Quantum computing software solutions, technologies, evaluation and limitations: a systematic mapping study

Abstract

Quantum computing has emerged as a promising field with the potential to revolutionize information processing and storage. Understanding its current state and research trajectory is crucial. This study aims to map the current literature on quantum computing, to identify how quantum computing is being used in solving quantum software problems and improving quantum algorithms. Moreover, it examines evaluation parameters and the challenges faced in this field. A systematic mapping study was conducted using the SCOPUS database, including all papers up to December 2024. Relevant articles were selected based on inclusion criteria, and a content analysis was performed to categorize and synthesize the findings. Information was gathered from 33 primary studies. Findings show that quantum computing is mainly applied in machine learning and optimization, though classical methods still outperform it. IBM Quantum is the most used platform. Research focuses on algorithm efficiency, verification over validation, and resource optimization, but lacks standardized evaluation methods. Key limitations include hardware constraints, noise, and scalability, while challenges involve integration, error correction, and problem formulation. Quantum computing continues to face significant technological challenges, but current research lays the groundwork for future advances. More practical studies on real hardware and strategies to improve the implementation and performance of quantum algorithms are needed. Effective evaluation, verification, and validation of quantum solutions must be further investigated.

1 Introduction

The field of quantum computing is advancing at an unprecedented pace. Gaining more and more notoriety as time elapses. Considering the global quantum computing market, it was estimated to be USD 866 million dollars in 2023, USD 1.3 billion dollars in 2024, and in four years, by 2029, it is set to reach USD 5.3 billion dollars [38]. The field of quantum computing is experiencing an exponential acceleration in its development, increasing the interest of the scientific and technological communities.

Using Google Trends data [24], we can analyse the temporal progression of interest in quantum computing, as reflected in search query frequencies. Figure 1 presents a temporal graph, depicting the fluctuating interest in the term “quantum computing” on a global scale. This spans from the earliest available data point in the tool, January 1^st, 2004, to December 31^st, 2024. It is important to note that the numerical values represented in this graph are indicative of the interest relative to the peak value recorded within the specified timeframe and geographical context. For instance, a value of 50 implies that the interest in the term at that particular time is half as prevalent as its peak popularity.

The figure shows that for over a decade, from 2004 to 2016, the overall interest in quantum computing was considerably low, with few fluctuations over time. As from 2017, there is an increase in interest. Finally, we can see a sharp surge in interest in quantum computing in 2024, marking the highest interest value recorded. Since Google Trends measures relative interest, this indicates that the search for the term “Quantum Computing” at this time greatly exceeds any previous point.

Fig. 1

Interest over time, from Jan 1^st 2004 to Dec 31^st 2024, of the term “quantum computing” obtained from Google Trends

Quantum computations are meant to exceed the limitations imposed on classical computation and redefine how we process information, as well as the problem solving approaches we employ. An example is the quantum walk, which demonstrates an exponential algorithmic advantage in locating an exit node on a graph from an entry node, using two balanced binary trees, as detailed in [12]. Because of this, a growth in quantum software solutions, algorithms, and work areas can be observed; some familiar algorithms involve Shor’s algorithm [64], Grover’s algorithm [26] or, for instance, quantum applications like quantum machine learning (QML). Quantum circuits and quantum annealing are among the quantum computing technologies that are being explored and researched nowadays. However, to date, no technology has emerged as the definitive leader in this field.

It must also be considered that quantum computers have limited capabilities and are error-prone. One of these limitations relates to the quantity of qubits they provide, which places a restriction on the amount of data that can be used as input when solving problems and reduces the range of problems that can be solved successfully. Another example is the so-called decoherence problem [73] that occurs when the quantum system cannot be properly isolated from its environment; this can lead to errors and inaccuracies in the results obtained.

The current state of quantum computers and the research performed on quantum computing led to the expression Noisy Intermediate-Scale Quantum or NISQ [51], which is what quantum computers nowadays are commonly referred to in the field. The term “Noisy” refers to the inherent imperfections and errors in quantum computations, while “Intermediate-Scale” notes the limitations in terms of the number of qubits and the error correction capabilities. Despite their limitations in terms of noise and size, they offer opportunities to explore quantum algorithms, study new physical phenomena, and develop techniques to manage and mitigate noise. Therefore these quantum computers represent a stage in quantum computing evolution, where scientists have to work around current technological limitations to effectively harness quantum effects for useful applications as well as develop and work toward the next generation of quantum technology that could be more stable and powerful.

Due to the growing interest and investment in quantum computing, as reflected in the data presented, it is essential to synthesise current knowledge in this field. In this study, we focus on research that directly uses quantum computing to solve software problems-from a theoretical or practical approach-, or improve quantum algorithms. We therefore exclude studies that focus on the hardware part of this technology. This paper presents an Systematic Mapping Study (SMS) [47] focused on current quantum software solutions, quantum technologies employed in research, and the evaluation, validation and verification processes applied to these solutions and the limitations and challenges associated with the use of this technology. A SMS involves observing and categorizing research results and reports within a specific field to provide a comprehensive summary of its history. The study aims to assess the current state of quantum computing knowledge, focusing on how it is being applied to solve software problems, whether through practical implementations or theoretical advancements, including efforts to improve quantum algorithms.

The remainder of this paper is structured as follows. The next Sect. 2 presents a discussion of the related work. After that, the SMS outline 3 is established, followed by SMS planning 3.1, execution of the SMS and data synthesis 3.2, and finally, a discussion of the obtained results 4. After that, there will be an analysis of the threats to the validity of the results obtained, 5, and a final Sect. 6 of the general conclusions and a discussion on future work and research possibilities.

2 Related work

The objective of a Systematic Reviews (SRs) is to explore and classify research, and critically analyse the quality of the found studies to ensure their validity. In the context of quantum computing, there are noteworthy SR articles.

There are SRs (Systematic Mapping Studies, Systematic Literature Reviews and Surveys) relevant to the field of quantum computing but not to the exact topics covered in this study. One of them introduces the general challenges of this field [4]. It outlines the most critical challenges quantum computation as a whole must confront. This article indicates that the knowledge gathered around the proposed research questions (see Table 1) contributes in three different ways to the quantum computing domain. First, they synthesize existing knowledge on the problems and challenges of quantum computing. Second, they employ a fuzzy analytic hierarchy process (F-AHP) to help identify the main obstacles to the adoption and development of complete quantum computing. Finally, they propose a methodological advance in quantum computing research by combining both the SLR and the F-AHP. This SLR gives an outline of the wide challenges experienced within the field of quantum computing, setting the arrange for a more profound investigation into particular topics.

Building upon the foundational understanding established by the previous SLR, [1] presents a SMS that delves into the emerging paradigm of Quantum Computing as a Service, or QCaaS, offering insights into the practical implications of quantum computing in cloud environments. We can observe the research questions proposed in this SMS, see Table 1. This work identifies and documents existing research on the application of both classical and quantum computing to Quantum Computing as a Service (QCaaS), and to explore and analyse the domain’s history to identify potential trends and gaps in the field.

Following the exploration of Quantum Computing as a Service, which highlights a specific yet broad viewpoint of quantum computing, it gets to be fundamental to further investigate the core principles supporting quantum software engineering. “The Talavera Manifesto” [52] establishes foundational guidelines for quantum software engineering. This document also discusses some of the issues for researchers when dealing with the quantum technologies previously mentioned; some of these issues include deficiencies of current quantum programming languages, lack of structure and control, the low level of abstraction that is still present, or the improvement of the existing integrated development environments (IDEs) to support, for instance, technology-agnostic design and development of quantum software.

Expanding upon the principles delineated in the Talavera Manifesto, in [50] they conducted a comprehensive survey aimed at defining the term “quantum software engineering” and establishing a structured lifecycle for quantum software development, composed of five phases: quantum software requirement analysis, quantum software design, quantum software implementation, quantum software testing and quantum software maintenance. As we can see, this life cycle is the counterpart of the classic software life cycle. In addition, the paper addresses the processes, methods, and tools related to quantum software engineering and provides a general overview of the field. It also includes some of the challenges encountered in the different phases they proposed.

Similarly, the objective of the article [14] is to investigate quantum software engineering (QSE) through the examination of research questions related to software testing, tools, and frameworks for quantum engineering. Moreover, the article explores the evolution of QSE as a novel research domain and the collaborative dynamics within the community. They analyze the collaborations and networks among researchers in QSE, focusing on the academic and professional community working to promote quantum software engineering as a proper discipline. Additionally, the analysis addresses the topics of QSE in material published in conferences and not only on thematic journals which reflects the still emergent character of QSE and the need of legitimization it faces. The research questions addressed in this study are presented in Table 1.

Based on the exploration of QSE, the article [60] identified unique problems in requirements definition and modeling for quantum systems, especially within hybrid quantum-classical architectures. The work emphasizes the importance of constructing frameworks that can incorporate methods of classical requirements engineering with quantum-specific elements like coherence and entanglement. The research questions that are addressed in this study are presented in table 1.

In relation to software design and development for classical computing, in [31] they have focused on software architecture, they discuss the importance of efficient modelling and creation of quantum software by employing a correct software architecture. We can observe the research questions proposed in this review, see Table 1. According to the study’s findings, quantum software architecture is still in its infancy and is developing quickly, as seen by the majority of the reviewed studies, which were published within the last four years (2018-2021). They discovered eleven tools and frameworks that can automate and personalize the quantum software architecture process, six reusable architectural patterns, and five architecture activities. In order to address new issues regarding architectural solutions for quantum software, fifteen emerging challenging factors were found and categorized.

In line with quantum software architecture, in [76] they analyse how specific design patterns and architectural models adapted to the quantum environment can improve the efficiency of the quantum software being developed. It introduces some architectures such as Q-UML and Quantum4BPMN, which are intended to adapt traditional design principles to handle quantum properties such as superposition and entanglement. It indicates how a well-structured architecture can optimise the performance and efficiency of quantum systems, underlining the need for design methods dedicated to this technology.

Furthering the discourse on quantum software development, [21] presents a SMS of the state of the art of quantum software testing and examines the current landscape of testing methodologies and practices within the quantum computing domain. The study’s research questions, outlined in Table 1, aim to collect data on the similarities and differences in testing and verification methodologies between classical and quantum software. The goal is to facilitate the adaptation of these techniques from classical to quantum computing, thereby enriching the Quantum Software Engineering (QSE) knowledge base.

Complementing the discussions on quantum software engineering, recent reviews, such as [61]’s comprehensive review on quantum components and platforms, provide essential insights into the infrastructure supporting quantum software development. Their objective is to investigate the most important components and platforms of quantum software and also to indicate a set of standards for the creation of quantum software platforms and the way their quality can be evaluated. If we observe their overview of quantum software technology we can find that they divide these technologies in 7 different groups:

• Quantum Programming Languages: shows the road map for quantum programming languages and discusses how these languages have been evolving since the beginning. It indicates two different types of quantum programming languages, imperative (with examples such as QCL, QASM, Cirq, Q#, or OpenQASM3), and/or functional (with examples like QML, Quipper, ProjectQ, or Qiskit).

• Quantum Software Simulators and Design Environments: in a field in which real quantum computers are prone to errors, and the cost of execution is high, the development of quantum simulators is growing. It also presents a list of devices and providers, some of them are Atom QASM Quantum Circuit Previewer, DDSIM, D-Wave, Microsoft Q# Quantum Simulators, or ProjectQ.

• Quantum Software Optimizers: these are needed to adjust the theoretical quantum circuit to the real-world quantum computer topology. Some optimizers that they mention are IBM QX, pQCS, or RevKit.

• Quantum Tools and Libraries: this article also discusses the quantum tools and libraries available. Some examples of the former are Drqubit, HOQST, or Q-Kit, and some examples of the latter are Linear AI, Q++, or Qubit4Matlab.

• Full Stack Software of Main Quantum Computing Vendors: there are many companies which have proposals and technologies in the field of quantum computing, as there is not just one quantum technology, each provider proposes a specific hardware, architecture, dependencies between software and hardware, and their own software solutions. They mention six vendors, D-Wave, Google with Quantum AI, Honeywell, IBM, Rigetti, and Xanadu.

• Quantum Software Development and Run Platforms: apart from the vendors mentioned earlier there are platforms that let users design, run applications, and interact with different quantum technologies. Some are 1QBit, Braket or QPath.

• Quantum Software Error Correction Tools: as it was mentioned earlier, quantum computers nowadays are prone to errors. Therefore, there exist error correction tools to mitigate defects. Some of these tools are Devkit, Boulder Opal, or Black Opal.

To address real-world quantum software development problems faced by developers, an empirical study [18] examines this issue by analyzing Stack Exchange forums and GitHub issue reports. Their focus is on the problems or challenges encountered in practical quantum computing projects.

Related to problems of quantum software development, there is a SMS related to techniques to employ in order to mitigate the inherent problems of NISQs [65], specifically the low number of qubits and their error-proneness. For this purpose, they designed and conducted an SMS to gather the existing knowledge and techniques proposed and used to serve as a starting point for further investigation. We can observe the research questions proposed in this SMS, see Table 1.

Similarly, the article [54] identifies and analyses some specific problems or obstacles that arise from the adoption of quantum computing in the industry environment, specially with the NISQ systems, observe the research questions of this study in Table 1. They perform a Multivocal Literature Review (MLR) in order to identify these problems and their possible practical solutions. This study identifies thirteen challenges and proposes fifty-five practical solutions for their resolution. One of these challenges is the lack of testing methodologies for quantum software. As its practical solution, they suggest adapting hybrid testing environments which combine both classic and quantum tools to detect which higher precision the errors.

Still on the subject of applications for industry, there is another article, [23], which details the use of quantum and quantum-inspired algorithms whithin the Operations Research field. Operations Research (OR) is a form of problem solving which helps the management of a company to make decisions based on scientific facts and data [25]. They perform a Systematic Mapping Study which provides a structured classification that identifies the most frequently utilized algorithms in Operations Research, including Quantum Particle Swarm Optimization, Quantum Genetic Algorithms, and Quadratic Unconstrained Binary Optimization (QUBO). Their specific research questions can be seen in Table 1. A notable feature of this mapping is the incorporation of the International Standard Industrial Classification (ISIC) to categorize the applications of OR, enabling both practitioners and researchers to easily recognize the specific uses of these algorithms in various industries. This method not only reveals clusters of existing evidence but also indicates areas where further research is needed, thereby providing a guide for potential future research on quantum algorithms within the Operations Research domain.

Moving on, there is an additional article, categorised as review, that discusses some concepts related to quantum computing. This article addresses the trends and the overall future of quantum computing [36], including quantum networks, quantum net surfing, quantum cryptography, and quantum gaming.

Table 1 Research questions of the articles of the related work

If we examine the studies most closely related to our research, along with the research questions they propose, we can identify significant differences between previous works and the present study. Although these works share similar themes, each offers unique approaches and specific perspectives that separate them. We will now delve into the primary focus of these works. We could observe in [21] that both proposed research questions include the verification process. In [4], they focus on the primary and great challenges concerning quantum computing and on the best method to model these challenges to be able to adopt quantum computing in the software field. In the survey [61], they obtain an overview of the existing quantum software technology and its types; quality requirements that target the development of quantum software platforms, and issues in this field focused on classified technologies.

In study [14], the author’s aim is to identify the most investigated topics within Quantum Software Engineering (QSE), along with the type and number of studies carried out, the most used technologies and the predominant tools as well as the researchers which work in this field and the collaborations that arise from the evolution of this discipline. They focus on QSE as a discipline, its academic collaborations and overall evolution.

Another work, [60], aims to explore the process of requirement gathering and analysis, focusing on the challenges that arise due to the unique nature of quantum computing, as well as the trends related to requirements elicitation specific to this field.

There is a survey, [75], which offers a broad overview once again of Quantum Software Engineering.

Additionally, there is an Empirical Study of the challenges of Quantum Software Engineering [18], it primarily focuses on practical quantum computing projects, by analysing Stack Exchange forums and GitHub issue reports, with a focus on problems or challenges encountered in practical quantum computing projects.

Another work which provides information about limitations and challenges is [54], its target is an industry practitioner trying to overcome immediate problems.

Compared to previous articles that address specific topics within quantum computing, our study distinguishes itself by focusing on the direct application of quantum computing to solve software-related problems or enhance quantum algorithms. We provide a comprehensive mapping of the current state of quantum software solutions by examining the quantum technologies employed, as well as the evaluation, verification, and validation processes implemented, and the limitations and challenges encountered in the selected studies. Furthermore, our research takes a broad approach to identify trends, gaps, and emerging areas in the practical implementation of quantum solutions, offering insights that benefit both academic research and real-world industry practitioners.

3 SMS outline

This systematic mapping study follows the guidelines proposed in [34, 48], ensuring a structured and reproducible approach. These types of studies include three distinct phases that we will explore in this section: planning the review, which includes defining the research questions, inclusion/exclusion criteria, search strategy and data extraction form; conducting the review, where relevant studies are selected and classified; and finally reporting the results obtained from the review. The replication package, which contains the dataset and analysis, is available in the repository of this article [15].

3.1 Planning the systematic mapping study

The planning phase includes defining the research questions and developing the search strategy, defining the inclusion and exclusion criteria, and the data extraction form.

This study formulates six research questions focused on exploring the use, technology, processes, and limitations of quantum software, which are the following:

• RQ1: What kind of software problems is quantum computing solving or trying to solve?

  • Motivation: To find the quantum algorithms or quantum solutions that are being used or proposed.

• RQ2: What approaches of technologies are being used within the quantum computing paradigm?

  • Motivation: To observe which quantum computing technologies are the most widely used, to find out if any of them have positioned themselves as dominant.

• RQ3: What parameters are quantum software solutions being evaluated with?

  • Motivation: To find out whether quantum algorithms or quantum software solutions are being evaluated, and if so, what criteria, such as performance, efficiency, sustainability, etc. and by which parameters or metrics.

• RQ4: What kind of verification or validation is being applied to quantum software solutions?

  • Motivation: To check whether quantum software solutions are being verified and/or validated and how. This question arises from the relative novelty of the field.

• RQ5: What are the limitations of quantum software?

  • Motivation: To observe the possible constraints and restrictions imposed to quantum software by the current state of quantum computers and quantum computing overall. The purpose is to highlight the limitations or unmet needs of present quantum software.

• RQ6: What are the challenges of quantum software?

  • Motivation: To observe the obstacles and difficulties encountered while developers or researchers work on quantum software. It focuses on issues related to the design, development, implementation and execution of quantum software solutions.

We limited the search of research papers to electronic form and only considered peer-reviewed journal articles and conferences so that their quality is more reliable. The electronic source selected for this study is the SCOPUS database, as it contains a curated collection of high-quality scientific material, ensuring reliable and rigorous content for the analysis [19].

The search string was defined as “TITLE-ABS-KEY(“quantum software”)”. The subject area was limited to Computer Science.

The documents found were then reviewed considering the following selection criteria. The inclusion criteria applied were as follows:

• 1. The article had to be published by December 2024.

• 2. The article must be written in English.

• 3. The article must be published in a journal or in a conference proceeding.

• 4. The article must address a software problem by directly employing quantum computation for its resolution- either from a theorical or practical approach- or for the improvement of a quantum algorithm.

The exclusion criteria applied were the following:

• 1. The article must not be limited to offering an opinion or a general perspective on the quantum computing field. It should present a concrete proposal or solution, which can be either theoretical or practical.

• 2. The article introduces a proposal, solution or software that is quantum-inspired, quantum-like or quantum-based proposals, as they draw inspiration from the idea of quantum but do not exploit this technology.

• 3. Published from January 2025.

• 4. Full-text access was not available.

To extract data from the identified primary studies, we developed the data extraction form shown in Table 2.

Table 2 Data extraction form

3.2 Conducting the review

In this phase, the relevant studies are selected and classified. After conducting the search employing the search string defined in the previous phase, we found 323 articles, we initially filtered out those that were clearly irrelevant based on their title and abstract. After this initial screening, 47 articles remained for further evaluation. For the remaining articles, we applied the inclusion and exclusion criteria defined in Sect. 3.1. As a result, two articles were excluded due to lack of full-text access, while twelve were discarded for not meeting Inclusion Criterion 4, as they did not directly employ quantum computation to address a software problem. After this process, we obtained a total of 33 primary studies. The table of selected articles is shown in the Appendix A, also located in the Repository [15].

If we analyse the articles obtained from the search, see Fig. 2, we see that the number of publications remained relatively low between 1990 and 2020. Since then, however, there has been an exponential increase in scientific production.

This tendency in scientific production could be related to the increased interest reflected in Google Trends, refer back to Fig. 1. As research in quantum computing advances and more results are published, the dissemination of knowledge leads to increased attention from the scientific community, industry and the general public.

After analyzing the temporal evolution of scientific production in quantum software, it is relevant to examine the nature of the published papers. Figure 3 shows the distribution between conference papers and journal articles, which provides information on the main ways of disseminating knowledge in this field. As we can see there is a clear dominance of conference papers over journal articles, with these making up 73% of contributions. This could be because of the rapid evolution within the field, which would encourage early publication at conferences to share progress and obtain feedback.

Fig. 2

Primary studies by year obtained from the search performed

Fig. 3

Documents found in the search by type

Beyond understanding the temporal trends and dissemination channels of research in quantum software, it is also essential to examine the nature of the contributions made in these studies. To achieve this, we classified the studies found using the scheme proposed in [68]. This classification provides a structured framework to categorize the studies efficiently, as summarized in Table 3, without requiring an exhaustive evaluation of each document.

Table 3 Summary of the classification scheme employed

Focusing now on the 33 selected studies, as we can see in Fig. 4, most of the documents fall into three types, “Evaluation Research”, and “Solution Proposal”, both with twelve articles; and “Validation Research”, with seven articles. We can also observe that there are two “Experience Papers”; and no “Philosophical Papers”, nor “Opinion Papers”. Since we focus on articles that directly apply quantum computing to solve software problems or improve quantum algorithms, it is reasonable that we do not find philosophical or opinion articles, as they do not fit solution-oriented nature of our inclusion criteria.

Fig. 4

Classification of the 33 primary studies found

If we consider both perspectives, time and the classification of the documents, we can observe through time the publication frequencies of the different categories listed. In the following Fig. 5, we can observe a bubble graph that illustrates this.

Fig. 5

Documents by category through time

In the last few years we can observe an increment of publication overall, but if we observe the categories, we can see that there appear more evaluation research, solution proposals and validation research. The increasing appearances of evaluation research and solution proposal articles can indicate that the field is maturing, with a growing emphasis not only on proposing theoretical concepts but also on demonstrating their practical applicability and providing concrete solutions to real-world problems.

In the following Table 4, we can observe the number of papers from the 33 primary studies encountered, that answer each of the questions. Note that one article might answer more than one research question. It can be seen that all primary studies answer the first question as it is directly related to one of the inclusion criteria. Also, almost all primary studies answered research questions two and three; more than three-quarters answered the fifth question. Finally, only 39% and 42% of the studies answered the fourth and sixth questions respectively regarding the verification and validation of quantum software solutions and the quantum challenges encountered.

Table 4 Number of papers that answer the research questions and their percentage to the total number of papers found

Figure 6 illustrates the mapping between primary studies classified according to our scheme and those addressing our research questions. We will initially examine the number of articles found per classification on the right side of the graph, followed by the number of articles addressing our research questions on the left side. Notably, from 1990 to 2018, no articles relevant to the focus of our study have been found.

From 2015 to 2020, specifically in 2019, we can find one paper classified as “Validation Research”, this paper addresses most of our research questions (RQ1, RQ2, RQ3, RQ5 and RQ6).

The vast majority of scientific contributions addressing our research questions, 32 out of 33 articles, were published between 2020 and 2024, highlighting the recent surge of interest in this field. If we analyse the contributions in relation to our research questions, most of them are well supported by the selected studies with at least 26 articles providing relevant information for each. However, if we observe RQ4 and RQ6, which focus on the verification and validation processes in quantum software and the challenges of this technology, we see that they have significantly fewer contributions, with only 13 and 14 articles, respectively (Table 4).

This disparity suggests that while significant progress has been made in various aspects of quantum software research, the evaluation of quantum software and the challenges of quantum computing remain relatively underexplored areas. In the following sections, we analyze the contributions of the selected studies to each research question individually.

Fig. 6

Number of primary studies by category and by response to the research questions proposed, over time

3.3 Reporting the results obtained

In this phase, each research question is examined individually, outlining the contributions made by the selected articles.

3.3.1 Quantum software solutions (RQ1)

Research Question 1 explored: “What kind of software problems is quantum computing solving or trying to solve?”.

In Table 6, also located in the Repository [15], in the Appendix B, we can observe the quantum algorithms and software solutions mentioned or proposed in the 33 primary studies. We used these words to generate a word cloud that illustrates the most frequently occurring terms accross the studies. This process was followed for each research question, generating a word cloud to highlight key concepts and recurring themes. However, for the last question, we did not find enough repetitions for the word cloud to provide meaningful value. Note that a word cloud is a visual representation of a group of words, where the most frequent used words appear larger.

To address this first research question, we analysed the contributions of the selected studies, classifying them based on two different dimensions:

• 1. Investigation/research area: the studies were classified considering their focus within quantum computing:

  • Quantum Machine Learning & Artificial Intelligence: this is an emerging area that explores the use of quantum algorithms to improve artificial intelligence models. In this category we find studies that apply quantum techniques to machine learning problems, artificial intelligence simulations or pattern recognition.

  • Quantum Algorithms: some of the studies analysed focus on the development and application of fundamental quantum algorithms. In this category, we group together those works that propose, improve or evaluate the performance of quantum algorithms such as Shor, Grover, QAOA, etc.

  • Quantum Optimization: quantum computing has great potential for solving complex optimisation problems. This group gathers studies that transform classical problems into quantum formulations, such as QUBO, as well as work that develops tools to facilitate quantum programming.

  • Quantum Security: this category includes studies that focus on security within the domain of quantum computing. It covers topics such as the protection and authentication of quantum software, security-related non-functional requirements, and cryptographic techniques designed specifically for quantum systems.

• 2. Problem nature: the studies were classified considering the type of software problem they address:

  • Classical problems → Solved with quantum: this category includes studies that address traditional software problems, historically solved by classical methods. The research focuses on the application of quantum techniques to solve these problems, seeking improvements in efficiency, speed or quality of the solutions.

  • Quantum problems → Require quantum computation: this groups together studies that deal with problems whose resolution is intrinsically dependent on quantum computation. These problems require exploiting phenomena inherent to the quantum world, such as superposition and entanglement, and cannot be tackled efficiently by classical methods. In this category, we seek to identify proposals that demonstrate how quantum computation can be used to solve challenges of the quantum computing field.

This classification can help us better understand the scope and direction of quantum software research, offering insight into how quantum computing might be being applied in different domains. The word cloud for this first research question, (see Fig. 7), was generated from the Table 6. As we can see on the word cloud, the most frequent terms are Grover, VQE, QAOA and Quantum annealing, among others.

Fig. 7

RQ1: Word Cloud of Relevant Keywords

Taking into account this classification, each article has been classified according to its research area and the nature of the problem it addresses. For each of the research areas presented, we will look at the nature of the problem addressed by the article.

• Quantum Machine Learning & Artificial Intelligence: we can find five articles which contribute in this research area. If we explore some of these insights we can see different ways to integrate quantum computing in this field. For instance, in article [37], they propose a Quantum-Train Toolkit which shows how Quantum Machine Learning (QML) can improve the training of classical neural network (NN) models for tasks such as image classification, reinforcement learning, or flood prediction, thereby optimising traditional models. In article [11], the authors present a hybrid quantum-classical-quantum workflow based on the algorithm Variational Quantum Eigensolver (VQE) to analyse quantum phase transitions in the Ising and XXZ models. To predict these transitions, it uses convolutional quantum neural networks (CQNN). Its application ranges across areas such as materials science and condensed matter physics. The process begins by preparing a quantum state that encodes possible solutions, then extracting its key features and transforming them into classical data, which is processed with conventional neural networks. Another article which focuses on a hybrid quantum-classical approach is [6] in which they propose a hybrid architecture for Smart City Security. They propose an integration between QBoost, a QML algorithm, and IBM QRadar, a security information and event management (SIEM) system, allowing the detection of threats to critical infrastructures using both quantum and traditional tools. We could observe that all the solutions proposed in terms of problem nature are applications of quantum software to classical problems.

• Quantum Algorithms: we could find some quantum algorithms being employed in the selected studies. For instance, one of the most used is Grover’s algorithm or quantum search algorithm, a quantum algorithm designed to search unstructured databases more efficiently than classical algorithms. Several publications explore implementations and applications of this algorithm, in article [63] they employ this algorithm to find a solution of a 2x2 Sudoku problem, illustrating how it can be used in the solution of combinatorial problems. Furthermore, in [56], they develop a Grover-based operation that computes multiples of a given number within a quantum state, constructing a specific oracle for this algorithm. Another well known quantum algorithm is Shor’s algorithm, an algorithm designed for integer factorisation in polynomial time. It is well known due to its implications in cryptography, as it can compromise the security of RSA-based systems. In [53], an implementation of Shor’s algorithm for integer factorisation is presented. Another approach within quantum algorithms is quantum walks, these are a generalisation of classical random walks. Unlike classical random walks, where the transition between states follows a probabilistic distribution, quantum walks use superposition and quantum interference to propagate information more efficiently, allowing certain search spaces to be explored exponentially faster in some cases. In article [10], they study the quantum Metropolis-Hastings algorithm in an optimisation problem applied to the N-queens problem. They evaluate three variants of the algorithm based on quantum walks and they compare then to their classical counterpart. The results obtained show that the usage of quantum walks allows for a more efficient exploration of the solution space, optimizing the speed and quality of the search in comparison to classical methods. Unlike the previous category, in this case we can find the use of quantum algorithms both to solve classical software problems and applied to quantum problems. Specifically, we can find instances of quantum computing applied to classical software problems, such as the case of the 2x2 Sudoku problem in paper [63]; and quantum computing applied to quantum problems, this is the case of the article [66], it analyses quantum algorithms, focusing on the k-local Hamiltonian problem, a QMA-complete problem with applications in quantum optimisation and simulation. Both exact and approximate methods are explored, including the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE), demonstrating how these algorithms can be used to solve intrinsically quantum problems.

• Quantum Optimization: this is an area focused on exploring algorithms designed to solve combinatorial and continuous optimisation problems employing quantum computing. Quantum computing offers techniques such as the Quantum Approximate Optimization Algorithm (QAOA) or Quantum Annealing (QA), which help us explore the solution space making use of the singularities of quantum computation such as superposition and entanglement. Some of these optimization problems are formulated employing Quadratic Unconstrained Binary Optimization (QUBO), which helps model them on quantum hardware. In article [59], they focus on the transformation of Polynomial Unconstrained Binary Optimization (PUBO) as a QUBO model. In article [5], they focus on a well-known software problem, the Knapsack Problem, and they analyse different approaches to solve it, they include QAOA, warm-start QAOA, Variational Quantum Eigensolver (VQE), quantum annealing, and classical heuristics such as simmulated annealing. Concluding that WS-Init-QAOA, which is a variant of QAOA that improves the initialization without changing the mixing Hamiltonian, and Quantum Annealing executed on a D-Wave device, offer the best results, being the latter faster than the other options detailed. Another article in which they employ quantum computing to solve classical well-known problems is in [7], the authors implement Quantum Annealing Search (QALS) for D-Wave devices to solve both the Number Partitioning Problem (NPP) and the Travelling Salesman Problem (TSP). They conclude that the quantum solution proposed is less efficient than hybrid and classical approaches. Although not currently competitive, its ability to process large problems suggests potential with improvements in hardware and algorithms.

• Quantum Security: the articles reviewed in this category address various dimensions of quantum security. An emerging approach within quantum security is to take advantage of quantum phenomena for encryption. In the field of image encryption, the use of Boson Sampling (BS) is explored. Boson Sampling is an experiment that uses photons to solve problems. These photons pass through a system of mirrors and light splitters, and their output distribution is so complex that only a quantum computer can simulate it effectively. In order to generate chaotic sequences of random numbers to serve as keys in an encryption scheme, the article [141] explores the use of Boson Sampling (BS) to generate chaotic sequences of random numbers to serve as keys in an encryption scheme. These keys are applied to reorder pixels and modify the greyscale using operations such as XOR and Arnold transforms, and a BS-based encryption and decryption scheme is proposed. Another aspect of quantum security relates to quantum cryptography and secure key generation, in [32] they evaluate E91, a protocol of Quantum Key Distribution (QKD) applied to the secure transmission of genomic data, while on [62] they explore an image encryption scheme based on Boson Sampling, using chaotic random sequences to modify and rearrange pixel values. There are also works focused on software protection and quantum authentication. Article [8] proposes a secure leasing scheme based on quantum principles, with protection mechanisms against unauthorised copying using Quantum Message Authentication Codes (QMACs), and [67] implements authentication schemes based on quantum computing, including protocols such as FrodoKEM-1344 and AES-256 with liboqs-python.

It is important to note that although we have classified the papers into different categories, they are not mutually exclusive. Many papers combine approaches of quantum machine learning, algorithms, optimisation and security, reflecting the interdisciplinary nature of quantum computing. The classification employed for this first research question is intended to improve the presentation of the information and to facilitate the analysis of the contributions of each article.

In the studies analysed in the different categories, there is a clear trend towards increasingly integrating quantum computing into traditional problem-solving approaches. Regarding Machine Learning and Quantum Artificial Intelligence, the studies reveal a growing trend towards using quantum methods to improve classical systems, demonstrating the effectiveness of the hybrid approach in optimising existing techniques to tackle more complex challenges. The papers also highlight the flexibility of quantum algorithms, which can be applied to both traditional software problems and those intrinsically linked to quantum computing, offering solutions to combinatorial and cryptographic problems while advancing quantum-specific challenges. We could also observe the potential of quantum computing to address optimisation problems, with current methods promising but falling behind classical and hybrid approaches in terms of efficiency. However, with further advances, quantum optimisation could become a key enabler for solving larger and more complex problems. Finally, we have seen how quantum principles are being applied to improve data protection, with innovations in secure key generation and authentication leading the way to more robust cybersecurity solutions, suggesting significant potential for quantum technology in the security domain as it matures.

As indicated above, we have also classified the selected studies according to the nature of the problem they address: on the one hand, we have the application of quantum computing to solve classical software problems, and on the other hand, the application of quantum computing to quantum computing problems. Of the selected articles, twenty-two belong to the first category and eleven to the second. Even though there is a noticeable difference, these findings suggest that quantum computing is being applied to both quantum and classical problems which could indicate a process of exploration in which quantum techniques seek to demonstrate their usability in multiple domains.

3.3.2 Technologies employed (RQ2)

Research question 2 explored: “What approaches of technologies are being used within the quantum computing paradigm?”

As stated in Table 4, there are 31 articles that include information relevant to this question. In Table 7 we can observe the quantum simulation platforms mentioned, and in Table 8 we can observe the real quantum computing platforms. Both of these tables can also be found in the Repository [15], in the Appendix B.

We have collected the number of times that various quantum platforms and technologies are employed in the selected articles, and from this information we have made a word cloud, see Fig. 8.  The same process described in RQ1 has been applied to make a word cloud from the Tables 7 and 8 for this second research question. Note that the terms appear in a larger or smaller size depending on the number of times they are used. Here we can see that numerous technologies have been employed in the selected studies. Some of the more notable are circuits (related to quantum gate-based computing), IBM Quantum, Annealing, or D-Wave.

Fig. 8

Word cloud of the responses to RQ2, technologies within the quantum paradigm

In this section, we will examine how quantum technologies are used from different perspectives. The selected papers focus on solving software problems or improving quantum algorithms, but they do so in different ways: some use real quantum computers, others focus on simulators, and some combine quantum and classical computing. All of these approaches will be explored below.

To begin with, let us consider the paradigm used in the selected articles. In the following Fig. 9 we can observe that out of the 31 articles that have relevant information, two of them do not employ a quantum technology, they propose quantum solutions only theoretically. The remaining 29 articles all employ quantum computing and, as we can see in the figure, 3 of them employ classical computing in combination with quantum computing. This is the case, as an example, of the article [6], the authors implement the QBoost algorithm, which uses a QUBO (Quadratic Unconstrained Binary Optimization) model to select the best combination of weak classifiers for a binary classification problem. This algorithm is run on the quantum computer Advantage_system5.3 from D-Wave and is combined with IBM QRadar, a classical Security Information and Event Management (SIEM) platform designed to detect, analyze, and respond to cybersecurity threats in real time.

If we observe Fig. 10, we can see that there are some instances in which articles employ both real quantum computers and quantum simulators, but, in general, they rely more on quantum simulators than on real quantum computers.

Fig. 9

Articles that employ quantum, classical computation, or none

Fig. 10

Articles that employ a real quantum computer or a quantum simulator

Before going into detail about quantum simulators and real quantum computers, it is important to understand the differences between the two. A quantum simulator is a software that mimics the behaviour of qubits in a classical computer, this means that they cannot employ real quantum phenomena such as superposition or entanglement. In contrast, a real quantum computer is a physical hardware that implements qubits using technologies such as superconductors or trapped ions, allowing to execute quantum algorithms natively and to take advantage of quantum mechanics and the quantum phenomena just mentioned.

Now, let us focus on the usage of quantum simulators. In this context, as can be seen in Fig. 11, we found that most quantum simulators emulate gate-based circuits. Followed far behind by quantum simulators that emulate quantum annealing. And that there are other simulators being vaguely used that emulate circuits based on photonic and ion trap technologies. On Fig. 12 we can observe some of the quantum simulation platforms employed by the selected articles. The most used platform is IBM Quantum, which offers circuit gate-based technology, followed by PennyLane, a library for quantum computing and machine learning that allows the integration of quantum algorithms with classical techniques; and D-Wave, a quantum annealing platform, designed to solve complex optimisation problems using superconducting qubits. Note that there are more quantum simulation platforms or quantum simulators being employed, they can be found in Table 7, and in the replication package (Appendix B) [15].

It is interesting to note that, although PennyLane is more recent, its greater adoption in the selected articles could be due to its flexibility in supporting multiple quantum and classical backends, allowing researchers to perform more general simulations. In addition, its active focus on Quantum Machine Learning and its integration with popular frameworks makes it attractive to those looking to explore quantum algorithms in artificial intelligence applications. While D-Wave has a longer track record, its specialisation in quantum annealing for optimisation problems may be more limited for general quantum simulations, making PennyLane a preferred choice for broader and more flexible simulations.

Fig. 11

Type of quantum simulator employed

Fig. 12

Quantum simulation platform employed

Finally, let us examine the use of real quantum computers. In the case of real quantum computers, as also happened in the case of quantum simulators, gate-based circuits are the most commonly used. However, in the case of real quantum computers, we can observe that the use of quantum annealing is gaining importance, with 6 articles using quantum annealing compared to 8 using gate-based circuits. In the case of the simulators, 18 used gate-based circuits and 3 used annealing (Fig. 13).

In this context, if we examine the quantum computing platforms that provide real computers, see Fig. 14, the most used platform is IBM Quantum, with their circuit gate-based computers, followed by D-Wave, which provides quantum annealing based computers. Finally, there are three platforms mentioned once which are Amazon Braket- a quantum computing service in the Amazon Web Services (AWS) cloud that allows users to explore and experiment with different types of quantum computers and simulators-, Xanadu Quantum- which specialises in developing photon-based quantum technologies-, and QSCOUT- a quantum processor based on ion traps developed by Sandia National Laboratories. It is important to note the usage of Amazon Braket, in this case, in article [53], they employ Amazon Braket to execute on a real quantum computer from D-Wave.

Fig. 13

Type of real quantum computer employed

Fig. 14

Real quantum computing platform/computer employed

As we have seen, the most widely used quantum computing platforms for running executions on real quantum computers are IBM Quantum and D-Wave. Figure 15 shows the evolution of their use in the selected articles over the years. D-Wave seems to have a stable evolution, while IBM Quantum has an upward trend. We have to keep in mind the purpose of the computers employed for each platform; IBM Quantum has quantum-gate based computers that are intented as multipurpose computers, while D-Wave has quantum annealing computers which are focused on optimization problems. Another possible reason why there has been an increase in the use of IBM Quantum in the last two years may be due to the evolution of the quantum computers offered by IBM Quantum. If we look at Fig. 16, we can see the evolution over the years of the number of qubits from different vendors, focusing on D-Wave and IBM Quantum, the former has been able to scale its number of physical qubits in a more remarkable way. However, we must bear in mind that these are quantum annealing qubits, with a very specific purpose. IBM is committed to universal quantum computing, which requires more control and fidelity, but potentially broader applications.

Therefore, it is possible that this increase in the number of qubits in real IBM Quantum computers has made it possible to solve new quantum problems and that is why we are seeing such an increase in their use.

Fig. 15

Evolution of the usage of D-Wave and IBM Quantum platforms for real quantum computers through time in the selected studies

Fig. 16

Evolution of the number of physical qubits from D-Wave, IBM, Google and Rigetti systems through time [28]

Overall, the selected articles show a clear preference for quantum simulators over real quantum computers, with 62% of the articles employing simulators versus 38% using real computers. Of the simulators, 18 use quantum circuits, while 3 use quantum annealing. On the other hand, real quantum computers are more evenly distributed, with 8 articles using quantum gate-based computers and 6 articles using quantum annealing computers. This could be caused by limitations or challenges associated with the use of real quantum computers, such as their type or other factors. This will be further discussed in the analysis of the data obtained for research questions five and six.

3.3.3 Quantum software evaluation (RQ3)

Research question 3 asked “What parameters or criteria are quantum software solutions being evaluated with?”

As stated in Table 4, there are 26 articles that include information relevant to this question. In Table 9 we can observe the terms mentioned in the context of the evaluation of quantum software and in Table 10 we can observe the classification of the terms. Both of these tables can also be found in the Repository [15], in the Appendix B.

If we analyse the most frequent terms employing the word cloud in Fig. 17, we can see that there are a few terms which are the most used, “Performance”, “Speedup”, “Number of qubits”, and “Complexity”. There are also other terms which are also notable such as “Accuracy”, “Depth”, “Efficiency”, “Quality”, or “Probability”.

Fig. 17

Word cloud of the responses to RQ3, quantum software evaluation

These terms can be classified as performance and efficiency, accuracy and quality of results, resources and complexity, probability and success, implementation, and other relevant terms. The entire table which contains the terms and their category can be observed in Table 10.

In the following Fig. 18 we can observe the percentage of mentions per category related to the total mention of terms within this third research question. From this plot, we can observe that the most frequently mentioned terms are those related to resources and complexity, accounting for 36% of the mentions, followed by terms related to performance and efficiency, which make up 30%. Terms associated with result accuracy and success, account for the 22%. Finally, other relevant terms contribute to the remaining 5%.

Fig. 18

Percentage of mentions per category for RQ3, evaluation of quantum software

If we analyse the categories:

• Performance and efficiency: these terms evaluate a system’s ability to execute computations achieving accurate results in an optimal time. In this category, the most mentioned terms are performance, speedup, efficiency, effectiveness, runtime, etc. These terms were normally employed in the selected studies to indicate how quantum computing could be improved over classical computing and to compare the two paradigms. This category has the second highest number of mentions, reflecting its importance in the field. It seems that research tends to spend part of its efforts or focus on improving speed, efficiency and reducing computational costs. These aspects are important to ensure that quantum algorithms can compete with classical solutions in practical applications.

• Resources and complexity: these terms focus on the quantity of resources employed, and the complexity of the algorithm to be executed. In this category the most mentioned terms are the number of qubits, complexity, depth, and scalability. This category reflects the challenges inherent in the physical and technical limitations of quantum computing that will be discussed in the analysis of research questions five and six. The number of qubits, the depth of circuits, and the complexity of algorithms are factors that directly determine the performance and viability of quantum algorithms. It can be noted that this category contains the highest number of mentions in the selected articles, which may indicate that resource constraints and computational complexity are central to current research efforts and technological advances in this field.

• Result accuracy and success: these terms focus on measuring the precision, stability, and reliability of results obtained from quantum computers, while also considering the probabilistic nature of quantum technologies. They assess how accurate and stable the results of a quantum algorithm are, taking into account potential errors. Key aspects in this category include accuracy, quality of results, stability, fidelity, probability, and success rate. Accuracy and quality of results are fundamental in any computational science field, including quantum computing, but may be secondary in the early stages of research, as efficiency and performance often take precedence. On the other hand, the probabilistic nature of quantum algorithms means that many executions do not guarantee an exact result on a single run. While success rate is important, it is often assumed in many studies, as researchers typically focus on optimizing efficiency and other parameters, leaving probability and success as underlying factors.

• Implementation: these terms evaluate technical aspects of quantum software implementation. In this category the most mentioned terms are reusability, composability, which focuses on combining small modules or circuits to form larger algorithms while preserving their individual functionalities and expected behaviours, and robustness, which describes the resilience of quantum software to errors, noise, and variations in hardware performance. The implementation of quantum algorithms is an importart aspect, but it is generally a more advanced technical matter that is usually dealt with at later stages of algorithm development. The relatively low frequency of mentions of implementation-related terms might indicate that, in the early stages of research, more attention is given to theoretical models and optimisation of algorithms in terms of resources and efficiency, while discussions on implementation tend to arise later in the development process, when the field has gained more maturity.

• Other relevant terms: this category gathers additional terms than can evaluate quantum software. In this category the most mentioned term is shots, it refers to the number of times an algorithm is runned in a quantum computer in each execution, as quantum results are not always accurate this is done to obtain statistically reliable measurement data. We can find other terms such as suitability, that evaluates how appropriate a quantum algorithm or approach is for solving a specific problem, counts, that represents the results obtained for each shot after an execution is done, or Quantum Utility. This last term, Quantum Utility [28], is defined as the practical advantage gained when an application requires less computation time, consumes less energy or produces more accurate results on a quantum computer or on a hybrid classical-quantum architecture compared to the best classical device of similar size, weight and cost.

The analysis of the different categories shows that research in quantum computing is mainly oriented towards overcoming the inherent limitations of resources and complexity, which is reflected in the high frequency of mentions in this area. In addition, attention is given to the performance and efficiency of algorithms, as well as to the accuracy and quality of the results, which highlights the importance of balancing probabilistic aspects. In addition, the relevance of terms related to implementation, such as reusability, composability and robustness, highlights the need to develop modular and resilient solutions that facilitate the integration and scalability of these systems. Collectively, these findings suggest that progress in quantum computing will depend on the ability to optimise both physical resources and design approaches, ensuring practical and effective applications.

3.3.4 Verification and validation (RQ4)

Research question 4 explored “What kind of verification or validation is being applied to quantum software solutions?”

As stated in Table 4, there are 13 articles that include information relevant to this question. In Table 11, also located in the Repository [15], in the Appendix B, we can observe the terms mentioned in the context of the verification and validation of quantum software.

If we analyse the most frequent terms by observing the word cloud in Fig. 19, we can see that there are a few terms which are the most popular, “Accuracy”, “Benchmark”, “Simulation testing”, “Effectiveness”, or “Success rate”.

Fig. 19

Word cloud of the responses to RQ4, quantum software verification and validation

In this context, we must distinguish between the concepts of verification and validation as applied to quantum computing.

• Verification: process of testing whether a quantum algorithm, circuit, or software has been implemented correctly according to its design and specifications.

• Validation: process of determining whether the quantum algorithm or software actually meets the purpose for which it was designed, providing correct and useful solutions to the problem at hand.

It is important to note that some techniques or terms can apply to both contexts. Although verification and validation have distinct objectives, certain metrics or procedures-such as accuracy-can serve both purposes. For instance, accuracy can be used to verify the correct functioning of a system within a specific test suite, as well as to validate its performance against other approaches. An example of this is [46], where the accuracy metric is used for validation. In this case, the authors employ the accuracy percentage of the executions to compare the results of classical and quantum approaches, demonstrating how it serves to assess the effectiveness of the quantum solution in relation to traditional methods.

This also applies to the use of benchmarks, we could find three different benchmarks. Two applied to the verification process: in [67], to verify the correctness of a password authentication scheme, the authors employ a simplified game based on Secure Software Leasing (SSL). In this game, a challenger generates a program protected with a secret key p, which is then sent to an evaluator along with a challenge input taken from a specific distribution. The evaluator’s task is to process the input and verify whether the scheme meets the expected security properties, even in the presence of partially dishonest adversaries. In [35] they employ the so called inverted pendulum, or the “Cart-Pole” problem, in this benchmark, the goal is to balance a pole on a cart by applying forces to the cart’s base, this problem serves to evaluate the performance of quantum algorithms in controlling physical systems. And one benchmark is applied to the validation process: in [10], they employ the N-Queen problem as a benchmark to validate their tool. This problem is widely used in AI as it is an NP-complexity search problem. The authors argue that any algorithm achieving good performance in solving the N-Queen problem can be easily adapted to other AI problems based on search, such as hypothesis search in machine learning algorithms.

One of the most commonly mentioned terms is success rate, which indicates how often a quantum algorithm returns the correct answer within a given number of runs. These runs are referred to as “shots” in the case of gate-based quantum computers, and “reads” in the case of adiabatic quantum computers (quantum annealing). In the selected articles, for example, [27], success rate is employed as a validation tool. This term is closely related to accuracy, as some articles use an accuracy percentage instead of success rate for the same purpose.

Some articles employ both verification and validation techniques, such as article [8]. In this study, they employ verification methods such as data authenticity checking, which confirms that a message or functionality is returned. Once this verification is confirmed, it is impossible to continue reading or using the encoded information due to the quantum non-cloning principle, which states that quantum information can be distributed and used, but cannot be duplicated. To verify the non-cloning property of the quantum copy protection scheme, computation and comparison circuits are modelled and tested in a simulated environment. Furthermore, quantum probability computation is used to demonstrate the system’s resistance to attacks based on imperfect cloning and partial measurements. In parallel, the paper also employs validation techniques, such as quantum message authentication code (QMAC) analysis, which are mathematically validated to ensure their ability to protect the authenticity and integrity of the software against unauthorised modification.

If we compare the different methods and techniques of verification and validation, including the frequency with which each is mentioned in the selected studies, we can observe that the verification process is mentioned much more frequently (73%) than the validation process (27%), see Fig. 20. Verification is crucial to ensure that quantum algorithms, circuits, and hardware work correctly, especially given the susceptibility of these computers to errors because of decoherence and noise. The concern for quantum error correction also leads to a more focused approach to result verification, while validation, although important, is more focused on evaluating the performance of systems in real applications or in comparison with other approaches.

Fig. 20

Percentage of mentions regarding verification or validation for RQ4

3.3.5 Limitations of quantum software (RQ5)

Research question 5 asked “What are the limitations of quantum software?” As stated in Table 4, there are 26 articles that include information relevant to this question. In Table 12 we can observe the terms mentioned in the context of the limitations of quantum software and in Table 13 we can observe the classification of the terms. Both of these tables can also be found in the Repository [15], in the Appendix B.

If we analyse the most frequent terms shown in the word cloud in Fig. 21, we can see that the most popular limitations are “Number of qubits”, “NISQ era”, “Error” and “Noise”. We can also observe other terms such as “Decoherence/coherence time”, “Qubit connectivity” or “Size”.

Fig. 21

Word cloud of the responses to RQ5, quantum computing limitations

These terms can be classified as hardware, operational, development and infraestructure related limitations. If we analyse the four categories:

• Hardware limitations: these limitations in quantum computing reflect the main physical and technological obstacles that impede its progress. Among them, the most frequent are the limited number of qubits, the NISQ era, noise, errors, and decoherence/coherence times. These constraints indicate that current devices cannot yet maintain stable quantum states over long periods, or execute interference-free operations. The NISQ era highlights that these systems suffer from a lack of error correction and limited inter-qubit connectivity, which affects scalability. In addition, problems such as qubit fidelity, state preparation and efficiency in the application of logic gates show that the manipulation of quantum information is still inaccurate. The presence of phenomena such as unintentional interference between qubits and the lack of fault-tolerance suggests that control over these systems remains weak.

• Operational limitations: these limitations focus on resources and constraints affecting user’s ability to interact with quantum systems. Machine time constraints and slow execution time (2 mentions) are the most recurrent limitations, they make it difficult to efficiently implement complex algorithms. In addition, usage queues indicate that access to quantum devices is limited due to high demand, delaying research progress. The limited RAM of simulators is also an important factor, as it restricts the ability to perform large-scale simulations, limiting the exploration of more complex systems. Finally, internal constraints imposing a maximum annealing time pose barriers in solving more complex problems within the time allowed, affecting the effectiveness of optimisation methods.

• Development limitations: this category reflects limitations related to the formulation and execution of quantum algorithms. A notable problem is that intermediate verification of results is not possible nowadays, which makes it difficult to validate results during the execution of the algorithms and may compromise the accuracy of the solutions obtained. In addition, the generation of random test cases is time and resource intensive, increasing the complexity of algorithm validation and testing. The fact that quantum algorithms and their parameters are architecture-dependent means that there is no universal approach that works well on all quantum systems, which limits the portability of the algorithms.

• Infraestructure and cost limitations: are closely related to the availability of resources, the technical complexity, and the cost of this technology. Lack of specialisation is a major limitation, as the skills needed to implement and manage quantum computing effectively are still limited, restricting expansion beyond highly specialised environments. In addition, the high production and operating costs of quantum computers represent a key limitation in terms of accessibility and adoption, making access to this technology restricted to sectors with high budgets. Infrastructure dependence is also a major problem, as a very specific environment and equipment is required to operate quantum computers, making their integration and large-scale use difficult.

In the Fig. 22 we can see the dominance of hardware limitations (77%) in quantum computing, this can be explained by the early progress advances in quantum technology, which faces barriers such as the limited number of qubits, high susceptibility to noise, coherence and decoherence problems. These physical limitations are fundamental to the development of quantum computing and limit progress in other areas. Development difficulties (13%) arise mainly because the creation of effective quantum algorithms and programming are directly dependent on the characteristics of quantum hardware, which makes quantum software design complex and device-specific. Finally, cost and infrastructure limitations are arising from the need to develop and maintain expensive and difficult to operate quantum devices, while operational issues focus on the difficulties of creating effective quantum algorithms and programming, which are largely conditioned by hardware advances or limitations.

Fig. 22

Percentage of mentions per category for RQ5, quantum computing limitations

We have also analysed the incidence of limitations through time. In Fig. 23 we can observe the top four most popular limitations during the period 2019 to 2024. Over the years, there has been a growing interest in increasing the number of qubits available, especially in 2023 and 2024, this would enable developers to solve more complex problems. At the same time, the NISQ era term has become more relevant, given the lack of fully functional quantum devices, which has highlighted the need to improve qubit fidelity and error correction. In addition, noise remains a key limitation, as evidenced by its increase in mentions in 2024. These results indicate that, while progress has been made, technological limitations remain significant barriers.

Fig. 23

Percentage of mentions per category for RQ5, quantum computing limitations

3.3.6 Challenges of quantum software (RQ6)

Research question 6 asked “What are the challenges of quantum software?”

As stated in Table 4, there are 14 articles that include information relevant to this question.

These terms can be classified as hardware, integration, error correction, problem formulation and representation, scalability, and adoption and usage. For each category we have compiled the information and identified what we believe to be the main challenges indicated in the selected studies. In Table 14 we can observe the terms mentioned in the context of the challenges of quantum software and in Table 15 we can observe the classification of the terms. Both of these tables can also be found in the Repository [15], in the Appendix B. If we analyse the six categories:

• Hardware challenges: as reflected in the selected articles, these challenges focus on the evolution of quantum hardware technology and its practical applications. Quantum computing is at an early stage of development, which implies that performance in NISQ (Noisy Intermediate-Scale Quantum) hardware is relatively low, making it difficult to implement complex quantum algorithms in real-world conditions. In addition, current quantum devices still face the challenge of complying with the laws of quantum mechanics in a consistent manner, which limits their reliability. The inherent fragility of qubits, which can suffer from decoherence and other unwanted quantum effects, complicates the stability and accuracy of quantum computations. Another major challenge is the scalability of quantum devices; creating quantum processors with millions of qubits remains a distant goal, making it unfeasible to tackle large-scale problems nowadays. Achieving fault tolerance in quantum gates is another crucial challenge, as quantum errors are common and require advanced correction methods to ensure the integrity of the results.

• Error correction challenges: quantum computers are subject to errors arising from various factors, such as noise, faulty hardware and decoherence, which can compromise the integrity of quantum information. The reliability and fidelity of quantum computations are essential for the results to be useful in practical applications, so achieving high-fidelity error correction is one of the main challenges. In addition, the development of fault-tolerant quantum correction codes faces technical challenges, as many existing methods are not fully compatible with the error mitigation techniques used in current quantum systems. An additional challenge is the detection of specific faults such as phase-shift faults, which are difficult to identify and correct without affecting the quantum state.

• Integration challenges: challenges in quantum software integration are mainly focused on the lack of adequate documentation on how to reuse quantum circuits, which makes it difficult to implement them in larger systems. The low level of abstraction in current tools, together with the lack of integration, deployment, quality control and security mechanisms, create significant obstacles to the development and maintenance of reliable quantum systems. The simulation of general-purpose quantum circuits and the lack of supporting tools for quantum software design further complicate the development of practical applications. Moreover, the need to adapt and extend classical software engineering to the quantum domain requires significant effort to adjust traditional approaches to the new quantum paradigms. At the software engineering level, compiler design and agnostic invocation of quantum programs are complex issues that must be addressed to facilitate the broad adoption of quantum computing.

• Problem formulation and representation challenges: these challenges are mainly the result of the need for multiple qubits to encode the input data, which adds additional computational cost and complexity. In addition, the computationally intensive requirement of technologies such as Quantum Machine Learning (QML) increases the demand for resources, which limits their feasibility in current systems. Improving the accuracy and extending the applicability of quantum algorithms are key areas where progress is needed. A fundamental challenge is the transformation of real-world problems into representations suitable for quantum computers, which involves a high-level formulation process. The conversion of problems into the QUBO format is another major challenge, as not all problems can be easily represented in this format. Finally, the identification of new quantum algorithms, such as those used in integer factorisation, presents additional challenges in terms of the efficiency and effectiveness of quantum solutions compared to classical ones.

• Scalability challenges: scalability challenges relate to the practicality of current quantum testing methods in real-world scenarios. Current test methods are not designed to work efficiently with error mitigation techniques, which makes them difficult to implement on large quantum systems. Moreover, when using real quantum computers, such as those from IBM, additional challenges arise including incompatible test cases and the need for full program specifications, which further complicates the testing process. The scale and complexity of quantum systems is also affected by the fact that qubits can exist in a linear combination of both 0 and 1 states simultaneously, which introduces difficulties in modelling and verifying large-scale systems.

• Adoption and usage challenges: these challenges are primarily focused on producing quantum software that is repeatable, efficient, maintainable, reusable, and cost-effective. Industry will not adopt quantum devices on a large scale until these standards are achieved in quantum software, which currently remains a significant barrier. In addition, the adoption of quantum technologies also faces the Quantum Interconnect Bottleneck (QIB), which limits the communication and processing capacity between quantum devices. Another major challenge is the vulnerability associated with the adoption of distributed quantum algorithms, which may expose systems to additional risks due to complexity and lack of mature security solutions.

In Fig. 24 we can see the percentage of mentions of challenges within each category. Hardware and Integration challenges are the categories with the highest percentage of mentions (22%), indicating that problems related to the development and integration of quantum technology are seen as the most critical. This may be because quantum hardware is still in its early stages of development. Error correction is also a prominent challenge (19%), as error correction in quantum systems is critical to ensure the integrity of computations. Problem formulation and representation (17%) reflects the difficulties in translating real problems into formats that quantum computers can handle, while scalability (11%) highlights concerns about the ability of quantum systems to handle large numbers of qubits. Finally, Adoption and usage (8%) has fewer mentions, probably because although it is a concern, it is seen as a challenge after the fundamental technical problems of hardware and integration have been solved.

Fig. 24

Percentage of mentions per category for RQ6, quantum computing challenges

4 Discussion

In this section, we analyze the key findings of our systematic mapping study, reflecting on the challenges and opportunities identified in the primary studies, and discussing their implications for the future of quantum software. After analyzing the primary studies, we can now address the key findings in relation to our research questions. This discussion focuses on the insights gathered from the articles that contribute to solving software related problems through the direct application of quantum computing-either from a theorical or practical approach- and to improve quantum algorithms.

Throughout our systematic mapping study, as described in Sect. 3.1, we identified a total of 323 papers after conducting our search and a total of 33 primary studies were selected.

Previously, we noted in the related literature that a recurrent application of quantum computing is machine learning, with works such as [17, 42, 49], or [3]. We also observed that, there are several well-known quantum algorithms, as for example, Grover’s algorithm which is frequently used or serves as a basis for constructing other algorithms [55, 57], or [77]. Another well-known algorithm according to [55] is Shor’s algorithm, which, despite its popularity, is mainly mentioned with an illustrative purpose instead of being a basis for proposing new approaches, such as in [17, 23, 49, 69].

Given this background, this systematic mapping has been focused on those articles that directly address software problem solving using quantum computing, either through the practical or theorical application of existing algorithms or by proposing new approaches that exploit the capabilities of quantum computing to improve traditional solutions or create new innovative solutions. Moreover, we focus on articles that propose improvements to quantum algorithms, as we consider that optimising and adapting these algorithms is key to advancing the effective application of quantum computing in solving software problems. The main insights obtained are:

• The first research question (RQ1) was related to quantum algorithms and quantum software solutions. Our results show a trend towards the integration of quantum computing into traditional approaches, particularly in areas such as Machine Learning and Quantum AI, where the hybrid approach improves on classical systems to address complex challenges. However, quantum methods still do not outperform classical and hybrid approaches in terms of efficiency, especially in optimisation problems. Despite progress, quantum computing is still at an early stage, and improvements in both algorithms and infrastructure are needed to enable it to handle larger-scale problems. Moreover, quantum principles show potential for improving security, but we are still in the early stages of their practical application.

• Regarding the quantum software technologies being employed (RQ2), the results reveal that the most common platforms for quantum simulation are still those based on quantum circuits. IBM Quantum is the most widely used quantum simulation platform, followed by PennyLane, which stands out for its flexibility and integration with classical techniques, especially in the field of Quantum Machine Learning. The third most used platform is D-Wave, although specialised in quantum annealing, its adoption is more limited in terms of general simulations, standing out more in solving optimisation problems. As for real quantum computers, the use of gate-based circuits remains dominating, with IBM Quantum once again being the leading platform. However, there is a growing interest in annealing-based quantum computing. Whilst quantum simulator platforms continue to be preferred (62% of the selected articles), there is a significant increase in the adoption of real quantum computers, which could be attributed to the evolution in the number of physical qubits available on their computers, allowing more complex problems to be approached.

• The most important conclusions from the information gathered for RQ3, which is related to quantum software evaluation, is that the selected efforts focus primarily on overcoming resource and complexity constraints. Special attention is given to the efficiency and performance of algorithms, as well as the accuracy and quality of results. In addition, terms such as reusability, composability, and robustness highlight the importance of developing modular and resilient solutions, facilitating the integration and scalability of systems. These findings suggest that progress in quantum software evaluation will depend on optimising resources and design approaches to ensure practical and effective applications.

• Verification and validation of quantum software (RQ4) remain crucial areas in the development of quantum systems. Verification receives more attention, it is a fundamental process to ensure the correct functioning of quantum algorithms and systems, especially due to decoherence and noise effects that affect their performance. It focuses primarily on confirming that the results generated are accurate and consistent. In contrast, validation is less frequently mentioned. Its purpose is to evaluate the performance of quantum systems in real applications or to compare them with other approaches, highlighting the importance of ensuring not only the internal correctness, but also the external effectiveness of quantum solutions. This trend reflects a priority to solidify the basic principles of verification before moving on to more complex stages of validation in application scenarios. It is possible that verification is seen as an essential first step in ensuring that quantum systems are robust and reliable, allowing researchers to ensure that the technology is ready to be validated in more complex and practical scenarios.Verification and validation are crucial in software engineering as they ensure the quality and reliability of software products and systems. These processes help improve software quality, minimize risks, ensure compliance with requirements, increase reliability, and save costs and time. By detecting and correcting errors early on, verification and validation contribute to the development of software that meets user expectations and functions correctly. The fact that the research found on these topics does not seem common can lead to problems related to the quality, reliability, adequacy, and correctness of the developed quantum software, among others.

• Most of the reported limitations (RQ5) are focused on the physical and technological obstacles that prevent faster progress in the field. Among the most common limitations are the limited number of qubits, the high susceptibility to noise, and coherence and decoherence problems, which affect the stability of quantum systems. Moreover, the poor ability of current devices to maintain stable quantum states over time, and the lack of error correction limit their ability to execute precise operations. Other major challenges include the complexity in developing efficient algorithms, the dependence on hardware architecture for the creation of quantum software, and the high resource demands that make both production and access to these systems expensive, restricting their adoption to sectors with high budgets. Observing these limitations over time, between 2019 and 2024, the main limitations of quantum computing have continued to evolve, but without disappearing. In particular, in 2023 and 2024 there has been a growing interest in increasing the number of qubits available to address more complex problems. At the same time, the concept of the NISQ era has become more relevant. In addition, noise remains one of the most frequently mentioned barriers, with an increase in 2024, indicating that, despite advances, technological limitations persist.

• In terms of the challenges of quantum software (RQ6), there are multiple aspects to consider, ranging from hardware development to the adoption and use of these technologies. The most frequently mentioned challenges in the studies reviewed are those related to hardware and integration, reflecting the difficulty of building scalable and reliable quantum systems. Error correction is also a key challenge, as current quantum systems are highly susceptible to noise and decoherence. Transforming real problems into formats compatible with quantum computing requires specialised approaches and remains a challenge. Furthermore, scalability and the adoption of quantum software present additional challenges, as the technology has not reached sufficient maturity for large-scale implementation. Addressing these issues will probably need ongoing efforts and may not produce solutions in the short term.

From all the evidence gathered, we can conclude that this technology is still at an early stage of development. However, the reported quantum software research lays the foundation for further research in a number of critical areas. These include improving integration with classical tools, optimising error correction methods, facilitating the formulation of complex problems and extending hardware scalability. Moreover, it is essential to promote strategies that not only enhance the technical capabilities of quantum software, but also encourage its adoption in industry, ensuring that it becomes a practical and cost-effective solution for real-world applications. These efforts will be crucial to accelerate the transition from theoretical advances to tangible, industry-ready quantum software solutions.

5 Threats to validity

This section tackles threats to the validity of the study by following the recommendations in [70]:

• 1. Construct and conclusion validity. We admit that it is impossible to read all that has been published about a subject in its entirety. We employed the SCOPUS database, as a result of its content being meticulously chosen under strict and superior scientific inclusion and exclusion standards. This ensures that the database only includes trustworthy, carefully chosen content, and verifies the accuracy of the information. Since much grey literature is secondary, we did not include supplementary documents, such as technical reports, books, etc. As a result, it is possible that any important papers were left out, but based on what we have seen, we consider it unlikely.

• 2. Internal validity. In assessing the internal validity of our systematic mapping study, it is important to consider the methods employed by the authors to verify the accuracy of the classification and other procedural activities. To ensure the reliability of our findings, all authors independently reviewed a subset of the data to confirm the consistency of the approach. This process involved cross-referencing of the classifications and assessments with their own interpretations, thereby enhancing the robustness of the methodology. By corroboring the results with a small sample, we mitigate the risk of misclassification or bias, thereby strengthening the internal validity of our study.

• 3. Threats to external validity. When evaluating the external validity of our systematic mapping study, we consider the generalizability of our findings beyond the specific scope of our research. Our study relied solely on SCOPUS as the primary database, which may limit the breadth of perspectives represented in our sample. However, we aimed to mitigate this limitation by employing a comprehensive search query and inclusion criteria to capture a diverse range of literature within the database. While our focus on SCOPUS ensures a rigorous approach to data selection and analysis, it is essential to acknowledge that our findings may not fully reflect insights from sources outside this database. Therefore, readers should interpret our results within the context of SCOPUS’s coverage and consider additional sources to complement our findings.

6 Conclusions and future work

Quantum computing has evolved rapidly these past few years; this technology is believed to tackle some of the limitations imposed to classical computing and redefine problem solving and data management, among others.

This paper studies algorithms and quantum solutions, technologies, evaluation, verification, validation, limitations, and challenges of quantum software discussed in the literature available in conference papers and journal articles up to December 2024. Only a 10% of the 323 documents found had relevant information that answered the six research questions proposed.

As it was indicated in [75], the field is evolving rapidly, but there is a need for research to gather more experience and increase the maturity of overall quantum computing.

This new paradigm imposes a great challenge, as the concepts and rules of classical devices are far from the quantum computing domain. But no matter the difficulty of the field, there are some aspects in which the computing community cannot fall behind; these were broadly discussed in Sect. 4, in which we discussed that something that cannot be measured cannot be improved. If we want to provide quantum solutions and services with a high level of quality, which was one term broadly repeated as a term for evaluating quantum software, we have to investigate towards solving the problems related to evaluating, testing, and the overall verification and validation of the quantum solutions being developed.

There are also some important topics that we may say are not being talked about enough, such as the security of the data employed or for instance a term which has not been even mentioned, energy consumption. We could observe terms related to efficiency and performance, but not related to the energy needed to run these devices. In a previous article [16], we were able to obtain empirical evidence of the power consumption required to run some of the gate-based quantum computers available during the study on the IBM Quantum platform. Energy efficient quantum solutions should be a topic further discussed and research upon; it is worth remembering that ICTs will account for 20% of global energy consumption by 2030 [20], and quantum computing cannot serve as a new source for boosting up this percentage.

Therefore, in the future, we want to keep investigating with a focus on the energy consumption of quantum computing. The main limitation observed in our results-the number of available qubits-was a restriction we faced in a previous study [16]. However, the quantum hardware landscape has significantly evolved, with devices now reaching up to 127 qubits, far surpassing the 5-7 qubit systems used in earlier studies. In this context, we plan to extend our previous research, performing additional case studies with different algorithms, exploring alternative encoding methods, and executing these experiments on real quantum computers. By doing so, we aim to contribute to the body of knowledge in quantum computing, helping to bridge the gap between theoretical advancements and practical applications, while also incorporating the critical aspect of addressing the energy consumption of quantum computing. This focus on energy efficiency will be essential as the field progresses, ensuring that future quantum systems are not only powerful but also sustainable in terms of their environmental impact.

Appendices

Appendix A: Selected articles

This appendix contains the table that summons the identification numbers, titles, and references of the selected articles (Table 5).

Table 5 Selected articles

Appendix B: Tables

This appendix contains the tables that summon the number of times each of the terms appeared in the six different research questions as well as the categorisation, if any, of terms within each research question (Tables 6, 7, 8, 9, 10, 11, 12, 13, 14, 15).

Table 6 Quantum algorithms, or software solutions mentioned or proposed and the number of times each one of them appeared (counts) in the 33 primary studies for RQ1

Table 7 Quantum simulators or simulation platforms employed, related to RQ2

Table 8 Real quantum computer platform employed, related to RQ2

Table 9 Parameters or other criteria to evaluate quantum software and the number of times each one of them appeared (counts) in the 30 primary studies for RQ3

Table 10 Responses for RQ3 categorized as performance and efficiency, result accuracy and success, resources and complexity, implementation, and others

Table 11 Verification or validation of quantum software and the number of times each one of them appeared (counts) in the 13 primary studies for RQ4

Table 12 Limitations of quantum software and the number of times each one of them appeared (counts) in the 26 primary studies for RQ5

Table 13 Responses for RQ5 categorized as hardware, operational, development, and infraestructure

Table 14 Quantum software challenges found in the 14 primary studies for RQ6

Table 15 Responses for RQ6 categorized as hardware, error correction, integration, problem formulation and representation, scalability, and adoption and usage