Quantum and quantum-inspired algorithms have not yet been systematically classified in the context of potential Operations Research (OR) applications. Our systematic mapping is designed for quick consultation and shows which algorithms have been significantly explored in the context of OR, as well as which algorithms have been vaguely addressed in the same context. The study provides rapid access to OR professionals, both practitioners and researchers, who are interested in applying and/or further developing these algorithms in their respective contexts. We prepared a replicable protocol as a backbone of this systematic mapping study, specifying research questions, establishing effective search and selection methods, defining quality metrics for assessment, and guiding the analysis of the selected studies. A total of more than 2,000 studies were found, of which 149 were analyzed in detail. Readers can have an interactive hands-on experience with the collected data on an open-source repository with a website. An international standard was used as part of our classification, enabling professionals and researchers from across the world to readily identify which algorithms have been applied in any industry sector. Our effort also culminated in a rich set of takeaways that can help the reader identify potential paths for future work.
1 Introduction
We live in a world where economic, social, and environmental challenges abound. Professionals and researchers working within operations research (OR) topics are facing ever-increasing pressure to make operations green, in the sense that they have to be sustainable from many viewpoints, including that of the environment and the social well-being of involved parties [57]. An example of this pressure is the recent Glasgow Climate Pact, supported by almost 200 countries that have committed to accelerating measures to keep global temperatures from rising more than 1.5 Celsius degrees [141]. All in all, these challenges culminate in an unprecedented set of objectives and constraints in OR problems that have to be tackled with creativity and innovation.
Optimization problems are becoming more complex with the addition of new objectives, variables, and constraints. For example, the increase in the installation of a large number of new solar and wind electricity generators has resulted in an increase in the number of variables in the electrical grid, as opposed to big conventional installations of large power plants or dams [31, 120]. This higher level of complexity is a challenge for computers, as they will require more time and energy to compute solutions for optimization problems of practical interest. This increase in time and energy is not sustainable, as: (i) (correct) solutions need to be provided in a timely manner; (ii) energy resources are limited and at some point may not scale; and (iii) one cannot cool computers from the heat generated from the increased energy usage [97, 166]. Consequently, quantum computers raise significant expectation that they will play an essential role in overcoming some, if not all, of these limitations, as they will be able to solve specific tasks in a much faster and more efficient way.
In the last decade, new advances in the manufacture of quantum processing units have generated a lot of buzz around this unconventional type of computing, with advantages over classical computers already being claimed [11, 177], although some claims are disputed [2]. In other words, the science of quantum computers was settled long ago, but only recently have they been physically realizable. These novel computers promise new ways to solve problems that classical computers would not be able to solve in a timely manner [95]. Algorithms executed in quantum computers are called quantum algorithms and are usually applied as part of a hybrid approach. That is, the general approach to using quantum computers is to use classical computers to control the flow of a program, to pre- and post-process data, and only submit specific tasks to a quantum computer, especially when there is an advantage in doing so [151].
Even before the first quantum computer was manufactured, researchers developed classical algorithms that were inspired by the theory of quantum computing. These algorithms take advantage of certain quantum computing properties to achieve higher performance and capabilities when solving problems. We call these quantum-inspired algorithms [96].
Considering the emergence of different quantum computing devices, researchers and practitioners are increasingly interested in leveraging them by designing/implementing quantum or quantum-inspired algorithms, especially to address real-world problems from OR. Namely, the industry is making consistent efforts in this line, with more and more companies investing in quantum-based commercial applications [84]. However, since the field is emergent and has not matured enough, we have found its literature to be sparse and not yet systematically classified.
We have identified two particular needs: First, professionals and practitioners from the industry who are interested in leveraging quantum or quantum-inspired algorithms in their areas need to know which algorithms have already been developed, if any, as well as which algorithms would present good potential solutions for their operational challenges. Second, researchers who are interested in joining the research effort want to know what has already been researched and what research questions are still open. Moreover, this need can also arise from researchers who are experienced in quantum algorithms but have not yet applied them to OR problems. In other words, both professionals and researchers need a map of what has already been explored and a list of potential paths for further exploration.
We believe that the best way to meet the identified needs is to perform a systematic mapping study (SMS) on applications of quantum or quantum-inspired algorithms in OR problems [110]. An SMS acts as a significant jump-start for newcomers since its outcome is a mapping of what has been significantly explored, of what has been vaguely addressed by the research community, and also of what has not been researched so far. Moreover, the mapping identifies potential paths for future work. All in all, these are the reasons that we believe a systematic mapping study is the best tool to address the needs mentioned above.
The SMS follows the guidelines from [65, 94, 110]. It is divided into six steps: (i) research questions, (ii) search, (iii) study selection, (iv) quality assessment, (v) data extraction, and (vi) classification and analysis. We designed a protocol that describes these steps in detail and allows the reader to reproduce the systematic mapping study. The protocol was also designed with validity threats taken into account.
While conducting the systematic mapping study, more than 2,000 studies were found,1 a number that decreased to 149 after the search selection step. We extracted and analyzed data from these studies, including features such as quality of the study, quality of the reporting, type of approach (classical, hybrid or quantum), algorithm name, whether the study uses a simulator or a real quantum computer, type of quantum computer, publication year, and publication source.
An important innovation of the SMS is the usage of the International Standard Industrial Classification of All Economic Activities (ISIC) as part of the data classification. This standard was used to classify the OR problems in terms of the industry activity that best relates to them. The classification enables us to identify which industry activities have been researched so far, as well as which industry activities lack any research effort.
The insights stemming from data analysis are discussed and synthesized into a list, itemized by feature. The reader can quickly navigate around the list, with the ability to immediately read the insights related to whatever feature they are interested in. We also developed direct answers to the research questions that we have posed.
Two additional outputs were created as part of the systematic mapping study: a decision flowchart that newcomers can use to find a potential path for future work, and a website with interactive charts that readers can use to explore the data related to the 149 studies.
To sum up the findings, we observed a trend of growth in the number of publications per year. Plus, we found a significant amount of primary studies that apply the following algorithms: Quantum Particle Swarm Optimization, Quadratic Unconstrained Binary Optimization, Quantum Genetic Algorithm, and Quantum-inspired Evolutionary Algorithm. We also found a significant amount of evidence surrounding applications on the following industry activities: “electricity, gas, steam and air conditioning supply”, “transportation and storage”, “manufacturing”, “professional, scientific and technical activities”, and “information and communication”.
We conclude that the results of the systematic mapping study meet the need that we identified in the beginning. Some findings negatively surprised us, such as the lack of applications involving gate-based quantum computers. Nonetheless, the results revealed clear paths for future work that newcomers can follow, irrespective of whether their background is in operation research or in quantum computing (or even in neither).
We list the innovations/contributions of this work as follows:
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An extensive search for primary studies was conducted among several search engines, including Scopus, IEEE Xplore, ACM Digital Library, and ScienceDirect. Snowball sampling was performed to increase the search depth. Selection criteria were applied to the search results, resulting in a list of primary studies that are relevant to our research questions.
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Data was extracted from the searched primary studies, culminating in a curated list of relevant primary studies that can be analyzed in several aspects, such as the quality of the study, and the quality of the reporting, among many other features.
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We used the International Standard Industrial Classification of All Economic Activities (ISIC) to catalog the selected primary studies regarding the economic activity that most relates to their OR problem. To the best of our knowledge, this type of classification has never been done before for OR problems.
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We analyzed and condensed all the data into an itemized list highlighting the main takeaways. This list serves as a practical map for researchers and practitioners to get acquainted with what has already been done and what are possible paths for future work.
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We developed a decision flowchart that helps researchers who wish to find a good research question, especially those who are interested in starting their first research work on an application of a quantum or quantum-inspired algorithm in an OR problem.
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We published a website that displays interactive charts, enabling readers to directly explore the data that was collected as part of the systematic mapping study. This website allows readers to gain more insights than they would gain by only reading the full-text.
The paper is outlined as follows:
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Section 2 explores related work in the literature and addresses whether all the conditions to undertake a systematic mapping study are reunited.
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Section 3 reveals the protocol of the systematic mapping study. The protocol specifies what are the steps of the systematic mapping study and details how each step is conducted. The protocol itself should make it possible for any reader to replicate the SMS.
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Section 4 shows the discussion of our findings, which synthesizes the data in an itemized list that facilitates reading. This section also includes direct responses to the research questions that guide the systematic mapping study. A decision flowchart is also included, which helps interested readers who wish to start working in the field.
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Section 5 presents the conclusion of the study. The conclusion includes a subsection dedicated to future work for any researcher who wishes to replicate the systematic mapping study in the future.
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The Supplementary Material, published electronically, contains the results of the systematic mapping study, including charts that illustrate the data that was collected, as well as our analysis of the data. We decided to place the results there due to length constraints.
Our paper caters to professionals and practitioners in the industry as part of their initial approach to the field of quantum computing, so that they can apply quantum or quantum-inspired algorithms in their sector. We provide relevant material, which could be part of the first step for any OR researchers interested in joining the field of quantum computing, whether they are first-year graduate students or experienced faculty. Last but not least, we also challenge interested quantum computing researchers to read this systematic mapping study as part of their initial move into the field of OR.
2 Contextualization
Before undertaking a systematic mapping study, it is necessary to perform a literature search to confirm that the need for the mapping has not already been addressed [110]. This confirmation prevents redundant work and unnecessary research effort. Moreover, this is necessary to find whether the literature is sufficiently mature to allow for a systematic literature review instead [65, 110]. Systematic literature reviews require a significant amount of primary, secondary, and possibly tertiary studies to derive findings and conclusions with sufficient validity and generalizability. On the other hand, systematic mapping studies are suitable for research topics that have a significant but small number of primary studies, lack secondary or tertiary studies, and benefit from a detailed mapping of what has been researched.
2.1 Conditions to Undertake a Systematic Mapping Study
We can classify existing studies in the literature as one of the following three types, according to the definitions from [65]:
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Primary studies, which are empirical studies that focus on a specific research question. We also refer to these studies as evidences.
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Secondary studies, that review all the primary studies pertaining to a specific research question. Their goal is to reunite and synthesize existing evidence on the research question.
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Tertiary studies, which are a higher-level type of study, usually only done in fields with a significant number of evidences and secondary studies. These studies review secondary studies that address a specific research question.
2.1.1 Initial Literature Search.
We searched on the literature using a set of relevant electronic databases that we are aware of (Google Scholar, ArXiv, SpringerLink, Scopus, IEEE Xplore, ACM Digital Library, Inspec, Compendex & Knovel, and ScienceDirect), as well as an extensive set of string queries. The results did not return secondary or tertiary studies that addressed applications of quantum or quantum-inspired algorithms in OR problems. Nevertheless, we found primary studies that each addressed a specific application of quantum or quantum-inspired algorithms in a specific OR problem. The number of primary studies that we found relevant was very low—we found fewer than 30 primary studies in this initial search. This number is far from sufficient to undertake a systematic literature review and supports the need for a systematic mapping study [65].
2.1.2 Similar Works.
There exists a significant number of works in the literature that survey quantum and quantum-inspired algorithms, especially given that quantum-inspired algorithms were suggested in the 90s by [99]. However, none specifically address their usage in the context of OR problems. We list some of the most closely related works that we have identified:
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[39] presents a detailed survey of quantum-inspired metaheuristics, including descriptions of the algorithms and their class of problems. This paper does not specify the OR problems to which these algorithms are applied, but rather indicates the types of problems these metaheuristics can solve (e.g., feature selection or optimization problems). Our SMS presents itself as a natural extension to this paper, contributing to a good view on the application (if any) of these metaheuristics in the different economic activities included in the ISIC.
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[96] presents another detailed survey of quantum and quantum-inspired algorithms, with descriptive pseudocodes and comparisons with regards to runtime, accuracy, and other complexity metrics. Similarly to the paper mentioned just before, this paper indicates the types of problems these algorithms can solve without listing any of their applications in OR problems. Therefore, our SMS also presents itself as an extension to this paper for the same reasons.
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[74] presents a detailed study of a specific set of quantum and quantum-inspired algorithms, including their descriptions. This paper shows experiments with each algorithm in their typical applications, mostly related to machine learning. These experiments, apart from one that solves a common combinatorial optimization problem, do not attempt to represent the population of possible applications of these algorithms. Our SMS attempts to provide this representation, as accurate and systematic as possible.
We find it necessary to reference [111] as another motivating work, which is a detailed systematic survey covering the extensively studied and established field of classical algorithms for OR. Our SMS, while not as exhaustive due to the nascent stage of quantum and quantum-inspired algorithms in OR, complements this paper by enumerating the applications of quantum and quantum-inspired algorithms.
All aspects considered, we believe that the conditions to design and conduct a systematic mapping study have been reunited. We have identified a need. We found no existing work that addresses that need. We came to the realization that a systematic mapping study is the best effort to meet that need, considering alternatives such as a systematic literature review.
2.2 An Introductory Overview of Quantum and Quantum-inspired Algorithms
In this subsection, we introduce the basic concepts present in many quantum and quantum-inspired algorithms, as well as short descriptions of the Quantum Particle Swarm Optimization, the Quadratic Unconstrained Binary Optimization solvers, and the Quantum Genetic Algorithm. The choice of those algorithms aligns with the three most popular approaches that we have identified in our SMS. For a more detailed and comprehensive reading, we refer to [39, 41, 74, 96, 138].
2.2.1 Quantum and Quantum-inspired Algorithms.
Quantum computing uses special and unique properties of quantum mechanics to perform computations. In this form of computing, the most common unit of information is the qubit, represented as a quantum mechanical system whose state space is an orthonormal basis formed by the unit vectors \(|0\rangle = \left[\scriptsize\begin{array}{l} 1\\ 0 \end{array}\right]\) and \(|1\rangle = \left[\scriptsize\begin{array}{l} 0\\ 1 \end{array}\right]\), such that
where \(\alpha\) and \(\beta\) are complex numbers that obey \(\left|\alpha \right|^2 + \left|\beta \right|^2 = 1\). This quantum mechanical system can be grouped together with other qubits to form systems of multiple qubits.
Together with quantum operations that apply reversible changes to the state of the quantum system, important properties such as superposition and entanglement can be leveraged to process information in a manner that is not possible in classical systems.
Algorithms that directly leverage quantum phenomena are called quantum algorithms. In contrast, algorithms that do not take advantage of quantum phenomena but implement classical subroutines inspired by these phenomena are called quantum-inspired algorithms. The rationale behind those algorithms is that, under certain assumptions, they perform better than their classical counterparts, especially in nonlinear optimization problems [39, 41, 74, 96].
2.2.2 Quantum Particle Swarm Optimization.
Particle Swarm Optimization is a metaheuristic that deploys multiple particles (also known as generating multiple solutions) in the domain of the optimization problem, and then uses a function to update the position of its particles iteratively. This function uses the aggregated information of all particles to move each particle in a specific direction, balancing exploration and exploitation. Over time, the particles will converge in the global optima of the optimization problem. PSO has a quantum-inspired variant called Quantum Particle Swarm Optimization(QPSO), which uses qubit representations for the particles and special functions inspired by quantum operators. QPSO has been empirically shown to converge at a faster rate than the standard PSO [39, 74].
There exist some quantum and quantum-inspired computers (and algorithms) that are specialized in solving a specific class of problems called Quadratic Unconstrained Binary Optimization(QUBO) [41]. These problems are represented as follows:
where \(x\) is a column-vector of binary variables and \(Q\) is a symmetric square matrix with diagonal elements representing the linear coefficients and off-diagonal elements representing the quadratic coefficients. This representation has been shown to be capable of representing any problem in the complexity class NP. Due to the straightforward representation, the usage of QUBOs has become increasingly popular among users of certain quantum and quantum-inspired computers, such as D-Wave Systems Inc. [42, 138]. In fact, most software stacks for universal quantum computers also include specialized utilities to solve problems in the QUBO form. In the remainder of our paper, whenever QUBO is mentioned, we refer to the entire approach of solving a problem with a QUBO.
2.2.4 Quantum Genetic Algorithm.
Genetic algorithms (GAs) are a specific class of algorithms that start with an initial population of solutions, which are iteratively refined until they reach the optimal solution for the optimization problem [39, 41, 74, 96]. These algorithms are inspired by concepts from the field of genetics, as the operators that iteratively refine these populations are called mutation and crossover operators. A mutation operator applies isolated changes to some elements of the new copies of the population. A crossover operator mixes different values amongst different elements of the population. Both operators allow GAs to explore the domain of the optimization problem without getting trapped in a local optimum, especially in nonlinear optimization problems. Quantum Genetic Algorithms are a variation of GAs that use quantum-inspired representations of the elements in the population, as well as quantum-inspired mutation and crossover operators. For example, some QGAs represent elements in the population with quantum states and apply quantum rotation gates as mutation operators, as well as crossover operators that emulate interference between quantum states [73].
3 Research Method
In this section, we present the protocol that we designed and implemented for our systematic mapping study, which follows the guidelines from [65, 110]. The main idea behind the protocol is to provide a way to facilitate the reproduction of the study, as well as to avoid or minimize the introduction of bias from the researchers conducting it.
First, in Section 3.1, we specify the research questions that drive the entire SMS. Second, in Section 3.2, we describe the search methods that are used to find studies that are related in some way to the specified research questions, as well as how primary studies are selected among the set of studies identified in the search. Afterwards, the selected studies pass through a quality assessment step, described in Section 3.3. Next, in Section 3.4, we describe which data is extracted from the selected studies. After that, in Section 3.5, we show how this data is analyzed and how the outcomes of the analysis are discussed. Finally, in Section 3.6, we discuss the validity and reliability of our study.
3.1 Research Questions
The goal of our systematic mapping study is to “summarize and disseminate research findings” [9], by giving a comprehensive overview over the application of Quantum and Quantum-inspired algorithms in Operations Research (OR). This overview will serve as an inventory for papers and as a good starting point for anyone who wishes to enter in this area, such as graduate students early during their Ph.D. studies [65, 110].
Concretely, our motivation is to assess what has been done with quantum algorithms and quantum-inspired algorithms in the field of OR. We expect to identify evidence clusters and evidence deserts in this area, which may guide or incentivize researchers to perform primary studies in specific sub-areas.
Considering our motivation and goal, as well as the structure of an SMS, we developed the following set of research questions that are to be answered as part of our SMS:
RQ1. What existing approaches apply quantum or quantum-inspired algorithms to the field of Operations Research?
RQ2. In which sources, in which years, and in which quantities were approaches that apply quantum or quantum-inspired algorithms to the field of Operations Research published?
RQ3. Which research questions related to approaches that apply quantum or quantum-inspired algorithms to the field of Operations Research are addressed by a significant amount of evidence (evidence clusters)?
RQ4. Which research questions related to approaches that apply quantum or quantum-inspired algorithms to the field of Operations Research are addressed by a scarce amount of evidence (evidence deserts)?
Having specified the research questions, we can develop the search strategy to be followed when conducting the study, which is presented in the next subsection.
3.2 Search and Study Selection
A systematic mapping study should strive for a selection of papers that constitute a good representation of the population. Hence, more papers is not necessarily better, as it may induce a distorted view of the overall research extant in the area. Nonetheless, if very few papers have been published so far in the area, it may be adequate to search for the maximum number of papers possible [110].
In our case, a simple and quick search in electronic databases suggests that very few papers apply quantum or quantum-inspired algorithms in the context of OR (e.g., by searching in IEEE Xplore for the keywords “Quantum” and “Operations Research”). Hence, during our search step, we will be targeting the maximum amount of studies while still keeping the number of studies feasible. Particularly, we should avoid steps in our search strategy that lead to an explosion in the number of findings.
For our SMS, three search strategies are used: electronic database search, reverse snowball sampling, and forward snowball sampling. In the following subsubsections, each search strategy is described in the order they are performed. We call the reader to take into account that those strategies are used while doing study selection between them, a step that is described in the next subsection. Figure 1 illustrates the order of these search and study selection steps during the SMS, which resulted in a list of 149 primary studies [3 –8, 10, 12–28, 30, 32–38, 40, 43–56, 59–63, 66–72, 75–83, 85–91, 93, 98, 100–109, 112–119, 121–137, 139, 140, 142–150, 152–165, 167–176, 178, 179].
Fig. 1.
Fig. 1. Illustration of the steps taken while searching and selecting studies and the number of selected documents after each step.
The search step is followed by the study selection step, where the collected studies pass through a selection process that decides which studies are candidates for answering some or all of the specified research questions. For this step, we read the title and abstract of each study.
The selection process consists of checking each study against a set of inclusion and exclusion criteria. This means that, by default, a study is excluded. To be included, the study must meet all the inclusion criteria. However, once the study meets any of the exclusion criteria, it is immediately excluded. With respect to our inclusion criteria, we only accept peer-reviewed studies written in English and published during or after 2011. We chose 2011 because it was in this year that the first commercially available quantum computer was released—D-Wave One, from D-Wave Systems, Inc. Moreover, a quick search in electronic databases also suggests that research on real-world applications with quantum and quantum-inspired algorithms only started to become significant during the years after 2011. With respect to our exclusion criteria, we avoid studies that are not a primary study or whose full-text is not accessible to us.
In order to validate the study selection, for each study, we find its ranking based on the SCImago Journal & Country Rank (SJCR) or on the conference rankings from the Computing Research and Education Association of Australasia (CORE), depending on whether the study is part of a journal or of a conference. For studies that belong to journals, we chose to use the SJR Indicator, as it is a good measure of the influence of the journal, enabling us to compare each study in terms of its journal’s influence [92]. We also take into account the SJR Best Quartile for the same purposes. For studies that belong to conferences, we chose to use the CORE rankings, since their method assigns each conference to a specific category with an associated prestige and impact [1]. However, we note that those indicators are limited and cannot be taken at face value.
Once we have determined the rankings associated with each of the studies, we validate the selection process by analyzing the distribution of selected studies among the rankings, since it provides an evidence on whether our process generally selects studies associated with journals and conferences with a higher prestige.
For a more detailed description of the search and study selection steps, we refer to Appendix A. There, reproducible instructions for all the different search and selection strategies are included.
3.3 Quality Assessment
Once the search and study selection steps are finished, the collection of included studies passes through a quality assessment. The objective of this step is to assess the quality of each study, such that we take into account the outcomes of the studies with their associated quality, giving more importance to studies with better quality.
For the purpose of our systematic mapping study, our quality assessment should focus on whether the evidence was well collected (quality of study) and on whether the evidence was well reported (quality of reporting). This assessment may help us identify evidence clusters that lack quality in their study or in their reporting.
To evaluate the studies on the quality of their study and their reporting, we developed two checklists. The first checklist, concerned with assessing the quality of the study, is shown in Table 1. The other checklist, concerned with assessing the quality of the reporting, is shown in Table 2. These checklists were made according to the guidelines of [65, 110] and are adjusted to our particular research questions. As is good practice, both tables include a checkmark (\(\checkmark\)) column to visually remind practitioners to ensure they answer all questions for each study.
Table 1.
ID
Question
\(\checkmark\)
SS01
Does the study have a comparison or control application?
SS02
If the study compares its application with another application, are both applications compared among different scenarios or settings?
SS03
Does the study use a real-world scenario or case study for its application, even if it is simplified?
SS04
If the study does not use a real-world scenario or case study for its application, does the study use benchmarks?
SS05
If the variables and/or scenarios in the study were defined randomly, does the study specify and justify the randomization?
SS06
If the variables and/or scenarios in the study were defined arbitrarily, does the study explain the reasoning behind the definitions?
SS07
Does the study use statistical analysis to assess the behavior of its application?
SS08
Does the study use statistical tests to check hypotheses?
SS09
Are validity and reliability threats addressed in the study?
SS10
Does the study debate on possible future work?
Table 1. Checklist to Assess the Quality of the Study
Table 2.
ID
Question
\(\checkmark\)
SR01
Does the study present in the beginning of the text its overall structure?
SR02
Does the study present the motivation behind the work?
SR03
Does the study indicate in the beginning of the text the research questions or research goals?
SR04
Does the study summarize in the beginning of the text the methodology?
SR05
Does the study summarize in the beginning of the text the conclusions?
SR06
Does the study describe its methodology in such a way that it is reproducible?
SR07
Does the study present the implementation of its application by means of a reproducible pseudocode listing, code repository, or any other specification?
SR08
If the study uses a comparison or control application, does the study present the implementation of its comparison or control application by means of a reproducible pseudocode listing, code repository, or any other specification?
SR09
If the study uses scenarios or settings, does the study present a reproducible specification of those scenarios or settings?
SR10
If the study uses statistical analysis, does the study detail this step in such a way that it is reproducible?
SR11
Does the study use images or tables to present its results?
SR12
Are all the tables and images referred to and explained in the text of the study?
SR13
Does the study use all its references in its text?
SR14
Does the study present the implementation of its application by means of a code repository?
SR15
If the study presents a code repository, is the code used in the study documented in any way?
SR16
Does the study describe the contributions that have been made?
Table 2. Checklist to Assess the Quality of the Reporting
For each checklist, the final score is calculated as follows:
\begin{equation} S = \frac{\text{Number of checkmarks ($\checkmark $)}}{\text{Number of applicable questions}} \times 100\%, \end{equation}
(3)
where \(S\) ranges from \(0\%\) to \(100\%\), and the higher the score the better is the assessed quality.
We use a Microsoft Excel file to fill these checklists for each study, calculating the study quality score (\(S_S\)–SS00) and the reporting quality score (\(S_R\)–SR00) for each study. The results are then exported to a comma-separated values (CSV) file which is publicly available and that can be analyzed with any statistical program.
To validate this step, we use the SJCR and CORE Rankings once again and then correlate those rankings with the study quality and reporting quality scores. We expect higher rankings to be associated with higher scores, which would serve as evidence that our quality assessment is representative of the quality of the studies.
3.4 Data Extraction and Classification
The data extraction step is responsible for extracting most of the data that will be fundamental to answer the research questions. Hence, we have to develop a form to be filled in for each selected study. This form should specify which data to be extracted and the instructions to extract each data should be as objective and concise as possible.
In Table 3, we present the extraction form of our study. The information extracted will constitute the majority of the data used to answer the research questions, with the remaining data coming from the previous steps, mainly the quality assessment step. Further instructions or clarifications of the extraction form are also presented together with the table, which should help with the reproducibility of this step.
Table 3.
ID
Query
Possible Values
D010
Type of Approach
Classical OR Quantum OR Hybrid.
D020
Algorithm Name
D030
Does the study use a simulator or a real quantum computer?
Quantum Computer OR Simulator OR Not Applicable
D040
Type of Quantum Computer
Gate-based OR Annealer OR Not Applicable
D050
ISIC Section
A to U.
D051
ISIC Division
01 to 99.
D052
ISIC Group
011 to 990.
D053
ISIC Class
0111 to 9900.
D060
Publication Year
2011 to 2021.
D070
Publication Source
Journal; Conference; Workshop; Other.
Table 3. Extraction form
The first query categorizes the approach of the study in respect of the algorithms it uses. We use this information as part of our answer to the research question RQ1. Concretely, a classical approach only contains classical and quantum-inspired algorithms; a quantum approach only contains quantum algorithms; and a hybrid approach contains classical, quantum and quantum-inspired algorithms. In other words, classical approaches only use classical computers, quantum approaches only use quantum computers, and hybrid approaches use both computers.
The second query extracts the name of the relevant algorithm of the study. Variants of the same algorithm should be attributed to the same name since we are not interested in the specifics of each application. Our goal with this query is to find which general algorithms have been used (RQ1).
For the particular cases in which the approach is categorized as hybrid or quantum, we categorize the study in respect of its usage of a real quantum computer or a simulated quantum computer. We also categorize the study in respect of the type of quantum computer it uses, such as gate-based quantum computer or quantum annealer. Both queries are relevant to answer the research questions RQ1, RQ3, and RQ4.
A piece of relevant information to be extracted, particularly for the research questions RQ3 and RQ4, would be the industry where the application is used or intended to be used. To retrieve this information, we decided to use the ISIC standard [29] for several reasons: first, this standard was developed by the Department of Economic and Social Affairs of the United Nations Secretariat, with the purpose of providing a single standard to be used by countries when collecting and reporting statistics about productive activities; second, the unified nature of this standard makes it possible to compare productive activities among different countries or studies; last, the standard enables us to find not only which industry sectors and activities have applications of quantum and quantum-inspired algorithms, but also which industry sectors and activities lack applications.
The procedure that is followed to choose an appropriate ISIC section, division, group, and class for each study is described as follows: given the study, we choose the ISIC section that is most relevant to the activity conducted by the application of the study. The same strategy is used for choosing the division, group, and class. If there are doubts about which choice to take, we keep the possible choices into account and then we use the information of the following sublevel to choose one of the choices. As an example, let us assume that for a study we are undecided between ISIC section M or P. We take both sections into account and check the following division. We then determine that the study actually is best corresponded to the division 73; thus, we discard section P.
Finally, we also extract the publication source and year from the studies, which are particularly useful to answer the research question RQ2. With this information, we can obtain insights such as which type of publication source has provided the most meaningful research so far, as well as the overall research trend over the years.
3.5 Analysis
Once the quality assessment and the data extraction and classification steps are finished, a significant amount of information is collected for each study. This information is composed of features, with each feature corresponding to the information collected about a specific query or data extraction.
These features need to be systematically analyzed such that we have good grounds for answering the research questions. For this reason, the analysis is divided into three parts: analysis of individual features (3.5.1), in which each feature is individually analyzed; analysis of relationships between pairs of features (3.5.2), in which we attempt to find relevant relationships between any pair of features; and exploratory data analysis (3.5.3), which concerns the exploration of the information with the goal of finding relevant relationships and insights that may not have been captured in the previous two parts of the analysis.
Before proceeding to the analysis, we have to define a set of features that are considered relevant and will be analyzed as part of the first two parts of the analysis. The relevance of the features is determined by their importance in answering the research questions and by their redundancy. Table 4 shows the features that we have selected as relevant, uniquely coded with an ID for future reference.
Table 4.
Table 4. Features Selected for Individual Analysis
3.5.1 Analysis of Individual Features.
In the first part of the analysis, the features are individually analyzed. The type of analysis depends on the possible values of the feature. The goal of this part of the analysis is to achieve in a clear and concise way a good visualization of how the studies are distributed among the values of their features, potentially identifying any trends or insights.
Features with a percentage value (SS00 and SR00) are analyzed with violin plots. These visualizations should help us understand what is the distribution of the studies with respect to their \(S_S\) or \(S_R\).
Features with a small number of categorical values (SS01 to SS10, SR01 to SR16, D010, D030, D040, D070) are analyzed with stacked bar charts that show their proportion. We can group some of the bar plots together for increased clarity. Particularly, we can group SS01 to SS10 and also SR01 to SR16. These stacked bar plots will be useful to get a general idea of which share of the studies belongs to a certain categorical value from a feature.
For the features D020 and D060, their different values will be plotted on simple bar charts. With this visualization, we will be able to compare the number of occurrences of each value among all the studies.
Finally, for the features D050 to D053, we will prepare a series of bar charts that follows the ISIC hierarchy. First, for the ISIC sections, we present a bar chart with the number of studies corresponding to each ISIC section. Next, we select the ISIC sections that are most relevant in number of publications (five or more). We present a bar chart with the number of studies corresponding to each ISIC division from the selected relevant ISIC sections. Afterwards, if we identify any bar that is relevant in number of publications, we select the corresponding ISIC section and prepare a bar chart with the number of studies corresponding to each ISIC group from the selected ISIC section. Once again, if there is still a bar that stands out from the others, we repeat the process by preparing a bar chart with the number of studies corresponding to each ISIC class from the selected ISIC section.
3.5.2 Analysis of Relationships between Pairs of Features.
In the second part of the analysis, the relationship between certain pairs of features is analyzed. The goal of this part is to find insights between any features, such as which industry sectors are more associated with quantum annealers.
Due to the large number of features, analyzing every possible pair is not feasible nor illuminating. Instead, we select which pairs will be analyzed, as illustrated in Table 5. A total of 34 pairs were selected according to their relevance to the research questions.
Table 5.
Table 5. Pairs of Features Selected for Analysis
A violin plot or a bar chart is developed for each pair, depending on the features. These charts enable us to analyze the relationship between the features of the pair. Afterwards, the relationship is verified with an appropriate statistical test, which also depends on the features. We expect the Mann–Whitney U test to be a statistical test that is very well suited to analyze these relationships, since we cannot assume the distributions are normal distributed.
3.5.3 Exploratory Data Analysis.
In the third and last part of the analysis, we perform a non-systematic data analysis that attempts to find any insight that has not been captured by the first two parts of the analysis. This may include any pair of features that was not deemed relevant as part of the protocol.
3.6 Threats to the Validity of the Study
Just like all research works, this systematic mapping study is subject to threats to its validity that come from us and from the data [65, 110]. Hence, we are responsible for planning, conducting, and reporting the SMS in a manner that minimizes bias and error. To address these threats, we placed several measures that are listed as follows:
Bias. To minimize systematic error, we made sure to use well-established code to extract and visualize the data, such as the Python libraries Pandas, numpy, and plotly. Potential bias may also arise from the set of identified electronic databases, data collection techniques, inclusion criteria, and exclusion criteria, which limits the exhaustivity of the data collection. We also avoided manually extracting the data from electronic databases as much as possible. Potential bias may still come from the quality assessment and the data extraction and classification steps, as they require assessment and classification from humans. All things considered, we believe that we have minimized the potential for bias as much as possible.
Internal Validity. The protocol was designed in such a way that the quality checklists and the extraction form are as concise and unambiguous as possible. We also made sure to label every response so as to avoid mislabeling errors and loss of data. We believe that this design is likely to prevent systematic error in the study.
External Validity. The systematic mapping study was designed to be reproducible and verifiable. The protocol is very detailed and the extracted data is publicly available. There is potential loss of generalizability due to the steps that involve human subjectivity. More concretely, different humans may return different outputs when doing the quality assessment and the data extraction and classification steps. Nonetheless, we believe that this potential loss of generalizability is addressed by the design of the protocol.
4 Discussion
In this section, the results are summarized and discussed in an itemized list of main takeaways, following the order of the features denoted in Table 4, as shown in Section 4.1. The list is followed by Section 4.2, which presents our answer to the research questions that guided the systematic mapping study.
4.1 Main Takeaways
The following itemized list presents the main takeaways from the systematic mapping study. Some of the takeaways will be highlighted in a pink box due to the relative importance we have given to them.
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Study Quality Score — \(S_S\) We believe this indicator is a good representation of the quality of the study. Compared with the other indicator, \(S_R\), the distribution of \(S_S\) among the studies appears more sparse. We would like to see new studies using hybrid approaches for a better \(S_S\). More concretely, hybrid approaches that use gate-based quantum computers have a very bad \(S_S\) and would benefit from new studies subject to much higher quality standards. On the other hand, certain classical algorithms, such as the quantum genetic algorithm, also lag behind in terms of \(S_S\) and would benefit from new studies with better quality. Regarding the ISIC sections, the section “Transportation and storage” has some studies with a very low \(S_S\) and would benefit from more studies subject to higher quality standards.
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SS01 The significant majority of the studies use comparison algorithms. We believe that this majority should be kept for future studies, as it enables us to evaluate novel algorithms in comparison to existing ones.
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SS02 We also observed a majority of the studies comparing their algorithm with existing algorithms across multiple scenarios. We believe that this majority enables us to accurately evaluate new algorithms among different conditions, which brings more conclusions and reliability to their findings. This majority should be striven for in the future.
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SS04 Considering the studies that do not use real-world scenarios, almost all of these studies use benchmarks. Despite not being as representative of the real world as real-world scenarios are, benchmarks are still a good assessment tool for when real-world scenarios are not possible or feasible. For this reason, we believe that future work should strive for benchmarks when real-world scenarios are not possible.
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SS05 We observed that very few studies used randomized variables and scenarios as part of their methodology, which was expected, since we also observed that a great majority of the studies use benchmarks or real-world scenarios. Among these studies, we observed that just over half of the studies explain and justify how the randomization process is done. We believe that this majority should be significantly larger in the future, as it would bring more credibility to their findings, help readers interpret the results, and bring more reproducibility to the study.
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SS06 A significant majority of the studies explain the reasoning behind variables that are defined arbitrarily in their methodology. However, there is still a noticeable portion of around 21% that does not. We believe that this portion should be decreased in the future, for increased credibility, reliability, and reproducibility.
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SS07 Almost all the selected studies use statistical analysis as part of their analysis. We believe that this trend should be kept in the future, because statistical analysis brings more credibility and reliability to the conclusions.
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SS09 A total of zero studies reported addressing any validity or reliability threats. We believe that this finding should be avoided in future work, since we believe that studies that address validity and reliability threats are more transparent and less prone to bias.
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SS10 A significant majority of the selected studies address future work. However, we believe that there is a significant margin for improvement in this feature. Future studies should address future work to illuminate possible paths for further research, as well as to provide readers with short-term goals for future studies.
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SR00 We also believe that this indicator is a good representation of the quality of the reporting. No significant difference was observed between the \(S_R\) of studies that use a purely classical approach and the \(S_R\) of studies that use a hybrid approach. However, among the studies that use a hybrid approach, those that involve gate-based quantum computers have a much lower \(S_R\) than those that involve quantum annealers. This is another evidence that supports our belief that future studies using gate-based quantum computers would benefit from much better quality standards.
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SR01 A significant majority of the selected studies present their overall structure in the beginning of their text. However, a significant portion of studies still do not. We hypothesize that this is due to text constraints placed by conferences (and possibly journals), as evidenced by the exploratory data analysis. We observed that almost 66% of the studies sourced from journals present their overall structure, while only 50% of the studies sourced from conferences do it. We believe that studies should present their overall structure if possible, as it helps reader to navigate quickly and easily.
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SR02 We observed that almost all of the selected studies present the motivation behind their work. We believe that future work should also follow this trend, since motivation is what justifies the need for conducting the work.
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SR03 A great majority of the studies present their research questions or goals in the beginning of the text. With a similar reasoning as in feature SR02, we also believe that future work should follow this trend.
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SR04 We observed that few studies summarize their methodology in the beginning of the text. We believe that this is detrimental for readers who wish to perform a quick lookup before reading the full-text, and also that future work should aim for summarizing the methodology in the beginning of the text. An example of this summary that was identified among the studies is a brief text that describes the methodology as a series of high-level steps.
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SR05 A significantly small portion of the selected studies does not summarize conclusions in the beginning of the text. We believe that this is also detrimental for readers who wish to perform a quick lookup before reading the full-text. Hence, we believe that future publications should aim to summarize the conclusions at the beginning of the text. One good example of this summary that was observed among the studies would be a bulleted list of the conclusions.
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SR06 Almost all of the selected studies presented their methodology in a way that it is reproducible. We believe that future work should strive to keep this trend.
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SR07 Among the selected studies, almost all of them present their application in a way that its implementation can be reproduced. This is another trend that we believe future work should strive for.
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SR08 We observed that very few studies present their comparison algorithms in a way that their implementation is possible. We believe that this is detrimental for readers who wish to reproduce or check the implementation, and that future work should strive to share the details of the implementation.
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SR09 A great majority of the studies share details that make it possible for readers to reproduce the scenarios, settings or variables used in their work. Once again, this is a trend that we believe future studies should aim for.
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SR10 Almost all of the studies describe their statistical analysis in a way that makes it reproducible. We expect future work to continue this trend.
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SR11 All the selected studies use images or tables when presenting their results. Images and tables are effective tools to summarize and display results, and we expect future work to keep relying on these visualization tools.
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SR12 Almost all the articles refer to and explain all the tables and images that are shown. The few articles that do not are usually associated with lower quality scores. There is margin for improvement, as we believe that there should not be as many articles not referring to and explaining their images. Hence, future work should improve this metric.
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SR13 All but very few articles use all their references in the text. The few that do not are usually associated with lower quality scores. We expect future studies to maintain this trend.
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SR15 Fortunately, the three studies with a publicly available code repository also documented their code. This is another expectation that we believe future work should attend to. Publicly available code repositories lose their effectiveness if readers are not capable of understanding the implementation details.
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SR16 A significant majority of the studies do not directly describe their contributions. We believe that future work should summarize their contributions, whether in the beginning or in the end of the text. A good example of this would be a bulleted list of the contributions shown in the introduction or in the conclusion, as some of the selected studies do.
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D020 We observed 26 different types of algorithms among the selected studies, which is a promising sign of quantum inspirations being applied in many different algorithms. However, all but six algorithms only have one or two studies. In fact, the Quantum Particle Swarm Optimization algorithm is used in around 43% of the studies, followed by the Quadratic Unconstrained Binary Optimization approach, which accounts for almost 17% of the studies. Other promising algorithms include the Quantum Genetic Algorithm, the Quantum-inspired Evolutionary Algorithm, and the Quantum Bat Algorithm. We also found that the QPSO is dominant in studies in the ISIC section “electricity, gas, steam and air conditioning supply”, whereas the QUBO approach is dominant in the ISIC section “transportation and storage”. The QUBO approach stands out from the other algorithms because it is a very recent approach that first appeared in 2017 and is already on its way to surpass the QPSO in number of publications per year, which is aligned with the availability of quantum annealers in the market. In terms of publication source, we found that the QUBO approach also stands out from other algorithms because it has more publications from conferences than from journals. We believe that future work should explore the application of QUBO in ISIC sections not yet explored, such as “electricity, gas, steam and air conditioning supply”. Moreover, we believe that there is opportunity to extend work on underexplored algorithms, so as to achieve both good quantity and good quality.
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D030 Among the studies that use algorithms that can be used in quantum computers, all but one used a real quantum computer. The remaining one used a simulator. For this reason, our analysis of this feature in comparison with other features was not feasible. Nonetheless, as simulators are becoming increasingly unable to keep up with the size of current quantum computers, we do not foresee a strong need for more studies using simulators. Therefore, we expect future work to follow this trend of using only real quantum computers.
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D051 When considering the ISIC divisions of the studies, we observed two divisions that reflected almost 45% of the studies: “electricity, gas, steam and air conditioning supply” and “warehousing and support activities for transportation”. Other divisions are not as expressive but are still significant, such as “public administration and defence; compulsory social security”, “architectural and engineering activities; technical testing and analysis”, “activities of head offices; management consultancy activities”, “water collection, treatment and supply”, “information service activities”, and “repair and installation of machinery and equipment”. The remaining divisions are not significant or have no associated studies. We refer the reader to the treemap chart and icicle chart in the webpage to explore the selected studies in terms of the ISIC hierarchy.
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D052 Examining the ISIC groups of the studies reveals that certain groups stand out due to their large amount of associated studies. Namely, “electric power generation, transmission and distribution”, “support activities for transportation”, “architectural and engineering activities and related technical consultancy”, “repair of fabricated metal products, machinery and equipment”, “data processing, hosting and related activities; web portals”, “activities of head offices”, and “water collection, treatment and supply”. The remaining ISIC groups have very few or no studies.
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D053 Considering the last subdivision of the ISIC, some classes that have significant expression, such as “electric power generation, transmission and distribution” with around 24% of the studies, “other transportation support activities” with almost 15%, “architectural and engineering activities and related technical consultancy” with around 8%, “data processing, hosting and related activities” with around 5%, “activities of head offices” and “repair of machinery” both with almost 5%, “service activities incidental to air transportation” with just over 4%, and “water collection, treatment and supply” with just over 3%, among others. For more details, we refer the reader to our webpage.
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D060 We have identified a growing trend in the number of publications per year. This growth is expected as the number of active researchers in the field of quantum computing has been growing, as well as the availability and capabilities of quantum computers. We believe that this growth in the number of publications will continue during the coming years.
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D070 Examining the last feature, we observe that a significant majority of the selected studies were sourced from journals, while all but two the remaining studies were sourced from conferences. There are two studies sourced from workshops. We noticed that the conference-sourced studies started becoming expressive in numbers only during recent years, with a big spike in 2021. We think that this growth spur in recent years is aligned with the market availability of quantum computers, as well as with the existence of conferences dedicated as a whole or in part to quantum algorithms. We also noticed a big spike in journal publications in 2020, many of which are related with the ISIC class “electric power generation, transmission and distribution”. The reason for this spike was not identified. All in all, we expect conference-sourced studies to gain more expression in comparison to journal-sourced studies during the coming years. We also expect more workshop-sourced works to be published, as motivation to bring new researchers to the field is currently increasing.
4.2 Answers to the Proposed Research Questions
Now that the findings have been analyzed and discussed, we are ready to formulate our answers to the research questions that were posed.
RQ1. What are existing approaches that apply quantum or quantum-inspired algorithms to the field of Operations Research?
We have identified two types of approaches that apply quantum or quantum-inspired algorithms to the field of OR: approaches that are purely classical, and approaches that are hybrid. Each type has its own associated set of possible algorithms.
Considering the approaches of the purely classical type, the following algorithms were applied to OR problems, listed from most applied to least applied (an identifier is placed next to each algorithm, for future reference):
Almost all of the hybrid approaches leverage quantum annealers, and the remaining few took advantage of gate-based quantum computers. More concretely, Quadratic Unconstrained Binary Optimization, Decomposition into QUBOs, and Quantum Generative Training were used in approaches that use quantum annealers, while Quantum Circuit and Quantum Alternating Operator Ansatz were used in approaches that use gate-based quantum computers.
RQ2. In which sources, in which years, and in which quantities were approaches that apply quantum or quantum-inspired algorithms to the field of Operations Research published?
For the second research question, we refer to Figure 50, which discriminates the selected studies by their source and their year of publication, while also illustrating the number of publications per year. A total of 149 studies were selected and analyzed, and the years 2020 and 2021 are the years with most publications.
RQ3. Which research questions related to approaches that apply quantum or quantum-inspired algorithms to the field of Operations Research are addressed by a significant amount of evidence (evidence clusters)?
Considering the possible research questions that may be posed when applying quantum or quantum-inspired algorithms to OR problems, our findings suggest that research questions related to certain algorithms and certain industry activity sectors have been addressed by a significant amount of evidence. That is, our findings suggest that research questions involving the application of each of the following algorithms in OR problems have been addressed by a significant amount of evidence: ALG1 (64 primary studies); ALG5 (25 primary studies); ALG2 (17 primary studies); and ALG3 (10 primary studies).
Moreover, research questions involving the following industry sections have also been addressed by a significant amount of evidence:
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Electricity, gas, steam and air conditioning supply (37 primary studies)
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Transportation and storage (35 primary studies)
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Manufacturing (22 primary studies)
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Professional, scientific and technical activities (21 primary studies)
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Information and communication (13 primary studies)
More specifically, these research questions also involve the following industry activities:
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Electric power generation, transmission and distribution
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Support activities for transportation
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Architectural and engineering activities and related technical consultancy
The remaining algorithms and industry sections do not have a sufficient amount of evidence for us to be comfortable with the validity and generalization of their findings.
RQ4. Which research questions related to approaches that apply quantum or quantum-inspired algorithms to the field of Operations Research are addressed by a scarce amount of evidence (evidence deserts)?
Any research questions that were not addressed with a significant amount of evidence, such as the ones identified in RQ3. are the answer to RQ4. Nonetheless, answering this research question with a response of the type “anything that is not...” is non-illuminating and not useful. To avoid this situation, we employed the ISIC standard to classify the OR problems addressed by the studies in a way that enables us to know which industry activities have been addressed and which have not.
Our findings show that research questions involving OR problems belonging to the following ISIC sections remain underexplored:
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Agriculture, forestry and fishing
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Mining and quarrying
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Water supply; sewerage, waste management and remediation activities
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Construction
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Wholesale and retail trade; repair of motor vehicles and motorcycles
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Accommodation and food service activities
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Financial and insurance activities
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Real estate activities
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Administrative and support service activities
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Public administration and defence; compulsory social security
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Education
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Human health and social work activities
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Arts, entertainment and recreation
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Other service activities
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Activities of households as employers; undifferentiated goods- and services-producing activities of households for own use
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Activities of extraterritorial organizations and bodies
Moreover, our findings show that research questions involving the following algorithms remain underexplored: ALG4 ; ALG6; ALG7; ALG8; ALG9; ALG10; ALG11; ALG12; ALG13; ALG14; ALG15; ALG16; ALG17; ALG18; ALG19; ALG20; ALG21; ALG22; ALG23; ALG24; ALG25; and ALG26.
We also found primary studies involving gate-based quantum computers to have very low quality and very low quantity. That is, only two studies were found, with subpar quality.
4.3 Decision Flowchart
We developed a decision flowchart that guides researchers in finding a new research question to tackle. It is aimed for researchers who are interested in starting their first work on an application of a quantum or quantum-inspired algorithm on an OR problem. The flowchart is shown in Figure 2. By answering the top and bottom questions, we receive suggestions for research ideas. Both research ideas can be joined to create a path for future work.
Fig. 2.
Fig. 2. Decision flowchart to help researchers who are interested in joining the field to find a good research question.
5 Conclusion
The increasing complexity of operations research (OR) problems and the emergence of quantum and quantum-inspired algorithms has brought a new research field: applications of quantum or quantum-inspired algorithms in OR problems. This field is still in its infancy, due to its low number of primary studies. We identified two necessities that arose from this situation: researchers starting research work in this field need to know what research has been done and what are the potential paths for future work; and professionals need to know what are the most promising applications for operational problems in their industry sector.
After a literature search, we reunited all the conditions to begin a systematic mapping study. We believe that this is the best way to meet the aforementioned needs, especially considering that the existing literature is still far from being sufficiently developed to perform a systematic literature review.
We designed a protocol that specifies every step of the systematic mapping study in detail. This makes it possible for any reader to reproduce the systematic mapping study. The protocol begins with a set of research questions that are aimed to be answered by the systematic mapping study. In short, the main goal is to find existing approaches that apply quantum or quantum-inspired algorithms to OR problems, while also detailing which applications are addressed by a significant amount of evidence and which applications are addressed by a scarce amount of evidence.
We resorted to several electronic databases as part of our search step, using string queries constructed with the aid of the PICOC criteria. We found 325 documents with the electronic databases. This search was extended with reverse snowball sampling and forward snowball sampling, which are methods that add the documents in the references and the citing documents. In the end, more than 2,000 documents were added and considered.
The study selection step is done in an alternate fashion at the same time as the search step. That is, we selected the studies right after the electronic database search, right after the reverse snowball sampling, and right after the forward snowball sampling. The selection step consists of the application of inclusion and exclusion criteria, with the goal of selecting only studies that are relevant to our research questions. The search and selection steps culminated in 149 studies, the majority coming from journals and conferences of high prestige.
The next step is the quality assessment, which passes each document through two checklists of “yes or no” questions, one related with the quality of the study and the other related with the quality of the reporting. Thanks to this assessment, we gained lots of insights related with the research methods and the structural features of the studies. The quality assessment was partly supported by the SJR Indicators of the journal-sourced studies and the CORE Rankings of the conference-sourced studies. We believe that the quality scores attributed to the studies are representative of their quality, especially when their attributed score is low.
The following step is the data extraction and classification, where a series of data features are extracted from the studies—more concretely, features such as Type of Approach, Algorithm Name, Type of Quantum Computer, and Publication Source, among others. Moreover, four additional features were inferred from the studies, using the International Standard Industrial Classification of All Economic Activities (ISIC), which enabled us to find to which industry activity each study belongs to. All the features considered, we have extracted a powerful set of data features that enables us to find useful insights not only about the features themselves but also the relationships between the features.
The most significant step in the systematic mapping study is the analysis, which is responsible for exploring the extracted data with the main goal of finding relevant insights that help us address the research questions of the mapping. Our analysis was divided into three parts. First, we focused on the individual features themselves. Next, we focused on the relationship between pairs of features. Last, we performed a non-systematic exploratory data analysis with the goal of finding any insight that was not captured by the other two parts. From a general standpoint, we were able to analyze in an effective way, with lots of useful findings that culminated into a productive discussion and useful answers to the research questions. However, two features were not subject to the same level of analysis as originally specified in the protocol due to lack of evidence. These features are D030 and D040.
In the end, we were able to produce an itemized list of insights, each item corresponding to each data feature. We believe that this lets readers navigate quickly among the synthesized insights. With the convenience of the reader in mind, we also produced a subsection with direct answers to the research questions that were posed. We believe that those outputs are particularly useful for readers who wish to go straight to the point, and also for those that wish to know what are the potential paths for future work.
To add to the two textual outputs, another two non-textual outputs were also produced, a decision flowchart and a website. The decision flowchart aims to help researchers that are interested in finding a research question to be addressed as part of a new research work. The website displays a series of interactive charts, letting users direcly explore the data that was extracted as part of the systematic study and acquire even more insights.
All the outputs considered, we believe that, by taking into account the convenience of the reader, we are maximizing the effectiveness of the systematic mapping study. That is, we believe that we have met our goal of providing a map of what has already been explored and a list of potential paths for further exploration in a way that is effective and easy to digest to readers who are interested in joining the research effort.
Considering the findings themselves, it was exciting for us to see that most of the applications focus on essential economy activities such as electricity supply, transportation, and manufacturing. However, we also would like to see more effort in other activities such as agriculture, forestry, and fishing, as well as construction, art, and mining. We believe that more applications in such sectors would be very valuable and impactful for the field. We also challenge newcomers to apply quantum algorithms in activities such as education, accommodation and food service, finance and insurance, and real estate. These activities, despite not being as explored as the others, are equally important and prevalent, and improvements in algorithms would still be very significant. Plus, we would learn about potential advantages that quantum or quantum-inspired algorithms may specifically bring to these activities. We also challenge newcomers who want to research in an already explored ISIC section to explore applications on ISIC divisions, groups, or classes that were not explored as much.
Now, shifting our view to the coming years, we expect to witness quick progress on the field, as quantum computers become more available and capable of larger problems. This means that we expect our systematic mapping study to become obsolete before the year 2030. However, we believe that our protocol can easily be replicated and modified for any future work that wishes to produce a new systematic mapping study or even a systematic literature review. Until then, we are open to submissions from authors who wish to add their primary studies to our website, to keep our insights up-to-date.
Footnote
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All the studies we analyzed are publicly available here.
Supplemental Material
Supplemental PDF
Supplementary Material for A Systematic Mapping Study on Quantum and Quantum-inspired Algorithms in Operations Research
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Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of ...
The quantum Fourier transform (QFT) is a key subroutine of quantum algorithms for factoring and simulation and is the heart of the hidden-subgroup problem, the solution of which is expected to lead to the development of new quantum algorithms. The QFT ...
Department of Informatics Engineering, Faculty of Engineering of the University of Porto, Porto, Portugal, Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, United States, and Laboratório de Inteligência Artificial e Ciência de Computadores, Universidade do Porto, Porto, Portugal
Department of Electrical and Computer Engineering, University of Coimbra, Coimbra, Portugal and Instituto de Telecomunicacoes Coimbra, Coimbra, Portugal
Cláudio Gomes, 
João Paulo Fernandes, 
