1/Ulyanov.html.

Программные продукты и системы / Software & Systems
37(1), 2024

9

Other known QA can be formulated and exe
cuted by similar scripts, and using the corresponding equations presented earlier [9–11].

Simulating QA as a dynamic system. In order to 

simulate dynamic system behavior with quantum 
effects, it is possible to represent QA as a dynamic 
system in the form of a block diagram and then 
simulate its behavior in time. Figure 5 shows an 
example of a quantum circuit Simulink diagram to 
calculate the fidelity <a|a> of the quantum state 
and to calculate the density matrix |a><a| of the 
quantum state. Bra and ket functions are from the 
common library. This example demonstrates using 
common functions to simulate QA dynamics. 

In Fig. 6, the input is ket function. The ket func
tion output is provided to the first input of the matrix multiplier and as the second input of the matrix 
multiplier. The input is also provided to the bra
function. The bra function output is provided to 
the second input of the matrix multiplier and as the 
first input of the matrix multiplier. The multiplier 
output is an input state density matrix. The multiplier output is the input state fidelity.

Figure 7 shows Simulink structure of an arbi
trary QA. 

We can use such structure to simulate a number

of quantum algorithms in Matlab/Simulink environment.

Dedicated QA emulator

Developments in QA algorithmic representa
tion are also applicable for designing QA software emulators. A key point is reducing multiple 
matrix operations to vector operations and the following replacement of multiplication operations. 
This may increase emulation performance, especially for algorithms that do not require complex 
number operations, and when a quantum state 
vector has a relatively simple structure (like 
Grover’s QSA).

The developed software can simulate 4 basic

quantum algorithms, e.g. Deutsch-Jozsa, Shor’s, 
Simon’s and Grover’s. The system uses a unified 
user-friendly interface for all algorithms providing 
3D visualization of state vector dynamics and 
quantum operators.

On the QA emulator launch window, you can 

choose to create a new QA model or continue 
modeling the existing one (http://www.swsys.ru/
uploaded/image/2024-1/13.jpg).

If we choose creating a new model, then algo
rithm selection dialog starts. Here we can choose 
QA and its dimensions. 

In fact, the system may operate with up to 50 

qubits and more, but, it is better to limit the number 
of qubits to 10–11 due to visualization problems.

Fig. 4. Structure of QA simulation software

Программные продукты и системы / Software & Systems
37(1), 2024

10

Once algorithm initial parameters are set, the 

system draws an initial state vector and selects an 
algorithm structure in the system’s main window 
(Fig. 7).

The main window (Fig. 7) contains all infor
mation of the emulated quantum algorithm and enables basic operations and analysis. The form 
menu has an access to involved quantum operators 
(Fig. 8), and it is possible to modify input functions. 

QAs have reversible nature; therefore, it is pos
sible to make forward and backward algorithm 
steps by clicking on arrows; currently applied algorithm step will be highlighted on the algorithm 
diagram.

The emulator menu consists of four compo
nents: 

1. Item File provides basic operations, such as 

project save/load, and new model creation interface access.

2. Item Model provides an access to the input 

function editor.

3. Item View provides an access to operator 

matrix visualizers, including Superposition, En
tanglement and Interference operators. It is also 
possible to get 3D preview of algorithm state dynamics (Fig. 9).

4. From Help menu there is an access to the 

program documentation.

Fig. 5. Simulink diagram for simulating the arbitrary quantum algorithm

Fig. 6. Simulink diagram for simulating an arbitrary quantum algorithm

Fig. 7. Main window of QA emulator software 

(3 qubit Grover QSA)

Программные продукты и системы / Software & Systems
37(1), 2024

11

Tabbed interface in the lower part of the win
dow provides an access to Shannon entropy chart 
and to 3D representation of the state vector dynamics, as well as to usual plain representation of the 
QA state. The tabbed area size can be modified by 
dragging a divider. A click on the divider middle 
point hides the tabbed area form the screen.

Buttons in the middle part of the main window 

allow making steps by the currently parameterized 
QA. As it was mentioned above, the system can 
make forward and backward steps.

If there are enough algorithm steps, then a click 

on “!” button extracts an answer from the current 
state vector. 

Depending on QA, an appropriate result inter
pretation routine appears.

The quantum operator visualizer displays the 

structure of involved quantum operator matrices in 
plain and in 3D representations. 

If an operator consists of a tensor product of 

smaller operators, it is possible to access subblocks of tensor products. The 3D visualizer enables zoom and rotation of the charts.

The input function editor permits to automate 

the entanglement operator coding process as it was 
described earlier. For Grover’s QSA it is possible 
to code functions that have more than one positive 
output. 

Figure 10 shows the results of Grover QSA 

simulation with entropy criteria termination. Figures 11 and 12 demonstrate initial and final states 
of the developed software emulator with DeutschJozsa, Simon’s and Shor’s QAs. 

The QAG simulation result for the case of defin
ing a constant or balanced function is shown in
Figs 13, 14, correspondingly. The QAG entropy 
evaluation for both cases is also shown there. These 
results are used for QA stopping criteria below.

The coding sample of input functions and the 

corresponding 3D representation of entanglement 
operators of Deutsch-Jozsa, Simon and of Shor’s 
algorithms are presented at the link http://www.
swsys.ru/uploaded/image/2024-1/14.jpg. 

Figure 15 demonstrates Shannon entropy be
havior of simulated quantum algorithms after several algorithm iterations. It is clear that its minimum is reached on minimum uncertainty states, regardless a simulated algorithm.

QA simulation results of simulated QA are pre
sented at the link http://www.swsys.ru/uploaded/
image/2024-1/15.jpg, after a result interpreter.

We presented a design method and hardware 

implementation of a modular system for implementing Grover’s QSA. We also developed hardware design of main quantum operators for 

Fig. 8. Plain representation 
of superposition operator

Fig. 9. 3D View of 3 qubit Grover’s QSA 
state vector after two algorithm iterations

Fig. 10. Shannon entropy dynamics 

after 31 steps of Grover’s QSA