Quantum Combinatorial Optimization in the NISQ Era: A Systematic Mapping Study

The application of quantum computing to combinatorial optimization problems is attracting increasing research interest, resulting in diverse approaches and research streams. This study aims at identifying, classifying, and understanding existing solution approaches as well as typical use cases in the field. The obtained classification schemes are based on a full-text analysis of 156 included papers. Our results can be used by researchers and practitioners to (i) better understand adaptations to and utilizations of existing gate-based and quantum annealing approaches and (ii) identify typical use cases for quantum computing in areas such as graph optimization, routing, and scheduling.


INTRODUCTION
Quantum Computing (QC) refers to the processing of information based on the phenomena of quantum mechanics [204].Continuous developments in QC raise the expectations that, in the near-term future, we could speed up simulations in quantum chemistry and improve optimization processes as well as machine learning approaches, compared to classical means of computation [164,175,187,200].A universal fault-tolerant quantum computer would require millions of qubits of highest quality [179].Whereas experimental realizations of such computers have not been achieved yet, so-called Noisy Intermediate-Scale Quantum (NISQ) computers already exist today and, therefore, may enable near-term superiority of QC with respect to classical computation [209].These devices operate with a limited number of noisy qubits.To utilize the existing resources in an efficient manner, specific algorithms have been proposed for various disciplines, e.g., quantum chemistry, combinatorial optimization, and machine learning.A promising strategy to cope with the existent resource limitations is represented by hybrid quantum-classical approaches, which comprise quantum as well as classical computation (e.g., References [164,176]).To leverage the potential of NISQ devices and according hybrid algorithms, great advancements have been accomplished, ranging from computer architecture [210] to software engineering aspects [214].
An application area of particular interest, especially concerning the near-term possibilities of QC, is Combinatorial Optimization (CO), which is referred to as quantum CO in the following.This research area aims to find the set of discrete decision variables that minimize a certain constrained or unconstrained optimization function [212].These optimization problems are usually non-convex (causing many local optima) and are often characterized by an exponentially growing search space, which makes these problems very hard to solve with classical approaches.QC is regarded to be a promising approach to account for these difficulties due to its fundamentally different computation principles such as, e.g., superposition and entanglement.In this regard, several algorithmic methods have been established that specifically account for the limitations of the NISQ era by using significant amounts of classical computational means [164].
As a consequence of the near-term potential of quantum CO, developing novel quantum solutions to such problems and investigating application areas have been popular research topics in recent years.Due to its interdisciplinary character, researchers from different fields contribute to quantum CO.The rise of scientific interest has led to a vast literature on quantum CO resulting in diverse approaches and research streams.The associated challenges of gaining a comprehensive view of the current state-of-the-art constitutes an obstacle for newcomers to this topic but also hinders the progress for experts.
Research interest.In this systematic mapping study, we focus on the problem domain of CO and review the current state-of-the-art concerning existing quantum solution proposals as well as use cases that have been solved with QC.Furthermore, we analyze the scientific literature in the field for bibliometric key facts, such as, e.g., the publication type and the research facet [219].Therefore, we provide answers to the following research questions (RQs) that are further described below.
-RQ1: Which approaches exist for combinatorial optimization with NISQ devices?We divide this research question into five subquestions.
-RQ1.1:Which mathematical problem formulation approaches for quantum CO exist?-RQ1.2:How can quantum algorithms for quantum CO be categorized?-RQ1.3 How can quantum algorithms for quantum CO be adapted?-RQ1.4 How can quantum algorithms be enhanced by and integrated into classical computation routines?-RQ1.5 Which properties of quantum algorithms for quantum CO  Concerning the subquestions of RQ1, we first study the primary literature on mathematical problem formulations that are feasible for quantum CO as well as their potential impact (RQ1.1).Regarding RQ1.2, we categorize available quantum algorithms for quantum CO and show possible adaptations of them (RQ1.3).This allows to describe possible classical means for enhancing quantum algorithms, e.g., by applying the mentioned adaptations (RQ1.4).Furthermore, it is outlined in RQ1. 4 how quantum algorithms can be embedded as a subroutine into classical approaches to tackle CO problems.In RQ1.5, we discuss works that explicitly study properties of quantum algorithms for quantum CO.
Article organization.Section 2 provides the background knowledge on CO as well as QC approaches for CO as a basis for understanding the upcoming sections.Thereafter, the need for a systematic mapping study is motivated, where our work is put into the context of already-existing literature reviews.The applied methodology of the review is illustrated in Section 3. Subsequently, Section 4 provides a condensed picture on our findings concerning quantum CO approaches and identified use cases for quantum CO and provides answers to the RQs.The threats to validity are stated in Section 5, before we conclude in Section 6.

BACKGROUND
The following section provides a short description on CO (Section 2.1) and how CO problems may be solved by the application of QC with NISQ devices (Section 2.2).We continue with a depiction of related literature reviews, which served as the basis for deriving our research questions (cf.Section 1).

Combinatorial Optimization
The goal when solving a CO problem is to find conditions that minimize a given function, the so-called cost function or objective function [150].In addition to these functions, certain equality or inequality constraints are generally also defined.Basically, these constraints must be fulfilled by the solution.A CO problem is formally defined as [222]: where x denotes the discretized vector that represents the decision variables, f (x) denotes the realvalued cost function, д i (x) represents the equality constraints, and h j (x) denotes the inequality constraints.These constraints constitute certain conditions to be fulfilled by the solution.
In the next section, the transformation of this formal general definition into a form that is acceptable for a quantum computer is explained.

Quantum Computing for Combinatorial Optimization
In the following, the problem transformation from (1) and (2) to a feasible format is introduced in Section 2.2.1, followed by a short description of the most prominent approaches for quantum CO: Variational Quantum Algorithms (VQAs) (Section 2.2.2) and Quantum Annealing (QA) (Section 2.2.3).

Problem Formulation.
The most prominent problem formulations for CO on quantum computers are the Ising-model [189] and the equivalent Quadratic Unconstrained Binary Optimization (QUBO) model [53,150,173].The QUBO model is defined as [221]: with x i ∈ {0,1} denoting the binary decision variables and Q being the real-valued n-by-n coefficient matrix that is supposed to be symmetric or in upper triangular form [53].The Ising model is motivated from physics and defined as follows: Given a graph G=(V,E) with weights h i on the vertices and J i j on edges, the goal is to find an assignment of spins s={s 1 , s 2 , . . ., s n }, where s i ∈ {−1, +1} to the vertices such that the Hamiltonian (i.e., the quantum mechanical operator that characterizes the system energy) is minimized [218].
The Ising model can be mapped to a QUBO formulation as shown in [221] Q with real-valued coefficients c i and d i j .Equations ( 4) and ( 5) are equivalent up to a linear transformation s i = 2x i − 1 [53].Neglecting the irrelevant constant, they are thus different representations of the same underlying NP-complete problem [221].Relating to (1), H (s) and Q(x) represent the cost function f (x) but do not consider the potential constraints given in CO problems as given in Equation ( 2).In case the original formulation of the problem has polynomial form and constraints, there are generally applicable standard techniques for transformation to Ising/QUBO-form.This requires to incorporate the constraints in the objective function and conducting a degree reduction from polynomial to quadratic form, which will be discussed subsequently.The equality constraints can be considered in the Hamiltonian as so-called penalty terms as [135]: where H cost is the original Hamiltonian representing the objective function, G is the set of all equality constraints, and λ is a hyperparameter that has to be set carefully and in a problem-specific manner to adjust the weight of the constraint.If λ is too low, then infeasible states are more likely to occur; if it is too high, then the quality of the obtained solutions deteriorates, because H constr aint dominates H cost .Inequality constraints can be mapped to equality constraints, where additional variables have to be introduced [53,135].This mapping replaces the inequality constraints with equality constraints and non-negativity constraints on the slack variables.As an example, Ax ≤ b can be transformed into Ax + y = b, where y ≥ 0 would be binary expanded and added to the constraint (refer, e.g., to References [53,135] for further details).Once the constraints are incorporated in the objective function, the remaining transformation to arrive at a QUBO/Ising model concerns the reduction of the objective function from higher-order polynomial to quadratic form [218], which requires the introduction of additional variables.The potentially large overhead in the number of variables lead to the development of several methods for degree reduction (e.g., References [160,167,168,180,188]), which are explained and compared in References [53,62,135,218].Finally, there are several already existent direct mappings for many NP-hard and NP-complete problems to QUBO/Ising form [199].According to complexity theory, these classes constitute problems to which any other problem that is solvable in polynomial time can be reduced to.
Quantum Combinatorial Optimization in the NISQ Era 70:5

Variational Quantum Algorithms.
VQAs represent an approach that utilizes gate-model quantum computers.Within this model of quantum computing, quantum operations (alias gates) and measurements are applied to qubits, the elementary quantum information carrier, resulting in a circuit-like model of computation in analogy to classical circuits [216].In VQAs, the quantum operations, which may be specified by parameters, are arranged in a certain manner that is referred to as the ansatz of the respective VQA.A main feature of VQAs is a classical optimizer that is deployed to train the parameterized quantum circuit, a procedure that is known as parameter optimization.VQAs appear to be a promising way to accomplish near-term quantum advantage, because these algorithms do not require fault-tolerant quantum computation.However, the parameterized quantum circuits often utilize multi-controlled operations to encode constraints into the ansatz, e.g., the Quantum Alternating Operator Ansatz [58].These multi-controlled operations are not natively supported by most available NISQ devices [201], limiting the practical applicability of approaches relying on this concept.Thus, these operations must be decomposed into operations natively supported by the target NISQ devices before execution [217].As this decomposition can increase the width and depth of the quantum circuits, this has to be taken into account when evaluating the executability of VQAs.Whereas comprehensive reviews on VQAs, in general, can be found, e.g., in References [164,171], the focus in this work lies on VQAs for CO.Therefore, the Variational Quantum Eigensolver (VQE) [206] and Quantum Approximate Optimization Algorithm (QAOA) [177] are introduced subsequently as prominent examples of VQAs for the domain of CO.

VQE:
The VQE [206] was originally proposed to find the lowest energy state of a quantum chemical system.However, with the mappings of CO problems to Hamiltonians stated in the previous section, besides quantum chemical problems, the same approach can be used to solve CO problems [10].The variational form of the parametrized quantum circuit, also called ansatz, is important in this regard, where standard approaches (e.g., References [191,202]) comprise the iterative alternating application of non-commuting single qubit rotations and controlled two-qubit operations.The choice of the right ansatz heavily depends on the problem as well as the available quantum resources and classical optimizer.Having fixed the variational form, the quantum computer applies the circuit with initial parameters.From measurement samples of the resulting quantum state, the expectation value of the system energy can be calculated classically.These results in turn guide the classical optimizer to iteratively find optimal parameters.Sampling the quantum state that minimizes the expectation value yields a probability distribution, which represents the solution bit-string for the CO problem.

QAOA:
The QAOA [177] has been specifically developed for CO problems.It can be regarded as a form of VQE with a specific choice of the variational form that is derived from the problem Hamiltonian [10].Furthermore, QAOA applies adiabatic evolution as the depth of the circuit goes to ∞.This means that theoretically the optimal solution to the CO problem can always be obtained under this condition and provided that the according parameters can be found.This property is also a reason why QAOA is often regarded a digitized version of QA [177].Given the problem Hamiltonian, the variational form of QAOA consists of two alternating unitaries, which are iteratively applied.The first one comprises the information from the problem Hamiltonian, i.e., the cost function of the CO problem.The second one, referred to as mixing unitary, is defined in a problem-agnostic way and drives the search space exploration.As in VQE, the expectation value is iteratively computed, parameters are classically optimized, and the algorithm returns the bitstring that minimizes the cost function.

Quantum Annealing.
QA [178] is a heuristic related to adiabatic quantum computation [178] that is inspired by simulated annealing [195].Here, quantum fluctuations take the role of the thermal fluctuations in simulated annealing.The effect of quantum tunneling is used to explore the energy landscape defined by the cost function.The initial state of the quantum system is prepared such that it represents the lowest-energy state of some initial Hamiltonian H 0 .The Hamiltonian of the system then continuously changes with time according to References [178,218]: where t a is the total annealing time, A(s) and B(s) are functions that obey A(s=0)=1, B(s=0)=0 and A(s=1)=0, B(s=1)=1.H again denotes the problem Hamiltonian according to the CO problem.
According to the adiabatic theorem [159], the system always stays in the lowest-energy state for sufficiently smooth changes.Therefore, after the annealing process the system is assumed to be in the lowest-energy state of the problem Hamiltonian.The solution to the CO problem can then be obtained by sampling from this final quantum state.As with gate-based approaches like VQAs, the efficient application of QA on current hardware requires a significant amount of classical computation.When a problem is actually run on a quantum annealer, a mapping from the logical problem structure to the underlying hardware has to be conducted, similarly to the compilation process from logical operators to hardware native operators for the gate-model devices.In quantum annealing, this mapping to the hardware topology is called embedding, where the standard technique in use is denoted as minor embedding [174].
Decomposition methods for QA comprise techniques that enable to treat larger problems than natively possible with a quantum annealer where qbsolv1 represents the standard technique for this purpose.To extend the size of feasible problems even further, hybrid quantum-classical solvers have been established where only a certain part of the problem is treated by a quantum annealer.We refer to existing reviews [62,159,218] for more information on QA.

Related Literature Reviews
Table 1 states the title of related studies, the year of publication, their focus, and an assessment concerning their overlap with quantum CO by stating the answer to each of the following questions based on the criteria explained by Kitchenham et al. [196]: • Q1: Does the review include a holistic illustration of the solution space for quantum CO? • Q2: Does the review include use cases, i.e., concrete problems solved with quantum CO?Quantum Combinatorial Optimization in the NISQ Era 70:7 The second criterium (Q2) was evaluated as being partially (P) fulfilled if use cases are at least mentioned in the review, but no further investigations have been conducted.The existing reviews in the field of quantum CO take the view from the solution space and therefore only treat a certain kind of quantum solution approaches.In this regard, References [164] and [171] focus on gate-based algorithms that can be implemented on NISQ devices and therefore may be applied to various application areas with CO just as one example of those.Specific studies on QA are provided by References [218] and [62].The former reviews basic concepts of QA with a special focus on the realization with D-Wave technologies, whereas the latter discusses QA techniques and possible future directions from a theoretical point of view.A similar theory-loaded review on the related field of adiabatic quantum computing is given by Albash et al. [159].Quantum-inspired algorithms, also known as quantum intelligent algorithms, have been covered by Li et al. [198].However, such algorithms are designed to be executed on classical rather than quantum hardware and, thus, are out of scope of this mapping study.
Furthermore, an additional unstructured search for studies that have been published before 2018 revealed another survey on gate-model quantum computing for CO [221].This review describes several fundamental quantum algorithms for gate-model devices.
In summary, none of the mentioned studies takes the view from the problem domain of CO and asks the question about existing quantum approaches in the solution domain.The focus of the listed reviews is on a certain type of quantum computational approach where the impact of the mathematical problem formulation and the combination with classical means of computation are not covered.
Furthermore, use cases of quantum CO mentioned in the considered studies, if mentioned at all, serve as demonstration cases, but no exhaustive summary of potential application cases is provided.
Finally, we highlight that none of the mentioned reviews represents a systematic study with its associated advantages of fairness, methodological rigor, and robustness [193].
Motivated by these missing aspects of related reviews and the growing scientific interest, we consolidate the existing state-of-the-art by viewing the field from the problem domain of CO in a holistic manner to (i) identify and classify according quantum solutions that are designed to be executed on NISQ devices, (ii) illustrate prominent use cases of quantum CO, and (iii) identify key bibliometric facts about quantum CO publications.

LITERATURE REVIEW METHODOLOGY
As a guideline, this study was performed following well-known principles of systematic studies: References [193,196,207,220]. Figure 1 shows an overview of our implemented methodology [207].
The first step was the definition of the research questions (label 1 ), which specifies the scope of the mapping study.Thereafter, the search for studies and the screening of identified papers followed a two-phase procedure.
In the first phase, we performed a search using a formal query to collect the papers for further analysis (label 2 ).By following Keele [193], we consider relevant electronic sources to perform an extensive search, including DBLP,2 ACM,3 IEEE Xplore, 4 Scopus, 5 and ArXiv. 6Here, Scopus has been regarded as a meta-platform due to its extensive list of indexed publishers (e.g., Elsevier, Springer), and DBLP for its broad coverage of computer science papers from various publishers and venues.ArXiv is considered because a relevant part of the quantum computing community publishes on this platform [157].
The search string ("quantum AND computing" AND "combinatorial AND optimization") was defined to answer the research questions that guide this study.This query was executed in each selected repository.The specific queries for each database can be found in the replication package [183].In addition, we just consider papers written in English that have been published and made available in the databases between January, 2018 and May, 2021.
We reduced the timespan to three years of research, given the fact that IBM made available in 2017 the first quantum system and also in this same year D-Wave Systems launched the D-Wave 2000Q quantum computer [157].Considering the fact that research starting in 2018 will primarily use these new available updated quantum technologies, we decided to use this year as our start date for the inclusion of articles.Furthermore, the concept of NISQ quantum computing has been introduced in 2018 [209].To select the relevant articles collected from the repositories, we established the following inclusion (✓) and exclusion (✗) criteria: ✓ Publications that cover CO on NISQ devices, ✓ Publications written in English, ✓ Publications that have been published and made available in the databases between January 2018 and May 2021.✓ Publications that are accessible as open-access or via institutional access, ✗ Publications that focus on compilation (gate-based) or embedding (QA) techniques, ✗ Publications that focus on error mitigation techniques, classical numerical solvers for parameter optimization, or classical post-processing techniques, ✗ Publications that explicitly cover quantum Monte Carlo, quantum walk, quantum machine learning, or quantum inspired approaches.
The first and second exclusion criteria refer to topics that are not specific to CO problems or quantum solutions to CO, but rather cover aspects regarding the underlying technology.Quantum Monte Carlo [221], quantum walk [225], and quantum machine learning [224] denote topics where quantum versions of classical approaches are developed that may be applied, among others, to CO problems, but are not associated with traditional CO approaches.Quantum-inspired approaches [198], finally, are not designed to be executable on real quantum hardware.
Figure 1 shows an overview of the selection of papers in each step.In total, 218 papers were initially retrieved, with ArXiv and Scopus being the repositories that represent 84% of all retrieved papers.
As a second step, we performed an initial assessment of all the papers (label 3 ).In this phase, we inspected the abstracts of all papers to discard the irrelevant works and the duplicated ones.At the end of this phase, we considered a first selection of 79 papers as relevant for the next phase.
The third step consisted of the final assessment of the selected papers (label 4 ).Every paper passing the initial assessment has been read in full detail to classify each of them as relevant or not relevant.In this way, we selected 63 relevant papers.It is worth mentioning that in both the initial assessment phase and also in the final assessment phase, we considered the inclusion/exclusion criteria mentioned above.
In the second phase of the search and selection process, we performed a snowballing procedure (label 5 ) [220].We accomplished a backward (i.e., use the bibliography to identify new papers) and forward (i.e., papers that cited the selected paper) snowballing.Based on the title and year of publication, 328 papers were selected for a subsequent initial assessment.After reading the abstract and the place of the reference in the original paper (label 3 ), 121 of the 328 papers were fully read (label 4 ).At the end of this phase, 93 papers from the snowballing process were finally selected as relevant, resulting in a total of 156 studies included in our review.
To validate the quality of our defined search strategy, we defined a set of 15 papers, which have been regarded as relevant to the field, prior to our search.The described initial database search yielded 11 out of these 15 papers, where the remaining 4 were found within the snowballing procedure.Furthermore, all related reviews mentioned in Section 2.3 have also been identified within the search and no additional ones have been found.
The development of the classification scheme (label 6 ) was based on the full-text of the 156 relevant studies, where the scheme has been iteratively updated throughout the reading process.The data extraction of bibliometric information happened for the following dimensions: time of publication, type of publication, publication channel, and research facet as defined by Wieringa et al. [219].Concerning the latter, we only consider solution, validation, and evaluation studies due to the stated inclusion and exclusion criteria.The results of our data extraction process (label 7 ) have been carefully documented and can be found in the replication package [183].

QUANTUM COMBINATORIAL OPTIMIZATION ON NISQ DEVICES
In the following, the obtained findings are presented by showing the state-of-the art concerning the solution approaches to quantum CO on NISQ devices as well as associated use cases.Based on these results, RQ1 is answered in Section 4.1 and RQ2 is answered in Section 4.2.Thereafter, Section 4.3 provides an analysis regarding bibliometric key facts and, thus, answers RQ3.

Solution Approaches to Quantum Combinatorial Optimization
Based on the background information given in Section 2, the current state-of-the-art concerning quantum CO is presented comprising various adaptations to the mentioned approaches as well as additional methods.The structure of this section follows the one of the research questions stated in Section 1.The studies on problem formulation are presented in Section 4.1.1,together with an answer to RQ1.1.We further illustrate prominent approaches and algorithms for quantum CO where a focus lies on the possible adaptations to those algorithms that may enhance their performance (Section 4.1.2).This allows to answer RQ1.2 and RQ1.3.Having categorized existing quantum CO algorithms as well as their potential adaptations, we further describe how means of classical computation can be used to enhance quantum algorithms for CO, e.g., for those adaptations (Section 4.1.3).Furthermore, identified possibilities of integrating quantum algorithms into classical routines for CO problems are depicted in Section 4.1.3to answer RQ1.4.Finally, RQ1.5 is answered after presenting the findings concerning studies on the properties of current quantum CO approaches in Section 4.1.4.

Problem Formulation.
A significant share of relevant studies covers methods to obtain a mathematical formulation of the CO problem that is acceptable for a quantum computer (QUBO-/Ising form), provided the problem is already stated in another mathematical formulation.Generic transformation techniques have been covered by Glover et al. [53] and Hadfield [57].The former covers several classical CO problems in increasing order of transformational complexity in a quantum computing-independent manner.The latter relies on fundamental mathematical concepts to provide a design toolkit for mapping real functions to Hamiltonians and to provide construction rules to better understand existing mappings stated in the literature (e.g., Reference [199]).Automated approaches for the problem formulation comprise formulation transformations, with the PyQUBO software tool [150], and penalty parameter setting utilizing reinforcement learning [9].PyQUBO allows to create a QUBO from the given objective function and the equality and inequality constraints of the optimization problem.The mentioned reinforcement learning approach [9] utilizes the quantum machine as the stochastic environment that provides feedback for the agent to adjust the penalty parameters accordingly.A different approach to deal with constraints has been suggested by Glover et al. [54] where the original QUBO and the constraints are handled separately.Here, the focus lies on a specific example of this extended QUBO formalism, namely, the Asset Exchange Problem.
Another sort of studies dealing with problem formulation techniques covers the direct formulation of CO problems in QUBO-or Ising-form without the intermediate step of having a preceding different mathematical formulation available.Some of these known direct mappings stated by Lucas [199] have been improved and novel ones have been introduced by Lodewijks [91], including bin packing and partitioning problems.Further problem mappings with subsequent validation using quantum technologies are stated in Section 4.2.
The above-mentioned techniques are hardware-agnostic.However, for most efficient quantum resource utilization, the type of utilized hardware should be already considered in the step of problem formulation as suggested by Lucas [93].For this purpose, improved problem formulations specifically for QA architectures are provided.Another QA-specific approach to the problem formulation is given by a transformation that yields just linear terms but fluctuating coefficients.The latter can be sampled with means of QA to efficiently treat constraints of the CO problem [105].Similarly, a problem-specific encoding for the graph coloring problem has been proposed regarding the gate-model quantum computation [134].Additionally, the class of treatable problems of VQAs has been extended to Mixed Binary Optimization (MBO) problems using appropriate problem formulations [20].
Several other studies investigate the impact of the problem formulation on the performance for certain problems where different backends have been utilized for the experiments.The CO problems that have been studied in this context are the vehicle routing problem with time windows [61] on gate-based devices; the job-shop scheduling problem [156], the maximum k-colorable subgraph problem [113], as well as the 3-SAT problem [80] on quantum annealers; and the graph partitioning problem [30] on the classical QCI sampler tool [19] where two alternative quantum-ready formuations have been evaluated.Therefore, the importance of regarding problem-and hardwarespecificities already in the step of problem formulation has been demonstrated.Answer RQ1.1: Which mathematical problem formulation approaches for quantum CO exist?The identified studies that go beyond the standard transformation procedures stated in Section 2 are summarized in Table 2. Whereas studies that consider problem formulation transformations range from conceptual prescriptions [53] to automated approaches [150], the ones dealing with performance dependencies are empirical in nature.
It can be seen from Table 2 that the majority of found papers deals with transformations of problem formulations (41%), followed by the effects that different formulations have on the performance of the computation (29%).

Quantum Algorithms for Combinatorial Optimization.
In the following, the presentation of identified studies is structured into approaches that relate either to QAOA/VQE or QA or represent other approaches without such a relation.Therefore, in the subsequent presentation, we build on the categorization of quantum solutions as mentioned in Section 2.
QAOA/VQE.Being the first step in solving a CO problem, the formulation and encoding of the problem affects all subsequent computations.Therefore, Lechner [82] presents a novel approach to parallelize QAOA for arbitrary complete graphs that uses single-qubit rather than two-qubit operators to encode the optimization problem.
Concerning the problem encoding, the question of how to account for given constraints is relevant for an approximation heuristic like the proposed quantum approaches.Besides the penalty method, which has been discussed in Section 2, there are solutions that aim to guarantee to stay in the feasible subspace.Hadfield et al. [58] propose the Quantum Alternating Operator Ansatz where the problem-agnostic mixing Hamiltonian of the original QAOA is replaced by a mixing Hamiltonian that enforces constraints and ensures that transitions between all pairs of feasible states are provided.In this context, the usage of XY-mixers, which allow rotations around the X-as well as the Y-axis, have been explored [116].Downsides of the proposed approach are the resulting deep quantum circuits and the complexity of the mixer design.To mitigate these drawbacks, a quantum walk-assisted approach has been suggested [95], a dynamic update of the ansatz based on the quantum outputs has been proposed [120], and constraint-encoding schemes for QAOA have been classified [117] into linear equality, linear inequality, and arbitrary constraints.For each category, according mixing operators are presented.Another possibility is to shift the complexity to the generation of the initial state [11], where the idea is to further utilize the state preparation circuit, which creates an equal superposition of all feasible states, for the design of the mixing Hamiltonian.Specifically for the flight-gate assignment problem, a method has been proposed to derive the initial state and possible mixing Hamiltonian where the construction principles can be extended for treating general graph coloring and scheduling problems [127].
An approach for network flow optimization problems like, e.g., routing problems, which is similar to the Quantum Alternating Operator Ansatz but inspired by quantum electrodynamics, has been proposed by Zhang et al. [153].They also provide an advanced, restricted version of their method that additionally avoids physically irrelevant solutions.
Continuing with the encoding of discrete problems, the encoding of constraints in the cost Hamiltonian rather than restricting the subspace via the mixing Hamiltonian has also been considered [48].After introducing a binary encoding as an alternative to one-hot encoding, a comparative empirical resource analysis is provided by Fuchs et al. [48].
Another way to increase the efficiency is to exploit symmetries in the optimization problem.Permutations of the variables in the objective function that leave the objective function invariant lead to equal measurement probabilities across states connected by such permutations [122].By exploiting such symmetries, the measurement costs can be significantly reduced by considering only those terms in the measurement process that are not connected by symmetries [125].
A further enhancement approach for VQE and QAOA is represented by using an objective function other than the expectation value.Proposals in this regard comprise the Conditional Value-at-risk (CVar) [10], ascending CVar [78], and Gibbs-objective function [86].All of these approaches serve to foster low-energy solutions and simultaneously avoid to get trapped in local minima.
A constraint within the approaches described in Section 2 is the fixed and rigid ansatz.Proposals to overcome these restrictions comprise methods that search in a generally unrestricted search space [97,151,152], but also approaches that take existing quantum algorithms as a starting point [86,155], as it will be further described in Section 4.1.3.Alternatively to classical search routines, the ansatz can be altered in a deterministic manner.In this regard, the Layer-VQE [89] iteratively builds the quantum circuit.The basic idea behind Layer-VQE is to start with a shallow ansatz, then, optimize the parameters and stop the optimization before convergence is reached.Thereafter, the procedure of (i) adding a new layer to the ansatz and (ii) parameter optimization until a certain point prior to convergence is iteratively repeated.The use of problem-specific parameterized quantum circuits of the VQE algorithm where the information on constraints is used for the circuit design has been suggested by DÃŋez-Valle et al. [96] to reduce the search space and accelerate convergence.Similarly, and in analogy to the Quantum Alternating Operator Ansatz for QAOA, variational forms for VQE have been proposed to ensure that the search is limited to domains of feasible solutions [112].
As repeated calls to the quantum computer for updating the parameters are considered a major bottleneck for VQAs in general, Streif and Leib [131] suggested a different approach to QAOA by introducing a technique that avoids the outer loop optimization and therefore repeated calls to get optimal parameters.The proposed method is based on the features of the according problem graph and makes use of tensor network techniques.Quantum Annealing.A comprehensive survey on QA on a conceptual level is provided by Hauke et al. [62], where a strong focus is on future perspectives.Promising improvement strategies comprise non-stoquastic Hamiltonians [170], spatially inhomogeneous driving of the transverse field, and reverse annealing [205].The method of counter-diabatic driving [190] circumvents the bottleneck of the adiabatic regime where proposals for implementation on current devices work with an approximate two-parameter Hamiltonian [111].An inhomogeneous driving of the transverse field is referred to as adapting the annealing schedule, which may reduce the total evolution time, but is complicated to apply, because the precise location of the minimal gap is in general unknown [223].Therefore, automated ways of finding such schedules are covered in the subsequent paragraph, which treats higher-level solutions concerning QA.
Other approaches.When it comes to approaches to quantum CO other than QAOA/VQE or QA, the majority is based on the gate-model of quantum computation.In this sense, Grover-based optimization algorithms [184] such as Grover Adaptive Search [161] have been proposed.An efficient way for the oracle construction within this algorithm that may allow it to be run on NISQ devices has been proposed [52].Another Grover-based approach suggests the automatic generation of the oracle specifically for the k-coloring problem [118].

70:13
The iterative feedback-based algorithm for quantum optimization represents a gate-based approach that is free from classical optimization [94] and shows monotonical improvement behavior.Turning to hybrid quantum-classical algorithms, the iterative quantum assisted eigensolver [16], as well as its predecessor, the Quantum Assisted Eigensolver [15], are designed to circumvent the barren plateau problem.
A variational quantum circuit for CO that is not derived from QAOA or VQE is represented by a quantum version of the shifted power method [211], where the rationale is based on finding the dominant eigenpair of a matrix [36].Highly problem-specific ansÃďtze to VQAs, which are not based on standard methods such as QAOA or VQE, and their effect on performance and resulting energy landscapes have been studied [83,88,151,152].
Other approaches to quantum CO comprise digital-analogue QC and photonic QC.Digitalanalogue QC can be understood as a mixture between continuous QC (e.g., QA) and discrete QC (e.g., gate-model QC) [197].This paradigm has been applied to implement QAOA, yielding superior results compared to the standard QAOA [63].Gaussian boson sampling, however, is a special-purpose photonic platform to perform sampling tasks that are classically intractable [186].CO applications of photonic QC include dense subgraph identification, Max-Clique, and graph similarity [24].Finally, when focusing on quantum-ready problem formulations, there is the possibility of not using quantum resources at all but to experiment with classical approaches like the QCI Qbsolv [19].

Answer RQ1.2: How can quantum algorithms for quantum CO be categorized?
The identified approaches to quantum CO are captured and classified in Figure 2. In the following, we will see how this categorization emerges naturally from the identified studies.Note that the subsequent categorization for answering the research question differs from the one used above.The latter categorization solely served to present the included studies in line with the knowledge provided in Section 2.
The first categorization can be made between gate-model, quantum annealing, and other approaches to quantum CO [19,63].Within the gate-model domain, a further distinction can be made between VQAs, Grover-based approaches [52,118], and other gate-based approaches that have been mentioned above, e.g., References [15,16,24,94].
Taking the feature model of Figure 2 as the underlying categorization scheme allows to classify all studies mentioned above.Here, the majority deals with gate-model approaches (80%), where only 16% treat QA and 4% other approaches, respectively.Within the gate-model architecture, VQAs seem to be the area of most scientific interest (85%).
Answer RQ1.3:How can quantum algorithms for quantum CO be adapted?Based on the findings regarding the categorization of existent quantum approaches to CO (RQ1.2),possible adaptations to those approaches are summarized and classified in Figure 2. The building blocks of any VQA are derived from existent literature [164,171] and can be used for classifying the respective adaptations for this kind of quantum algorithms.In this sense, CVar [10], ascending CVar [78], and the Gibbs objective function [86] can be regarded as adaptations of standard QAOA or VQE that affect their underlying objective function.Likewise, architecture search approaches that adapt the ansatz of the variational approach may take an existing ansatz as a starting point [86,89,155] or be more generally created [88,96,97,151,152].Besides the mentioned ansatz search, as the driving force behind ansatz adaptation, there is also the possibility of having these adaptations as a byproduct, like it happens, e.g., within the Quantum Alternating Operator Ansatz where the primary goal is to find a mixing Hamiltonian that ensures to stay in the feasible subspace [11,58,95,112,116,117,120,127].Several other approaches have been identified where the ansatz adaptations happen as a secondary effect, e.g., References [36,48,82,131,153].Methods to conduct the classical parameter optimization of a VQA that go beyond the standard classical optimizers (cf.[166]) comprise machine learning approaches [51,75,76,141,143,146] and sequential minimal optimization [99].These studies are described in more detail in Section 4.1.3,together with the explicit classical means for parameter optimization.Furthermore, a technique has been proposed to gain efficiency in the final measurement by taking advantage of symmetries [122,125].
In QA, the cost Hamiltonian, the driving Hamiltonians, and the annealing procedure must be specified.Adaptations to the standard QA mentioned in Section 2 comprise the utilization of timedependent Hamiltonians [136], non-stoquastic Hamiltonians [62], transverse couplers [136], and counter-diabatic driving terms [111].The annealing procedure has to be specified in terms of space, i.e., action on different qubits, and time.In this regard, adapted annealing schedules [27,88], reverse annealing [62], as well as spatially inhomogeneous driving [136] represent according possible adaptations.
Analyzing the feature model given in Figure 2 allows to identify the underlying ansatz as the most prominent possible adaptation for VQAs (64%), whereas changes to the Hamiltonian represent the most studied adaptation for QA approaches (50%).
The considered studies associated with the leaves of Figure 2 are given in Table 3.Note that the included studies only cover adaptations to VQAs and QA.Studies on adaptations for Groversearch-based algorithms or other approaches have not been identified.

Enhancement by and Integration into Classical Computation Routines.
Building on the categorization of quantum solutions provided in Section 2, the following presentation of identified studies is structured into approaches that relate either to QAOA/VQE or QA or represent other approaches without such a relation.QAOA/VQE.Many enhancement proposals refer to a close interaction between quantum and classical computation.In this regard, the option of not starting from an equal superposition state but rather from a state that is already near the optimum has been explored, a procedure that is known as warm-starting [42].A relaxed QUBO is first solved via classical means, and the obtained results are used as initial states for the QAOA.However, the initial state represents the lowestenergy state of the mixing Hamiltonian in QAOA, making adaptations to this operator a necessity.Experiments utilizing an Semi-Definite Programming (SDP)-algorithm as the classical solver have revealed that the procedure causes a flattening of the parameter landscape that reduces the Quantum Combinatorial Optimization in the NISQ Era 70:15
To overcome fundamental limitations of QAOA, like the ones mentioned in Section 4.1.4(e.g., References [43,44]), a drastically different, non-local version of QAOA is proposed by Bravyi et al. [23], which is called Recursive QAOA (RQAOA).In this context, the quantum device is used to reduce the problem size to a feasible one for classical solvers.Besides this approach, the problem of many local optima in the energy landscape has been addressed in two ways: utilizing a APOSMM multi-start approach and reusing good parameters that have been found for similar problems [123].Similarly, patterns in the parameters have been explored during advancement to higher depths of the circuit for parameter initialization [6,154], e.g., via different regression techniques [6].Additionally, neural networks have been trained on small simulatable problem instances to learn parameters that can then be applied to larger problems.The utilized machine learning approaches for parameter optimization comprise reinforcement learning [51,75,76,141,146] and kernel-density estimation [76] as well as Long Short TermMemory (LSTM)-networks [143].The experimental results suggest a high degree of transferability and robustness to noise as well as an increased solution quality.Besides such machine learning techniques, also analytical solutions to parameter optimization have been suggested, e.g., sequential minimal optimization [99].
Machine learning approaches have not only been tested for finding optimal parameters but also for finding an appropriate mixer Hamiltonian for the Quantum Alternating Operator Ansatz [58].In this sense, the mixer unitary can be learned based on two cost functions and subsequently be directly inserted into the quantum circuit [114], allowing for a flexible control concerning the trade off between accuracy and the depth of the learned unitary.
Continuing with methods for ansatz search problems, the approach of differentiable quantum architecture search, which is based on the ideas of differentiable neural architecture search for neural nets [151] has been proposed.Similarly, reinforcement learning can be used to generate hybrid quantum-classical programs where the agent iteratively builds the quantum circuit by choosing from a given set of quantum gates [97].An ansatz search method that is more specific to QAOA is the ADAPT-QAOA [155], where the ansatz is grown iteratively and tailored to a specific problem.Furthermore, it has been proposed to search for the solution space near the original QAOA ansatz, e.g., with a greedy heuristic [86].
In the following, studies regarding a higher-level utilization of QAOA or VQE are discussed, which allow either to extend the problem size or the supported problem types.Although further hybrid approaches that utilize quantum computing as a subroutine are subsequently mentioned when discussing other approaches, the specific design or evaluation of the following methods justifies to present them within this section.In this sense, QAOA has been utilized as a local neighborhood sampler for a Tabu-driven search [98], and a divide-and-conquer version of QAOA has been proposed to cope with larger problem instances [84].
To not only extend the size of feasible problems but also their type, an extension to VQAs that allows to tackle MBO problems has been suggested [20].Similarly, Alternating Direction Method of Multipliers (ADMM) heuristics [169] have been utilized for such extensions to MBO problems, where the constraints are treated classically [50].
An applied problem-specific hybrid algorithm has been proposed for financial indexing [46].A further specific application of variational algorithms within a classical framework is a proposed quantum version of case-based reasoning [158].In this context, the classifier and synthesizer parts are substituted by according quantum algorithms [8].
Quantum Annealing.Applied QA techniques are subsequently discussed.These include, e.g., problem decomposition methods, hybrid applications where specifically QA represents a subroutine and the combination of QA with machine learning.
Regarding problem decomposition, an alternative to qbsolv is represented by the quantum tabu algorithm [109] and its extended version [108], which perform similar to qbsolv in terms of solution quality but require less QA resources.Specifically for one-hot encoded problems, two decomposition methods are suggested: multivalent and binary partition [107].Further problem-specific decomposition methods have been proposed for the item listing problem [102] and the Max-Clique problem [110].
Similar to those decomposition techniques, there are several hybrid approaches that utilize QA, with or without the additional use of decomposition, to solve a sub-problem within a classical framework.Although some of the methods may also work with gate-model algorithms, they have been specifically stated in the context of QA.These approaches include the combination of QA with Simulated Annealing (SA) [38] and with a molecular dynamics algorithm [72], which serves as a preconditioner for QA.Problem-specific hybrid approaches have been applied to the capacitated vehicle routing problem [45] and the community detection problem [115].
Hybrid approaches have also been proposed to treat inequality constraints [149], allowing to tackle larger problems compared to using standard approaches that introduce slack variables (cf.Section 2).An adaptation to standard QA, which deals with local constraints by global movement of decision variables, is called the sweeping quantum annealing algorithm [145].Another approach to deal with constraints within the QA regime is based on a transformation of the problem formulation, which yields linear terms but fluctuating coefficients.Then, QA is used to compute those coefficients via sampling [105].
Finally, as it has been the case for the gate-model VQAs, machine learning techniques also have been applied to QA to enhance its performance.Reinforcement learning [88] and a Monte Carlo Tree Search algorithm in combination with a neural net [27] have been proposed to find optimal annealing schedules.Reinforcement learning has also been applied to find appropriate penalty terms of the Hamiltonian where the quantum annealer serves as the stochastic environment that provides feedback for the agent [9].
Other approaches.The subsequently presented hybrid quantum-classical approaches are designed to augment the capabilities of current quantum-only methods by tackling larger and/or more generic problems.In this sense, a QUBO solver, whether gate-model-based or QA-based, is applied as a subroutine within a classical framework.A problem-specific example is the Quantum Local Search (QLS) algorithm, which is designed for the community detection problem [124].The QLS is further combined with a multi-level

Hamiltonian definition
Machine learning [9] Parameter initialization and optimization Analytical solver [99] Knowledge base [123,154] Machine learning [6,51,75,76,141,143,146] Preconditioning for QA Analytical solver [72] Preconditioning for VQAs (warm-start) Solver for relaxed CO problem [42,120,137] approach (e.g., Reference [182]) to tackle even larger problems [139].The proposed method allows to formulate and evaluate sub-QUBOs without the need of constructing the QUBO for the entire problem.The rationale of QLS has also been adapted to extend the range of feasible problems to arbitrary QUBOs [90].Kalehbasti et al. [74] proposed the generalizable idea of replacing the greedy heuristic within the Louvain algorithm [165], a classical heuristic for the k-community detection problem, by a QUBO solver to consider a larger search space in each iteration [74].
Further proposed hybrid approaches allow to extend the feasible problem classes.Examples comprise the application within Bender's decomposition [26] for mixed integer problems, and the combination with an ADMM heuristic for MBO problems [50].In this context, a certain amount of noise in the quantum solver has turned out to be even beneficial for the final solution quality [50,61].QUBO solvers have also been applied within the Branch-and-Price method [162] to solve the Restricted Master Problem within this approach [133].Finally, Ajagekar et al. [2] presented four examples of large-scale optimization problems solved with problem class-specific hybrid decomposition techniques: molecular conformation, job-shop-scheduling, manufacturing cell formation, and vehicle routing [2].
Answer RQ1.4:How can quantum algorithms be enhanced by and integrated into classical computation routines?Answering RQ1.4 comprises two parts.First, possible enhancements of quantum algorithms for CO together with the according classical means of computations are outlined in Table 4. Second, the cases where a quantum solver is used by a classical framework as a subroutine are summarized in Table 5.
Concerning the former, a preconditioning of the quantum algorithm for VQAs is provided by warm-starting with classical solvers for relaxed CO problems [42], e.g., an SDP solver [137].Quantum Annealing, however, may be preconditioned by a molecular dynamics simulation [72].Initial parameters for VQAs may be taken from previous, similar problems [123,154] or be obtained by machine learning approaches, which additionally can be used for finding patterns in parameters and parameter optimization.Specific techniques concerning the latter include: reinforcement learning [51,75,76,141,146], regression techniques [6], and LSTM networks [143].Besides the vast amount of available numerical optimizers, solvers for sequential minimal optimization [99] represent another alternative for parameter optimization.It shall be noted at this point that, according to our exclusion criteria, we refrain from taking into account the former in our analysis, as these approaches deserve a survey on their own [166].Similarly, we do not consider pure error mitigation techniques in our study.More information on the latter may be taken from Endo et al. [176].Besides finding reusable parameters and optimizing parameters, machine learning techniques have been applied for ansatz search purposes, e.g., for finding mixer Hamiltonians for the
The second part of the answer to RQ1.4 is summarized in Table 5, where a distinction is made concerning the specificity of the underlying quantum solver.Whereas some studies explicitly utilize a certain quantum CO approach, others are agnostic in this manner and just require any solver that is feasible for QUBO problems.It can be seen that the majority of studies considers quantum architecture independent solvers for the subroutine (42%).However, another significant part deals with decomposition methods (17%) that have been designed in the context of QA approaches (33%).Approaches that utilize QAOA or VQE only constitute 25%.

Properties.
Practically important theoretical and empirical properties of the presented methods are subsequently outlined.In this section, we only include studies that solely investigate properties of existing quantum solutions without proposing a new approach.Applying this restriction, only papers relating to QAOA/VQE and QA, respectively, have been found and are presented.
QAOA/VQE.Starting with theoretical properties of the gate-model approaches, an analysis of the locality of QAOA revealed that the circuit depth has to exceed a certain value, which depends on the degree and size of the problem graph to yield correlated measurement results and to "see" the whole graph [43,44].The strong limitations concerning the reachability of QAOA have been numerically validated [4].However, a lower bound for the required circuit depth has been theoretically derived and found to be the chromatic index of the problem graph plus one [64].
Regarding the Max-Cut problem, it has been theoretically and numerically shown that for fixed parameters of QAOA the resulting objective values for different graphs of a given family are similar, provided the number of variables is large [21].This property can be exploited for parameter initialization.
Although it has been shown that for infinite depths and optimal parameters the QAOA corresponds to a discretized adiabatic computation that yields optimal results [177], the large but finite depth regime is not well understood.Studies on QAOA as a bang-bang protocol, like the ones by Liang et al. [87], allow to derive its properties for such large circuit depths.In this context, a bangbang protocol is an important part of optimal control theory, where in general a system switches abruptly between two different modes [208].Finally, a theoretical comparison between QAOA, QA, 70:19 and SA identified a class of problems that is only feasible for QAOA due to high and broad walls in the energy landscape [130].
Continuing with empirical studies on gate-based quantum CO approaches, a large-scale evaluation on QAOA [92,121] validated the theoretical properties [4,21,43] and yielded further empirical performance bounds.Other investigations highlight the strong dependence of QAOA on the problem instance, the number of parameters, and their initialization [81,142].Concerning the impact of the graph type, correlations between graph metrics and QAOA performance metrics have been identified [65], where symmetries in the graph have been found to be strong success predictors.
Further studies put their focus on the effects of noise on the performance of QAOA [5,56,144].The results of these studies are affected by the utilized hardware as well as the conduction time of the experiment.Thus, as an example, the optimal depth as a tradeoff between ansatz accuracy and error rates has been found to be p=1 and p=3, respectively [5,60].With such peculiarities in mind, one should also view the results of Guerreschi and Matsuura, who found the point of quantum supremacy for QAOA to be in the order of several hundred to thousand of qubits [56] and further raised scaling issues [60].An empirical evaluation of recursive QAOA has demonstrated its superiority compared to the standard version as well as its competitiveness compared to the Newman benchmark algorithm [203] for the Max-Cut problem [23].
All the studies mentioned above cover the evaluation of QAOA.Only two papers specifically consider benchmarking VQE in the context of CO.The limitations of VQE under noiseless settings have been investigated [100], whereas the focus of others lies on the role of entanglement within VQE [37].Experiments utilizing CVar and ascending CVar as the underlying objective function suggest that VQE is more promising than QAOA for shallow circuits due to the reachability issues of QAOA [10,78].
Investigations on the impact of different classical optimizers for parameter optimization in general advocate for gradient-free global optimizers when dealing with real quantum hardware [37,47,100,121].The importance of the optimizer is further highlighted by a time reduction of two orders in magnitude regarding experiments on various graph instances [81].
Further evaluation studies concerning the impacts of enhancement possibilities of VQAs focus on their combination with machine learning techniques.In this sense, the effects of parameter symmetry exploitation and parameter regression have been investigated [85], and the training of parameters on a whole batch of randomly generated graph instances to reduce training costs [33] has been studied, yielding promising results.
Last, comparative evaluation studies between VQAs and QA have demonstrated (i) the reachability issues of QAOA compared to VQE [10], (ii) the ability of QAOA to harness diabatic processes in contrast to QA [154], (iii) the superiority of QA in terms of performance [142], and (iv) the character of QAOA as a digitized version of QA [141].However, one has to consider the different experimental setups in this regard, e.g., different backends and embedding costs of graph instances.
Quantum Annealing.Tang and Kapit [136] provide an empirical benchmark study on mitigation techniques concerning the shrinking energy gap.The methods comprise inhomogeneous driving, added transverse couplers [213] and RFQA (Random Field Quantum Annealing or Radio Frequency Quantum Annealing) [192].The impact of various quantum controls, such as annealing time and number of spin reversals [194], on the success probability and chain breaks has been studied by Grant et al. [55], yielding a saturation effect for both controls.Furthermore, the effects of problem formulation, embedding scheme, and annealing schedule on the QA performance have been evaluated [156].
It has been suggested that, in contrast to classical complete algorithms, increasing problem hardness causes a loss of precision for QA instead of detrimental solution times [49].A comparison with

Effects of noise
The effects of noise constitute an optimal depth for QAOA [5,60].Ideal parameter values do not deviate when noise is not serious [144].Noisy computation leads to scaling issues for QAOA [56,60].

Parameter concentration
Optimal parameters for QAOA are similar for similar problem instances.This feature can be used for good parameter initialization.It has been theoretically shown [21] and empirically validated [121].

Properties of adapted versions
The benefits of using RQAOA [22], CVar [37,78], and machine learning [33] have been empirically validated in dedicated studies.

QAOA metrics correlate with graph metrics
The highest correlation has been identified between the expected cost of QAOA and the number of edges, diameter, clique size, number of cut vertices, and number of odd cycles [65].

QAOA reachability
The reachability issue of QAOA refers to its limited usefulness for low circuit depths.It has been theoretically proven [4,43,44] and empirically validated [121].For this reason, VQE is preferable for shallow depth circuits [10].

Quantum Annealing
Comparison to QAOA QAOA is preferable when the energy landscape shows high and broad walls [130].QAOA can harness diabatic processses [130].

Dependence on problem hardness
In contrast to classical complete algorithms, increasing problem hardness causes a loss of precision instead of detrimental solution times [49].
Scaling QA shows superior scaling properties compared to SA, where both are significantly inferior to simulated QA [7].
Shrinking spectral gap Proposals to overcome the problem of a shrinking spectral gap: inhomogeneous driving, RFQA-M, RFQA-D [136].

Success rate saturation
The success rate of obtaining the optimal solution saturates with increasing annealing time and number of spin reversals [55].
A comparison between various decomposition techniques, i.e., principal component decomposition [185], free-and-anneal [163], qbSolv, and iterative centrality halo [172] has demonstrated the superiority of qbsolv in terms of solution quality [12].Answer RQ1.5: Which properties of quantum algorithms for quantum CO have been studied?The main properties that have been extracted from the theoretical and empirical evaluation studies above are summarized in Table 6.Note that the mentioned properties are derived from dedicated studies on existing approaches.Thus, properties of adaptations or new prototypes are not considered here but mentioned in Section 4.1.2and 4.1.3,respectively.
Overall, 75% of the identified studies cover a property of QAOA with a special focus on its reachability issues (33%) and performance under noise (27%).Each property of QA, however, is covered only by a single study.

Use Cases for Quantum Combinatorial Optimization
In the following, some identified use cases of the standard as well as the above-mentioned approaches are stated.Thereafter, an answer to RQ2 is given.Note that the presented applications of quantum technologies to optimization problems should not be considered as a complete list but rather a representative sample obtained from the included literature of this study.Furthermore, studies that only concern general or arbitrary QUBO/Ising formulations that do not belong to a Others: [90] certain problem are not covered.The included studies apply a quantum computing approach either on real quantum hardware or according classical simulators for validation.Therefore, they do not comprise solely mappings from certain problems to according QUBO or Ising forms.After a short depiction of studies that have not been mentioned so far but rather apply a standard approach to certain problems, Tables 7, 8, and 9 show typical use cases and their associated solution approaches.
The stated problems are structured into problem classes (e.g., graph optimization).Table 7 shows instances where QC has been applied to standard CO problems, whereas Table 8 lists real-world problems that have been mapped to such standard CO-problems.Table 9 covers real-world problems that have been tackled directly without such a mapping.A more detailed depiction including the various quantum CO approaches is provided in the replication package [183].Dalyac et al. [35] utilized tailored implementations of QAOA to solve two problems arising in the field of smart charging electric vehicles, which can be mapped to Max-k-Cut and Maximum Independent Set problems, respectively [35].Whereas a parallelized version of QAOA [82] is used for the first problem, the second one requires a transformation of the original graph into a unitary disk graph to be embeddable on Rydberg QC architectures.The Max-k-Cut problem has also served for a comparative study concerning standard QAOA and RQAOA [22].
Standard QAOA has been applied to: the cluster head selection problem [28], the wireless scheduling problem [29], the vehicle routing problem [14], multi-coloring graph problems [104], the graph isomorphism problem [66], and a satellite scheduling problem [128].An adaptation to QAOA, which does not require exhaustive access to the quantum processing unit [131], has been used to solve the binary paint shop problem [132].The Maximum Independent Set problem has been tackled with the Quantum Alternating Operator Ansatz [119].VQE has been applied to a QUBO formulation for the social workers problem, which explores problem specificities to avoid excessive auxiliary variables [1], and to multi-coloring graph problems [104].
QA, which may include standard decomposition using qbsolv or simulated QA often serve for the validation of problem formulations, which is usually accompanied by an empirical comparison to classical solvers like, e.g., SA.In this sense, QUBO formulations for the graph coloring problem [126], the k-community detection problem [101], the Maximum Independent Set problem [148], the Minimum Multicut problem [34], the most frequent itemset problem [103], consensus clustering [32], the online advertisement allocation problem [135], traffic signal optimization on a square lattice [70], the Multi-depot Capacitated Vehicle Routing problem in its static and dynamic version [59], and the Vehicle Routing Problem including the concepts of time, capacity, and state are provided [73].Furthermore, QA has been applied to the problem of distributing transactions using a 2-phase locking protocol in an optimal manner to reduce waiting times [17,18].A simplified version of the air-traffic management conflict-resolution problem has been tackled with QA after extraction of feasible subproblems [129].Additional QUBO-formulations for scheduling or routing problems that have been validated with QA comprise: the workflow scheduling problem [138], conflict management on a single-track railway [40], real-time multi-robot routing [31], and controlling automated guided vehicles on a factory floor [106].Ajagekar and You [3] apply QA to three problems from energy systems, namely, facility location allocation, unit commitment, and heat exchanger networks, whereas Ding et al. [39] used QA for the prediction of financial crashes [39].In this context, comparing the initial formulation of the problem as a Higher-Order Unconstrained Binary Optimization Problem (HUBO) to the QUBO-form highlights the importance of multi-qubit connected annealers to natively treat HUBO problems.The problem of election poll forecasting has also been formulated as a HUBO and QUBO for QA, respectively [68].Further problems that have been treated with the standard QA approaches include: max-sum diversification [13], model-predictive control [71], the tiling puzzle problem [41], and image acquisition planning for earth observation satellites [128].Finally, Koshikawa et al. [79] utilized and benchmarked QA within black-box optimization.
When it comes to more advanced quantum annealing methods, forward as well as reverse annealing have been applied to the nurse scheduling problem [69] and portfolio optimization [140].Hybrid quantum-classical solvers have been applied to the garden optimization problem [25] and traffic optimization [67].QA has also been utilized for traffic optimization within the Web Summit Conference in Lisbon, where it has been shown how QA can be applied within a classical application to enable real-world utility [147].
An example for the so-called quantum-ready approaches is represented by the graph partitioning problem [30] solved with a purely classical sampler.

Answer RQ2: What are investigated use cases for quantum CO?
To answer RQ2, Tables 7-9 have been analyzed concerning problem classes of CO with the results being stated in Figure 3. Problems of Table 7 have been assigned to standard CO problems, whereas the ones of Table 8 to real-world problems.All problems of Table 9 have been classified as real-world problems without mapping.In general, the assignment of certain problems to such CO classes is not unambiguous and problem classes or forms may be transformed into each other, e.g., the Vertex Cover problem may be regarded as a graph optimization or covering problem.Therefore, the classification has been conducted according to the definition of the problems in the original literature.An exception is the Quadratic Assignment Problem, which constitutes a general form of CO problems and, therefore, is not included in Figure 3.
In the right part of Figure 3, the problems mentioned above are grouped to according problem classes where for each class a further distinction is made on whether the problem under study constitutes a standard CO problem or a real-world problem with an according mapping to a CO problem.Finally, problems that represent real-world instances but where no mapping to standard CO problems is available have been taken into account in the last column.Whereas this  classification has been made for the number of problems in the right part of Figure 3, the left part illustrates the classification concerning the number of studies.For example, taking into account the graph coloring problem in Table 7 would lead to an increase of the standard CO problem part of graph optimization by 1 in the right part of Figure 3, but to an increase by 6 in the left part.From Figure 3 (right) it can be seen that most of the problems that have been solved with QC techniques are real-world instances with no given mapping to a standard CO problem in the original literature.This is mainly due to studies where the aim is to validate a QUBO-formulation of a particular problem with QA.Apart from this, the graph optimization and routing/scheduling problems turn out to be the most frequently tackled ones.A similar picture is shown in the left part of Figure 3 where a significant amount of studies on standard graph optimization problems can be seen.This is due to the Max-Cut problem being the most popular validation problem especially for gate-based approaches (cf.Table 7).

Bibliometric Key Facts
This section shows the results of the bibliometric analysis of the included studies regarding (i) the publication type, (ii) the research facet, (iii) the topic area, and (iv) the publication channels.The information upon which this analysis has been conducted can be found in the replication package [183].
Answer RQ3: What are the bibliometric key facts of quantum CO publications?The distribution of papers by publication type and year is shown in the left part of Figure 4.The increase of works on CO in NISQ devices can be observed as of 2018.In addition, 38% of the considered studies have been published in ArXiv, which is the publication type that contributes second most after journal articles found on peer-reviewed repositories (47%).These findings are in accord with the report of Scopus [157], which conclude that there is an increase of work in the area of CO with NISQ devices and a large amount of papers are published in non-peer-reviewed repositories.The distribution of the research facet is shown in the right part of Figure 4.The relative frequencies for the research facets as introduced by Wieringa et al. [219] are shown for each year.The contribution of pure solution papers has steadily decreased from 17% to 0%.Furthermore, the relative frequency of studies that provide a solution proposal together with some validation has steadily increased from 50% to 73%, whereas the pure validation papers make between 20% and 36%.There have not been any evaluation papers identified that would require the according technique to be implemented in practice to identify problems in the industry.The high contribution of solution proposals shows that quantum CO is still an emerging field.However, the increasing importance and interest in validating the solutions also reflects the maturing process of the technologies related to quantum CO.The topic area of highest scientific interest are gate-based approaches to quantum CO (53%), followed by QA (35%).Studies have been assigned to other approaches (4%) or problem formulation (8%) only, if the given method has been explicitly designed in a quantum solution-independent manner.Additional information that further refines the topic areas by year can be found in the replication package [183].Taking a closer look at the published articles and conference papers, Tables 10 and 11 present the most frequently used channels for publication.Journals that are not listed in Table 10 show a number of published articles of ≤ 3, whereas the non-listed conferences and workshops only published one of the studies considered in this review.Within the journals, 58% of the listed studies have been published in journals that explicitly focus on quantum computing, whereas the percentage is 45% for the conferences and workshops.The significance of publication channels of related fields highlights the interdisciplinary character of quantum CO.

THREATS TO VALIDITY
To minimize the variables that could compromise our study, we (i) conducted our study following accepted guidelines of systematic studies [181,193,207,220], (ii) carefully designed our study in advance, and (iii) assessed, validated, and discussed the potential threats in each phase of the study.Nevertheless, the following threats have been identified: External validity: The generalizability of the obtained findings may be threatened by a nonrepresentative study sample.To mitigate this threat, we (i) decided to include potentially non-peer reviewed literature from e-print repositories as an important scientific contribution source and (ii) conducted a two-phase search procedure including forward and backward snowballing.The threat due to limiting our studies to the English language can be considered as minimal, because it is the most widely used scientific language.Other exclusion criteria, like the limitation to studies that have been published since 2018, are discussed in Section 3.
Internal validity: An extraneous variable that may have influenced the design of our study is the objectivity concerning the decision of study inclusion/exclusion.This potential threat has been mitigated by the required agreement of the researchers regarding the selection of relevant papers.
Construct validity: Within a systematic study, the main threat to the validity with respect to the research questions regards the sample of included studies and how well it represents the current state-of-the-art in light of the research questions.We encountered this threat by (i) using multiple electronic databases for the automated search to avoid potential biases, (ii) deriving the search string from the research questions, (iii) conducting a snowballing procedure, and (iv) relying on inter-researcher agreement in all phases of the data extraction and analysis.

Conclusion validity:
We would like to emphasize that other researchers may derive different concepts and attributes compared to the ones we presented in our classification framework.We took four countermeasures to mitigate this potential threat: (i) not starting with fixed concepts, but letting the concepts emerge from the selected studies by refining them throughout the analysis process, (ii) performing the data extraction process conducted by several researchers, (iii) relying as much as possible on the concepts and terminology provided in the primary studies, and (iv) limiting our discussion on results to information that is directly derived from extracted data.

CONCLUSION AND FUTURE WORK
Quantum CO refers to the use of quantum computing technologies to solve complex combinatorial optimization problems, where already-existing NISQ devices enable the first experimental steps in this domain.We presented a systematic mapping study with the goal of identifying, classifying, and understanding existing approaches of quantum CO approaches that utilize NISQ devices.Applying a rigorous selection procedure, we end up with 156 papers included in this review.After analyzing and discussing the extracted data, we end up with the following results: (i) Quantum CO already starts with an according problem formulation that is feasible for current NISQ devices and takes into account the properties of the same.Whereas the majority of studies investigates methods to transform between different mathematical problem formulations, only one paper directly states 70:27 feasible formulations for CO problems.(ii) Current solutions to quantum CO comprise adaptations to and utilization of standard quantum algorithms via classical computational means.The topic of ansatz design for VQAs is of predominant interest in this regard.Furthermore, we see a high interest for using machine learning techniques to enhance quantum solutions.From our point of view, this approach has to be monitored carefully, as we see its great potential, on the one hand, but possible scaling issues, on the other hand.(iii) Typical use cases for quantum CO can be found, e.g., in the areas of graph optimization, routing, and scheduling.The Max-cut problem has been found to be the most popular problem for validating solution proposals.Given the popular application of QA to real-world problems, we identify a lack of research interest concerning its fundamental properties as well as adaptation possibilities in the considered timeframe.(iv) We observe a steeply increasing research interest in the emerging field of quantum CO in general, where recent studies more frequently cover gate-based, rather than QA-based approaches.
Future Work.Based on the provided dataset [183], the evolution of research interests and use cases can be extracted.In this regard, the annual publications concerning QA, gate-based approaches, and also their integration into classical routines can be analyzed.Furthermore, the problem classes of the investigated use cases can be analyzed further regarding their evolution.Moreover, as quantum computing is in its transition from a purely scientific research topic to a technology for real-world industry applications, the study corpus may be extended with grey-literature, e.g., by vendors and early adopters of quantum solutions.

3 -
have been studied?Quantum Combinatorial Optimization in the NISQ Era 70:RQ2: What are the investigated use cases for quantum CO? -RQ3: What are the bibliometric key facts of quantum CO publications?

Fig. 2 .
Fig. 2. Feature diagram for categorization of quantum algorithms for CO and their adaptations.(Numbers below leaves denote absolute number of identified studies and percentage regarding parent node.).

Fig. 4 .
Fig. 4. Left: Distribution of papers by year and publication type; Right:Distribution of papers by year and research facet.

Table 1 .
Existing Reviews on Quantum CO

Table 2 .
Studies Treating Problem Formulation within Quantum CO

Table 3 .
Identified Studies Concerning Quantum Algorithms for CO and Their Adaptations

Table 4 .
Identified Possibilities for Enhancing Quantum Algorithms with Classical Means of Computation

Table 5 .
Utilization of Quantum CO Approaches within Classical Framework

Table 6 .
Theoretical and Empirical Properties of Quantum CO Approaches

Table 7 .
Treated Standard CO Problems and Associated Solution Approaches

Table 8 .
Treated Real-world CO Problems with Given Mappings to Standard Problems and Associated Solution Approaches

Table 9 .
Treated Real-world CO Problems and Associated Solution Approaches

Table 10 .
Most Frequently Used Journals for Publications

Table 11 .
Most Frequently Used Conferences and Workshops for Publications