Focusing on the Hybrid Quantum Computing -- Tabu Search Algorithm: new results on the Asymmetric Salesman Problem
Eneko Osaba, Esther Villar-Rodriguez, Izaskun Oregi, Aitor Moreno-Fernandez-de-Leceta
FFocusing on the Hybrid Quantum Computing - Tabu SearchAlgorithm: new results on the Asymmetric Salesman Problem
Eneko Osaba
TECNALIA, Basque Research andTechnology Alliance (BRTA)20009, Donostia-San Sebastian, [email protected]
Esther Villar-Rodriguez
TECNALIA, Basque Research andTechnology Alliance (BRTA)20009, Donostia-San Sebastian, [email protected]
Izaskun Oregi
TECNALIA, Basque Research andTechnology Alliance (BRTA)20009, Donostia-San Sebastian, [email protected]
Aitor Moreno-Fernandez-de-Leceta
Instituto Ibermatica de Innovacion.Parque Tecnologico de Alava,01510 Miñano, [email protected]
ABSTRACT
Quantum Computing is an emerging paradigm which is gather-ing a lot of popularity in the current scientific and technologicalcommunity. Widely conceived as the next frontier of computation,Quantum Computing is still at the dawn of its development beingcurrent solving systems suffering from significant limitations interms of performance and capabilities. Some interesting approacheshave been devised by researchers and practitioners in order to over-come these barriers, being quantum-classical hybrid algorithms oneof the most often used solving schemes. The main goal of this paperis to extend the results and findings of the recently proposed hybrid
Quantum Computing - Tabu Search
Algorithm for partitioning prob-lems. To do that, we focus our research on the adaptation of thismethod to the Asymmetric Traveling Salesman Problem. In overall,we have employed six well-known instances belonging to TSPLIBto assess the performance of
Quantum Computing - Tabu Search
Al-gorithm in comparison to QBSolv – a state-of-the-art decomposingsolver. Furthermore, as an additional contribution, this work alsosupposes the first solver of the Asymmetric Traveling SalesmanProblem using a Quantum Computing based method. Aiming toboost whole community’s research in QC, we have released theproject’s repository as open source code for further application andimprovements.
KEYWORDS
Quantum Computing, Metaheuristic Optimization, Traveling Sales-man Problem, Transfer Optimization, DWAVE
ACM Reference Format:
Eneko Osaba, Esther Villar-Rodriguez, Izaskun Oregi, and Aitor Moreno-Fernandez-de-Leceta. 2021. Focusing on the Hybrid Quantum Computing -
Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for components of this work owned by others than ACMmust be honored. Abstracting with credit is permitted. To copy otherwise, or republish,to post on servers or to redistribute to lists, requires prior specific permission and/or afee. Request permissions from [email protected].
GECCO ’21, July 10–14, 2021, Lille, France © 2021 Association for Computing Machinery.ACM ISBN 978-1-4503-7127-8/20/07...$15.00https://doi.org/10.1145/3377929.3398084
Tabu Search Algorithm: new results on the Asymmetric Salesman Problem.In
Proceedings of the Genetic and Evolutionary Computation Conference 2021(GECCO ’21).
ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/3377929.3398084
Quantum Computing (QC, [49]) is deemed as the next frontier incomputation. The current scientific community has high expecta-tions on this specific paradigm, being this the main reason for thefast-growing popularity that the field is gaining in recent years.Among all its potential benefits, QC has arisen as a promising al-ternative for solving optimization problems. Thus, QC providesa revolutionary approach for tackling this kind of tasks, offeringclear theoretical advantages ranging from significant computationspeed to efficient search ability [1].Two QC architecture types are available today: annealing-basedquantum computers and gate-model quantum devices [28, 53]. Thispaper is devoted to the first of this architectures, which is mostused one for optimization purposes.Although scientific community has devoted many efforts in re-cent years to the advancement of QC paradigm, QC is still at thedawn of its development and current commercial QC systems sufferfrom huge computational and performance limitations [2, 15]. Inthis paper, we focus our attention on the quantum-annealer pro-vided by D-Wave Systems [56], which has emerged as the mostemployed commercial QC device on these days. In any case, eventhis architecture has its own drawbacks despite being the mostused QC machine in the current community [22]. Problems suchas decoherence, limited control to quantum resources or poor er-ror correction suppose unavoidable obstacles for researchers andpractitioners for the formulation of efficient and reliable purelyquantum solving approaches.The objective of overcoming current limitations of QC have ledresearchers to the design and development of hybrid quantum-classical hybrid solving methods. A representative example of theseapproaches can be found in [14] proposing a hybrid technique basedon the 2-Phase-Heuristic for solving the Capacitated Vehicle Rout-ing Problem using D-Wave Quantum Annealer. Another quantum-classical scheme is described in [11] for solving multi-robot routingon a grid in real time. Specifically, the implementation counts on a r X i v : . [ qu a n t - ph ] F e b ECCO ’21, July 10–14, 2021, Lille, France E. Osaba et al. a classical-based mechanism for generating candidate paths and aD-Wave quantum-annealing-based strategy for choosing the op-timal combination of paths. In the recent [1] a hybrid techniqueis proposed for dealing with large-scale optimization problems,both continuous and discrete. The major asset of that paper is itsextensive experimentation, focused on four heterogeneous appli-cations: vehicle routing, manufacturing cell formation, job-shopscheduling and molecular formation. Additional remarkable workis also present in [51] with the development of a quantum-classicalgradient descent-based optimizer.In this context, authors of this paper have recently proposed in[44] a Hybrid Quantum Computing - Tabu Search Algorithm (QTA)for solving Partitioning Problems. Thus, the main objective of thispaper is to solidify the preliminary concepts described in that paper.To do that, first we describe our previously introduced method andthen we investigate its application on the well-known AsymmetricTraveling Salesman Problem (ATSP, [10, 40]). In a nutshell, theATSP is characterized by the particularity of having asymmetriccosts, which means that traveling from a node 𝑖 to another node 𝑗 may not equal the reverse trip. Despite being a feature valuable forreal-world applications by bringing realism and complexity to theoptimization problem, asymmetric costs have been historically lessstudied by the community. With all this, the main contributions ofthis work are threefold: • With this work, we extend the findings obtained in our previ-ously published work [44]. Thus, we elaborate on the applica-tion of our QTA on another partitioning problem: the ATSP.Also interesting is the size of the instances considered in thiswork. Up to now, most investigations focused on solvingdiscrete optimization problems through QC deal with verysmall instances. In this paper, and taking advantage of one ofthe greatest virtues of our QTA, we dedicate our efforts to thefacing of bigger instances. More concretely, we conduct anexperimentation using six different instances obtained fromthe well-known TSPLIB Benchmark [46], composed by 17 to48 nodes. Although this size is indeed small when comparingwith classical computing ATSP solvers, they are much largerthan the instances often included in QC research. • As far as we are concerned, this work supposes the firstsolver of the ATSP using a QC-based method. Through thiscontribution, we attend some recent calls made by the com-munity in papers such as [55] and [54]. These studies ex-plicitly evince the scientific interest in ATSP instances ex-tracted from the TSPLib. In line with this, and with the aimof strengthening the value of this work, we release the wholedeveloped source code which embraces mechanisms for deal-ing with both TSP and ATSP instances. Our final goal withthis initiative is to favor the advancement of the related field. • As mentioned in [44], by using our QTA we also contributeto the field of Transfer Optimization (TO, [26]). Specifically,we employ a multiform multitasking technique as initial-ization procedure for the QTA. The algorithm used for thispurpose is a multiform variant of the Coevolutionary Vari-able Neighborhood Search Algorithm (CoVNS) for DiscreteMultitasking proposed in [43]. The remainder of this work is organized as follows: we introducesome background related to both QC and ATSP in Section 2. InSection 3, we briefly detail the main features of our QTA. After that,we describe the experimentation carried out in Section 4 along withthe results and discussion. We finish this paper in Section 5 withmain conclusions and further work.
As stated in the introduction, this section is meant to provide a briefbackground on two main concepts studied in this paper: QuantumComputing (Section 2.1) and the ATSP (Section 2.2).
QC is gaining an increasing popularity, even breaking the barriersof scientific-world and bringing the attention of many generalistnewscasts and magazines. High hopes are placed in this computa-tion paradigm, mainly because its potential capabilities in termsof speed and efficiency. As aforementioned, the optimization ofcomplex problems is one of the fields expected to benefit from theadvances of QC [4, 38].Quantum computers are specific machines which can performcomputation by means of quantum mechanical phenomena as en-tanglement and superposition. These machines work with a unitcoined as quantum bit, or qubit [29]. In a nutshell, a qubit is con-ceived as the quantum version of the classical bit, being able to storemuch more information by virtue of its superposition condition.Precisely, a qubit can simultaneously depict properties of both 0 and1 [11], surmounting the obstacle of classical binary representations.More concretely, the superposition of qubits favors the possibilityof deeming infinite quantum states before collapsing into a classicalbasic state (0 or 1) [1]. A further essential feature for properly un-derstanding QC potential is the so-called entanglement. Throughthis characteristic, states may encode correlations between differ-ent particles, allowing for interactions and subsequent influencesamong chained qubits.As commented, two kinds of QC architectures can be currentlydefined: gate-model quantum computers and the annealing-basedquantum computers. On the one hand, gate-model quantum com-puters are featured by employing quantum gates for the state ma-nipulation of the qubits. The operation of these gates can be slightlycompared with the traditional logic gates, and they are applied toqubit states in a sequential way, evolving them up to their finalstate [27]. Today, commercial devices have from 10 to 50 qubits,and some significant applications are search methods (Grover’s Al-gorithm) [24], optimization problems [13] or integer factorization(Shor’s Algorithm) [48].On other hand, the main rationale behind annealing-based quan-tum computers is the search of a minimum-energy state of a givenquantum Hamiltonian (i.e., energy function). To do that, this sortof devices employ quantum related concepts such as entanglementand superposition. On this regard, the Hamiltonian is implementedas the objective function of the optimizing problem, represent-ing its optimal solution the lowest energy state [33]. Today, theleading provider company for this type of device is the CanadianD-WAVE, which currently counts with the named as
D-Wave Ad-vantage_system1.1 computer. This machine, located in Vancouver esults of QTA on the ATSP GECCO ’21, July 10–14, 2021, Lille, France and openly accessible though LEAP cloud service , works with agraph composed by 5436 qubits distributed in a Pegasus topology.Specifically, this topology improves the previously one based onthe well-known Chimera. Thus, each qubit is connected to other15 qubits, instead of being linked to 6, as in previous versions ofD-Wave quantum devices. Thanks to this recently deployed fea-ture, D-Wave Advantage_system1.1 is the most connected of anycommercial quantum device in the world .Going deeper, in this QC machine the Hamiltonian is representedas an Ising model H Ising = (cid:205) 𝑛𝑖 = ℎ 𝑖 𝑞 𝑖 + (cid:205) 𝑗 > 𝑖 𝐽 𝑖,𝑗 𝑞 𝑖 𝑞 𝑗 , where 𝑞 𝑖 ∈{− , } depicts the 𝑖 -th qubit, ℎ 𝑖 represents the linear bias associatedto this variable, and 𝐽 𝑖,𝑗 is the coupling strength between 𝑞 𝑖 and 𝑞 𝑗 qubits. Furthermore, D-Wave QC devices allows its users to usean alternative mathematical formulation, based on the well-knownQuadratic Unconstrained Binary Optimization (QUBO, [23]). QUBOcan be formally expressed as: z ∗ = min z ∈{ , } 𝑛 z 𝑇 · Q · z . (1)Where Q is a programmable upper triangular matrix containing thecouplers and the bias needed by the Ising model. Specifically, thenonzero off-diagonal elements represent the coupling coefficientswhile diagonal terms are the linear biases. In this way, z depicts a 𝑛 -length binary variable array, and z ∗ the state that minimizes thequadratic function, namely, the problem solution.As pointed before, limitations of the current QC devices, bothgate-model-based and annealing-based ones, have led the com-munity to rely on hybrid models as good alternatives for solvingcomplex problems [17]. We have spotlighted some of these alterna-tives in the introduction of this paper, but many more can be foundin the literature. The study proposed in [9], for example, is devotedto the Multi-Service Location Set Covering Problem using a hybridtechnique considering D-Wave as the QC component. Authors in[32] propose a quantum-classical hybrid solver for facing the quan-tum optimal control problem whereas a research on communitydetection on graphs is conducted in [36]. Hybrid classical-quantummethods can also be found in [16] and [7] for operating on financialindex tracking and graph coloring problems, respectively. Furtherexamples of hybrid solvers can be found in [20, 25, 35]. Finally, wewant to highlight the work proposed by Warren in 2020 in [55],which is strictly related to the research we are describing in thispaper. In that study, Warren accurately outlines the current stateof the QC research community regarding the TSP. Also, the authorposes some challenges and opportunities which should guide fu-ture efforts to be made by the community. Among these challenges,ATSP on a QC device is incorporated.It is also appropriate to mention in this section the solving toolknown as QBSolv. This strategy is the most employed one by thescientific community for solving large-sized optimization problems,and it consists on an algorithm able to overcome the current D-Wave hardware limitations. Thus, D-Wave can face optimizationproblems with a higher size. Some interesting examples of theapplication of this tool can be seen in [31, 34, 39, 47]. In brief, QBSolvis a partitioning solving tool offered by D-Wave for splitting large complete QUBO into smaller-sized subQUBOs, tackling them inan independent and sequential way. More specifically, QBSolvebreaks the complete QUBO matrix using a tabu search heuristic,and it can be run in a pure local way or using the D-Wave hardware.In any case, and despite being a promising method, QBSolve hasits own limitations, as highlighted in recent studies such as [52].One of these limitations is the high usage of the QC resources.Besides, and despite that it is slightly configurable, the fine tuningof QBSolv demands a significant access to quantum services, notbeing possible for the user to fully control the usage of the D-WaveQC hardware. We refer interested readers on this tool to [5]. As introduced before, this work supposes the first solver of thewell-known ATSP employing QC technologies. Being a specificvariant of the canonical Traveling Salesman Problem (TSP), theATSP can be formulated as a complete graph 𝐺 = ( 𝑉 , 𝐴 ) , where 𝑉 = { 𝑣 , 𝑣 , . . . , 𝑣 𝑁 } represents the group of 𝑁 = | 𝑉 | nodes, and 𝐴 = {( 𝑣 𝑖 , 𝑣 𝑗 ) : ( 𝑣 𝑖 , 𝑣 𝑗 ) ∈ 𝑉 for 𝑖 ≠ 𝑗 } the set of arcs interconnectingthese vertices. Moreover, each connection has an associated 𝑐 𝑖 𝑗 cost,which in this case differs from its reverse path, that is 𝑐 𝑖 𝑗 ≠ 𝑐 𝑗𝑖 . Inthis way, the main goal of the ATSP is to calculate a complete route 𝑇𝑆𝑃 ∗ visiting each node only once, minimizing the total cost andfinishing in the same point in which the route starts.Despite having some crucial advantages in comparison to the ba-sic TSP, such as a more realistic nature, ATSP has been less studiedby the community along the years. Nonetheless, many remarkablearticles can be found in the literature using this specific problem. Inworks such as [18, 42, 45], the ATSP is employed as a benchmarkingproblem along with the TSP for developing and testing evolutionarycomputation methods such as a Genetic Local Search Algorithm,Water Cycle Algorithm and Bat Algorithm, respectively. Furtherexamples can be found in [6] and [50] opting for a Harmony Searchmethod and a Constant-factor Approximation solver. Theoreticalstudies have also been published, as those in [10, 19, 40]. Further-more, diverse real-world oriented variants of the ATSP have alsobeen formulated, aiming at adapting the problem to logistic-relatedenvironments [3, 21, 41].Like every problem within TSP family, the ATSP implies a hugeoptimization challenge due to its NP-Hard nature. This is the mainreason since a myriad of intelligent solvers have been designedfor solving this problem . Additionally, ATSP recurrently servesa benchmarking purpose. Embracing this philosophy, the mainobjective of this paper is not to solve the ATSP up to its optimality,but to provide the first ever approach in the literature for beingtackled with QC devices.Lastly, it is important to highlight that the QUBO formulationadopted in this research for the ATSP is inspired by the canonicalATSP defined by Feld et al. in [14]. We refer interested readers tothat paper for obtaining additional details on its QUBO expression. As mentioned in the introduction, one of the main objectives ofthis papers is to extend the outcomes obtained in our previous
ECCO ’21, July 10–14, 2021, Lille, France E. Osaba et al.
Figure 1: Workflow of QTA employed in this paper for theATSP resolution. work published in [44], where we introduced our hybrid quantum-classical QTA. Accordingly, we do not give deep details of the QTA,encouraging interested researchers to read [44] for delving deeperinto more specific aspects.However, it is important to highlight the key principles thatguided the design of our QTA: i) to reduce the non-profitable callsto the QC device and ii) to have a greater control over the accesses toQC resources . Following these pillars, the QTA indirectly reaches areduction in the involved economical cost, associated with the usageof quantum computational resources. These economical expensesusually constitute obstacles for researchers to contribute to thisfield and community .Consequently, our QTA is divided into three different main stepsfor addressing large partitioning problems: i) the calculation of prob-lem partitions, ii) the solving of the created subinstances throughthe use of D-Wave Advantage_system1.1 and iii) the automatic merg-ing of the independently-faced subproblems and the evaluation ofthe obtained unique global solution. We show the main architec-ture of QTA in Figure 1 . Furthermore, with the aim of properlyunderstanding the main phases of QTA, we briefly describe themost important aspects of these stages. • Initialization procedure : the initialization phase of the QTAis carried out by the above mentioned multiform CoVNS. In few words, CoVNS is a multitasking method composedby as much subpopulations as tasks to solve. Each of thesesubgroups is responsible for the solving of one sole task, ac-commodating the coevolutionary nature by means of period-ical migrations of solutions between subpopulations. On thisregard, and considering that the optimum graph partitiondoes not compulsory entail a good decomposition scheme forthe given optimization problem, we explore three differentclustering solutions using three well-known metrics: Calin-ski Harabasz [8], Davies Boulding [12] and Modularity [37].Therefore, the multiform CoVNS offers the best partitionfound for each of these metrics. Lastly, if any practitionerdecides not to employ any initialization problems, we sug-gest to begin the running of the technique with a randomlygenerated feasible solution. • Optimization module : first, note that the encoding employedfor representing the clustering proposed for each ATSP datasetis the well-known label-based [30]. Thus, the algorithm startsby extracting all partitions (cid:101) G that compose the initial solu-tion. Then, for each partition (cid:102) G 𝑖 , QTA calculates the routeusing the above mentioned D-Wave Advantage_system1.1 device. Once the algorithm calculates the route of a specificsubproblem, it store this route into a
Tabu Dictionary , collect-ing in this way all previously considered clusters and theiroptimized solutions. . In this way, QTA can resort to thesepreviously calculated solutions in subsequent iterations ofthe search process, thus avoiding redundant accesses to theremote D-Wave machine. In other words, the
Tabu Dictio-nary is a strategy conceived to prevent the repeated andnon-profitable calls to the QC cloud service for already cal-culated subgraphs, drastically reducing the amount of QCdevice accesses. We show in Figure 2 an illustration of thisspecific module using the ATSP as example problem. • Merging module : this module follows a greedy strategy seek-ing to compose the complete final solution
𝑇𝑆𝑃 ∗ taking asinput the pool of (cid:157) T SP subsolutions. In few words,
𝑇𝑆𝑃 ∗ is composed by all arcs that comprise each 𝑇𝑆𝑃 𝑖 with theexception of those links needed to open up the closed loops-subroutes. The formation of the final cycle is conductedthrough the creation of 𝐶 bridges, each responsible for link-ing two different partitions in (cid:101) G . Once the merging is per-formed and the 𝑇𝑆𝑃 ∗ is built, QTA measures the total fitnessusing the cost-based ATSP objective function. Then, the cur-rent solution is replaced if the new generated one representsan improvement. Finally, if the maximum amount of D-Wavecalls have been reached, the QTA returns the best found so-lution. Otherwise, the current partition is evolved using an insertion function [45] and re-introduced in the optimizationmodule. As mentioned before, six different ATSP instances have been em-ployed for this experimentation, each belonging to the TSPLIBBenchmark and composed by 17 to 48 nodes. More concretely, cho-sen instances are br17 , ftv33 , ftv35 , ftv38 , p43 and ry48p . Each esults of QTA on the ATSP GECCO ’21, July 10–14, 2021, Lille, France Figure 2: Optimization and Merging modules workflow us-ing as example a 10-node ATSP dataset. instance have been executed 20 times in order to obtain statisti-cally representative results. Furthermore, and due to the stringentlimits on the access to the QC device, we have also made use ofthe local QBSolv alternative offered by D-Wave. This way, we havebeen able to conduct a representative experimentation comprisinginstances up to 43 nodes. Thus, and embracing the same strategyas remarkable papers such as [14, 52], the default setup of QBSolvhas been kept .Additionally, the maximum clusters size for the QTA has beenestablished on 10. Because of the QUBO matrix-based representa-tion requirements for efficiently dealing with ATSP problems andthe 5436 qubits available, this number is the maximum size that thecurrent D-Wave Advantage_system1.1 can assume. 40 calls to QChardware have been defined as termination criterion.With the aim of contributing to the TSP and ATSP community, aswell as to QC research field, we openly offer the QTA code describedin this paper. Thus, the Python implementation of our method,together with the scripts needed for solving ATSP instances, is pub-licly available in github . It is also worth-mentioning that the mainalgorithmic base used for developing our method is also availablein GitHub, openly offered by user mstechly .Last but not least, it is important to bring to the fore that themain objective of our QTA is to obtain promising outcomes while At the time of writing, QBSolv’s current version is 0.3.2. https://github.com/EnekoOsaba/QAT4ATSP https://github.com/BOHRTECHNOLOGY/quantum_tsp overcoming one of the major concern voiced by the QC community.Thus, our motivations with this research are a) to offer practition-ers and researchers a fully configurable method which providesabsolute control to QC remote accesses, and b) to drastically re-duce the number or non-profitable calls to the quantum resources,avoiding possible connection problems, and undesirable waste ofthe committed quantum resource budget.With this in mind, Table 1 shows the results (average/standarddeviation/best) yielded by both QTA and QBSolv. We have alsoadded to this table the difference between the best performingmethod and the runner-up represented by a percentage. Addition-ally, for faithfully measuring the key objective of our QTA in termsof the reduction of QC device accesses, Table 1 also contemplatesthe average calls needed by the QBSolv technique for achievingeach outcome. Table 1: Obtained results (average/standard deviation/best)using QTA and QBSolv. Best results have been highlightedin bold.
QTA QBSolvInstance Avg Std Best Avg Std Best AvAcc br17 ftv33 ftv35 ftv38 p43 ry48p
Several conclusion can be drawn by analyzing the results de-picted in Table 1. First, it can be confirmed that both algorithmsperform similarly. On the one hand, QTA reaches better resultsin 3 out of 6 datasets ( ftv33 , ftv35 and ftv38 ), whereas QBSolveslightly outperforms QTA in two of the considered instances ( p43 and ry48p ). Finally, both solving schemes work equal in b17 in-stance. In any case, performance gaps are certainly non-significant.This situation brings, indeed, a great advantage for our introducedQTA. This is so since it can obtain similar results using remark-ably less QC resources. Specifically, QTA just requires 40 calls toD-Wave’s QC device, while QBSolv shows a considerably increasingdemand of QC accesses, ranging from 184.0 calls for the smallestinstance to 1036.6 for the bigger one. This aspect, together with thecompetitiveness in terms of results, evinces that our method pro-vides a considerable advantage. These results, along with the onesobtained in [44], help us to confirm that QTA is a novel classical-quantum hybrid solver scheme which offers a good performancefor dealing with partitioning problems ensuring a less demandingbehavior in terms of QC resource calls. This paper has been devoted to extending the overall findings drawnin our previously published work [44], focused on the design andimplementation of a quantum-classical hybrid optimization methodcoined as Hybrid Quantum Computing - Tabu Search Algorithm(QTA). To do that, we have adapted our method to the well-knownAsymmetric Traveling Salesman Problem, representing the firstefforts for tackling this problem through the quantum computing
ECCO ’21, July 10–14, 2021, Lille, France E. Osaba et al. paradigm. Thus, for measuring the quality of our method on theATSP, we have employed 5 different well-known TSPLIB instances,made up by 17 to 43 nodes. Experimental outcomes confirm theresults of our previous work, illustrating that our developed QTAis a promising method for solving partitioning problems. As fu-ture work, we have planned the application of the QTA for largerproblems, closer to real-world contexts. Extensive research on themerging module of QTA will be also undertaken by covering theapplication of quantum-based solvers in this stage.
ACKNOWLEDGMENTS
Eneko Osaba, Esther Villar-Rodriguez and Izaskun Oregui wouldlike to thank the Basque Government for its funding support throughthe EMAITEK and ELKARTEK programs.
REFERENCES [1] Akshay Ajagekar, Travis Humble, and Fengqi You. 2020. Quantum computingbased hybrid solution strategies for large-scale discrete-continuous optimizationproblems.
Computers & Chemical Engineering
132 (2020), 106630.[2] FF Al Adeh. 2017. Natural Limitations of Quantum Computing.
Int J Swarm IntelEvol Comput
6, 152 (2017), 2.[3] Anna Arigliano, Gianpaolo Ghiani, Antonio Grieco, Emanuela Guerriero, andIsaac Plana. 2019. Time-dependent asymmetric traveling salesman problem withtime windows: Properties and an exact algorithm.
Discrete Applied Mathematics
261 (2019), 28–39.[4] Christian Ayub, Martine Ceberio, and Vladik Kreinovich. 2020. How quantumcomputing can help with (continuous) optimization. In
Decision Making underConstraints . Springer, 7–14.[5] M Booth, SP Reinhardt, and A Roy. 2017. Partitioning optimization problems forhybrid classcal/quantum execution.[6] Urszula Boryczka and Krzysztof Szwarc. 2019. The Harmony Search algorithmwith additional improvement of harmony memory for Asymmetric TravelingSalesman Problem.
Expert Systems with Applications
122 (2019), 43–53.[7] Sergey Bravyi, Alexander Kliesch, Robert Koenig, and Eugene Tang. 2020. Hybridquantum-classical algorithms for approximate graph coloring. arXiv preprintarXiv:2011.13420 (2020).[8] Tadeusz Caliński and Jerzy Harabasz. 1974. A dendrite method for cluster analysis.
Communications in Statistics-theory and Methods
3, 1 (1974), 1–27.[9] Irina Chiscop, Jelle Nauta, Bert Veerman, and Frank Phillipson. 2020. A Hy-brid Solution Method for the Multi-Service Location Set Covering Problem. In
International Conference on Computational Science . Springer, 531–545.[10] Jill Cirasella, David S Johnson, Lyle A McGeoch, and Weixiong Zhang. 2001.The asymmetric traveling salesman problem: Algorithms, instance generators,and tests. In
Workshop on Algorithm Engineering and Experimentation . Springer,32–59.[11] James Clark, Tristan West, Joseph Zammit, Xiaohu Guo, Luke Mason, and DuncanRussell. 2019. Towards real time multi-robot routing using quantum computingtechnologies. In
Proceedings of the International Conference on High PerformanceComputing in Asia-Pacific Region . 111–119.[12] David L Davies and Donald W Bouldin. 1979. A cluster separation measure.
IEEEtransactions on pattern analysis and machine intelligence arXiv preprint arXiv:1411.4028 (2014).[14] Sebastian Feld, Christoph Roch, Thomas Gabor, Christian Seidel, Florian Neukart,Isabella Galter, Wolfgang Mauerer, and Claudia Linnhoff-Popien. 2019. A hybridsolution method for the capacitated vehicle routing problem using a quantumannealer.
Frontiers in ICT arXiv preprint arXiv:2007.01966 (2020).[16] Samuel Fernández-Lorenzo, Diego Porras, and Juan José García-Ripoll. 2020.Hybrid quantum-classical optimization for financial index tracking. arXiv preprintarXiv:2008.12050 (2020).[17] Daniel Stilck Franca and Raul Garcia-Patron. 2020. Limitations of optimizationalgorithms on noisy quantum devices. arXiv preprint arXiv:2009.05532 (2020).[18] Bernd Freisleben and Peter Merz. 1996. A genetic local search algorithm forsolving symmetric and asymmetric traveling salesman problems. In
Proceedingsof IEEE international conference on evolutionary computation . IEEE, 616–621.[19] Alan M Frieze, Giulia Galbiati, and Francesco Maffioli. 1982. On the worst-caseperformance of some algorithms for the asymmetric traveling salesman problem.
Networks
12, 1 (1982), 23–39. [20] Laura Gentini, Alessandro Cuccoli, Stefano Pirandola, Paola Verrucchi, andLeonardo Banchi. 2019. Noise-assisted variational hybrid quantum-classicaloptimization. arXiv preprint arXiv:1912.06744 (2019).[21] Amir Gharehgozli, Chao Xu, and Wenda Zhang. 2020. High multiplicity asym-metric traveling salesman problem with feedback vertex set and its application tostorage/retrieval system.
European Journal of Operational Research
Nature News arXiv preprint arXiv:1811.11538 (2018).[24] Lov K Grover. 1997. Quantum mechanics helps in searching for a needle in ahaystack.
Physical review letters
79, 2 (1997), 325.[25] Gian Giacomo Guerreschi and Mikhail Smelyanskiy. 2017. Practical optimizationfor hybrid quantum-classical algorithms. arXiv preprint arXiv:1701.01450 (2017).[26] Abhishek Gupta, Yew-Soon Ong, and Liang Feng. 2017. Insights on transfer opti-mization: Because experience is the best teacher.
IEEE Transactions on EmergingTopics in Computational Intelligence
2, 1 (2017), 51–64.[27] Laszlo Gyongyosi. 2018. Quantum Circuit Designs for Gate-Model QuantumComputer Architectures. arXiv preprint arXiv:1803.02460 (2018).[28] Philipp Hauke, Helmut G Katzgraber, Wolfgang Lechner, Hidetoshi Nishimori,and William D Oliver. 2020. Perspectives of quantum annealing: Methods andimplementations.
Reports on Progress in Physics
83, 5 (2020), 054401.[29] Tony Hey. 1999. Quantum computing: an introduction.
Computing & ControlEngineering Journal
10, 3 (1999), 105–112.[30] Eduardo Raul Hruschka, Ricardo JGB Campello, Alex A Freitas, et al. 2009. Asurvey of evolutionary algorithms for clustering.
IEEE Transactions on Systems,Man, and Cybernetics, Part C (Applications and Reviews)
39, 2 (2009), 133–155.[31] Hasham Hussain, Muhammad Bin Javaid, Faisal Shah Khan, Archismita Dalal,and Aeysha Khalique. 2020. Optimal control of traffic signals using quantumannealing.
Quantum Information Processing
19, 9 (2020), 1–18.[32] Jun Li, Xiaodong Yang, Xinhua Peng, and Chang-Pu Sun. 2017. Hybrid quantum-classical approach to quantum optimal control.
Physical review letters
Frontiers inPhysics arXiv preprint arXiv:2011.14268 (2020).[35] Giacomo Nannicini. 2019. Performance of hybrid quantum-classical variationalheuristics for combinatorial optimization.
Physical Review E
99, 1 (2019), 013304.[36] Christian FA Negre, Hayato Ushijima-Mwesigwa, and Susan M Mniszewski. 2020.Detecting multiple communities using quantum annealing on the D-Wave system.
Plos one
15, 2 (2020), e0227538.[37] Mark EJ Newman and Michelle Girvan. 2004. Finding and evaluating communitystructure in networks.
Physical review E
69, 2 (2004), 026113.[38] Michael A Nielsen and Isaac Chuang. 2002. Quantum computation and quantuminformation.[39] Shuntaro Okada, Masayuki Ohzeki, Masayoshi Terabe, and Shinichiro Taguchi.2019. Improving solutions by embedding larger subproblems in a d-wave quantumannealer.
Scientific reports
9, 1 (2019), 1–10.[40] Temel Öncan, İ Kuban Altınel, and Gilbert Laporte. 2009. A comparative analysisof several asymmetric traveling salesman problem formulations.
Computers &Operations Research
36, 3 (2009), 637–654.[41] Eneko Osaba, Javier Del Ser, Andres Iglesias, Miren Nekane Bilbao, Iztok Fister,and Akemi Galvez. 2018. Solving the open-path asymmetric green travelingsalesman problem in a realistic urban environment. In
International Symposiumon Intelligent and Distributed Computing . Springer, 181–191.[42] Eneko Osaba, Javier Del Ser, Ali Sadollah, Miren Nekane Bilbao, and DavidCamacho. 2018. A discrete water cycle algorithm for solving the symmetricand asymmetric traveling salesman problem.
Applied Soft Computing
71 (2018),277–290.[43] Eneko Osaba, Esther Villar-Rodriguez, and Javier Del Ser. 2020. A CoevolutionaryVariable Neighborhood Search Algorithm for Discrete Multitasking (CoVNS):Application to Community Detection over Graphs. In
IEEE Symposium Series onComputational Intelligence (IEEE SSCI) . ToBePublished.[44] Eneko Osaba, Esther Villar-Rodriguez, Izaskun Oregi, and Aitor Moreno-Fernandez-de Leceta. 2020. Hybrid Quantum Computing–Tabu Search Algorithmfor Partitioning Problems: preliminary study on the Traveling Salesman Problem. arXiv preprint arXiv:2012.04984 (2020).[45] Eneko Osaba, Xin-She Yang, Fernando Diaz, Pedro Lopez-Garcia, and Roberto Car-balledo. 2016. An improved discrete bat algorithm for symmetric and asymmetrictraveling salesman problems.
Engineering Applications of Artificial Intelligence
ORSAjournal on computing
3, 4 (1991), 376–384. esults of QTA on the ATSP GECCO ’21, July 10–14, 2021, Lille, France [47] Masahiko Saito, Paolo Calafiura, Heather Gray, Wim Lavrijsen, Lucy Linder,Yasuyuki Okumura, Ryu Sawada, Alex Smith, Junichi Tanaka, and Koji Terashi.2020. Quantum annealing algorithms for track pattern recognition. In
EPJ Webof Conferences , Vol. 245. EDP Sciences, 10006.[48] Peter W Shor. 1999. Polynomial-time algorithms for prime factorization anddiscrete logarithms on a quantum computer.
SIAM review
41, 2 (1999), 303–332.[49] Andrew Steane. 1998. Quantum computing.
Reports on Progress in Physics
61, 2(1998), 117.[50] Ola Svensson, Jakub Tarnawski, and László A Végh. 2020. A constant-factorapproximation algorithm for the asymmetric traveling salesman problem.
Journalof the ACM (JACM)
67, 6 (2020), 1–53.[51] Ryan Sweke, Frederik Wilde, Johannes Jakob Meyer, Maria Schuld, Paul KFährmann, Barthélémy Meynard-Piganeau, and Jens Eisert. 2020. Stochasticgradient descent for hybrid quantum-classical optimization.
Quantum arXiv preprint arXiv:2009.01373 (2020).[53] Yuanhao Wang, Ying Li, Zhang-qi Yin, and Bei Zeng. 2018. 16-qubit IBM universalquantum computer can be fully entangled. npj Quantum Information
4, 1 (2018),1–6.[54] Richard H Warren. 2017. Small traveling salesman problems.
Journal of Advancesin Applied Mathematics
2, 2 (2017).[55] Richard H Warren. 2020. Solving the traveling salesman problem on a quantumannealer.
SN Applied Sciences
2, 1 (2020), 75.[56] Zhangqi Yin and Zhaohui Wei. 2017. Why quantum adiabatic computation andD-Wave computers are so attractive?