A Novel Quantum Algorithm for Ant Colony Optimization
Mrityunjay Ghosh, Nivedita Dey, Debdeep Mitra, Amlan Chakrabarti
AA NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 1
A Novel Quantum Algorithm forAnt Colony Optimization
Mrityunjay Ghosh,
QRDLab India, University of Calcutta,
Nivedita Dey,
QRDLab India, University ofCalcutta,
Debdeep Mitra,
University of Calcutta and Amlan Chakrabarti,
University of Calcutta
Abstract —Ant colony optimization is one of the potential solutions to tackle intractable NP-Hard discrete combinatorial optimizationproblems. The metaphor of ant colony can be thought of as the evolution of the best path from a given graph as a globally optimalsolution, which is unaffected by earlier local convergence to achieve improved optimization efficiency. Earlier Quantum Ant ColonyOptimization research work was primarily based on Quantum-inspired Evolutionary Algorithms, which deals with customizing andimproving the quantum rotation gate through upgraded formation of the lookup table of rotation angle. Instead of relying on evolutionaryalgorithms, we have proposed a discrete-time quantum algorithm based on adaptive quantum circuit for pheromone updation. Thealgorithm encodes all possible paths in the exhaustive search space as input to the ORACLE. Iterative model of exploration andexploitation of all possible paths by quantum ants results in global optimal path convergence through probabilistic measurement ofselected path. Our novel approach attempts to accelerate the search space exploitation in a significant manner to obtain the bestoptimal path as a solution through quantum parallelization achieving polynomial time speed-up over its classical counter part.
Index Terms —Quantum Computing, Quantum Algorithm, Ant Colony Optimization, QACO, Quantum Ant, Quantum Circuit Synthesis. (cid:70)
NTRODUCTION
Neo Darwinism theory indicates the complex, unexpected,emergent behavior of biome communities found in diversenature. Any complex system like nature or distributedcomputing system exhibits uncertain behavior for betterresilience and adaptation. Swarm intelligence (SI) via arti-ficial synapses is a conglomeration of interdependent sub-systems where collective behavior is irreducible to individ-ual behavior as predicting evolution of a complex systemincurs irreducibly large computation. Swarm engineeringinvolves designing of self-organizing systems working inlarge numbers, at small scales with underlying swarmstrategies like ant colonies, bird flocks, fish shoals etc. Artifi-cial Swarm Intelligence (ASI) helps in amplification of groupintelligence of a formed network to enable mere accurateforecasts, insights, evaluations and assessments. [27]An ant colony is a natural system which as a whole,is capable of engaging in complex behaviors like buildingnests, foraging for food, raising aphid livestock, wagingwar with other colonies etc. The concept of Ant ColonyOptimization (ACO) was first proposed by an Italian scholarM.Dorigo in 1991, where foraging behavior of real worldant colonies is mapped as a meta-heuristic for solving dis-crete combinatorial optimization problems. [1] [2] Real antsrelease a chemical on their path from nest to food calledpheromone, which is used by following ants so as to find • M. Ghosh is with A. K. Choudhury School of Information Technology,University of Calcutta and Head of Quantum Computing of QRDLab,Kolkata.E-mail: [email protected] • N Dey is with QRD Lab, Kolkata and University of Calcutta. • D Mitra is with University of Calcutta. • A Chakrabarti is with A. K. Choudhury School of Information Technology,University of Calcutta.Manuscript received 1 March, 2020; revised October 15, 2020. the shortest path to food source via the pheromone trail.This metaphor in artificial ant colonies is able to solve opti-mization problems with a diverse range of applications likeTravelling Salesman Problem (TSP), [21] Vehicle Routingproblem [10], Assignment problem, [18] Job-shop problem,[25] 0-1 Knapsack [19] and many more. ACO has provenitself to be a promising one by exhibiting tremendousgrowth in solving discrete optimization problems. [20] ButACO, unless combined with evolutionary algorithm (EA),remained lacked behind in terms of better optimization.Classical evolutionary algorithms are algorithmic upgrada-tion over traditional, calculus-based, enumerative optimiza-tion strategies due to robustness, scalability and domain-specific applicability which is free from any heuristic con-straint. Stochastic search optimization strategies of EAs arebased upon randomized search heuristic that successivelygenerate potential candidate solutions depending on localbest solution obtained so far, which in turn, goes through asampling procedure called mutation. [28] [29]Quantum computation follows quantum-mechanicalphenomena like principles of superposition, entanglement,and quantum tunnelling to bring a paradigm shift in thedomain of computing by achieving substantial speed-upover its classical counterpart. It gained computational sig-nificance after research works of Peter Shor on Large IntegerFactorization and work of L.K.Grover on Quantum Searchalgorithm for unstructured database, opening up new mile-stones and inclusion of quantum theory into algorithms.[3] In maintaining a trade-off between exploration andexploitation, quantum based EAs outperform conventionalevolutionary algorithms. In 1990s, formulation and studyof quantum analogous of randomized search heuristics hadaccelerated the search for improved offspring includingQuantum Local Search (QLS) and Quantum EvolutionaryAlgorithms (QEAs). [3] [30] [7] a r X i v : . [ c s . ET ] O c t NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 2
The idea invented so far regarding Quantum Ant ColonyOptimization (QACO) is based on QEA, where Q-bit andQuantum Rotation strategy are used to represent and up-date the pheromone respectively in discrete binary combina-torial optimization domain. In this paper, we have presenteda novel quantum ant colony optimization algorithm basedon hybrid implementation of Grover's amplitude amplifi-cation technique [3] and time driven quantum evolutionto encode the possible paths by updating the distributionof pheromone in terms of probability density function.The amount of pheromone deposited on the shortest pathwill eventually lead to global convergence of the optimiza-tion problem in the solution space. The implementationand simulation results exhibit better efficiency, improvedcomputation speed and enhanced optimization capabilityof ACO making it a robust solution. To the best of ourknowledge, the QACO proposed in this paper is the firstattempt towards a full scale quantum technique where allthe basic operations of Ant Colony Optimization like pathexploration, pheromone deposition and pheromone evapo-ration are performed in quantum registers.The rest of this paper starts with literature review sectionas Section 2. Section 3 covers the basics of combinatorialoptimization (CO) problems and classical ACO to solveCO problems along with its variations, section 4 illustratesquantum gates and circuits as prerequisites in our proposedMNDAS (Mrityunjay-Nivedita-Debdeep-Amlan-Subhansu)algorithm. Section 5 sheds light on conventional quantum-inspired evolutionary technique for ACO, section 6 eluci-dates our novel QACO algorithm along with its probleminitialization, path exploration and pheromone updationmodules. Section 7 provides a mapping between the pro-posed algorithm and QACO problem in terms of realizingthe approach on an ant colony example. Section 8 presentssimulation result showing global path convergence and timecomplexity analysis. The last section illustrates conclusionand scope for future research growth on related areas. [29]
ITERATURE R EVIEW
ACO has a diverse range of applications and thus has al-ways been a fundamental topic of theoretical interest. Sinceour work aims to propose a novel ’quantum algorithm’for ACO, we will mention fewer insights into workingprinciples of classical ACO through their evolution andhighlight the notion of quantum ant colony optimizationbased Quantum-inspired Evolutionary Algorithms (QEAs).During the course of evolution starting from classicalACO metaheuristic by M. Dorigo, the first rigorous theo-retical investigations on ACO was proposed by Neumannand Witt in 2006, where authors had presented -ANT al-gorithm. [14] The algorithm operates by constructing a newsolution and performing pheromone updation only if thecurrent solution is better than the best solution obtained sofar. Pheromone updation in -ANT algorithm is controlledby evaporation factor ( ρ ) [ < ρ < ]. The larger valueof ρ is associated with increasing the impact factor of thecurrent solution over the previous best obtained solution.Later, investigations made on -ANT ACO have shown theperformance degradation of the algorithm for a very small value chosen for evaporation factor, as the expected timeto achieve optimal solution is exponential. [14] [12] Furtherwork on classical ACO was made by reinforcing the bestsolution obtained so far in each iteration using best-so-farupdate strategy. This concept was first coined in MAX-MIN Ant System (MMAS) algorithm. [12] [13] [15] Anotheralgorithmic advancement was made in this direction byreinforcing the best solution created in the current iteration,known as iteration-best update which is found to work welleven with a small value of evaporation factor. [16]Transformation of a discrete optimization problem asa ‘best path’ problem is the intuition behind originationof quantum ant colony optimization, which is inspiredby theories of Quantum-inspired Evolutionary Algorithms(QEAs). Unlike classical ACO, QEA based ACO is formu-lated with the help of Q-bit and quantum rotation gate.As pheromone updation strategy is dissimilar to updationstrategy of QEA, Ling Wang et. al first proposed a rotationangle updating strategy to update the pheromone trails overthe existing pre-determined updating strategy in 2007. [30]Their proposed work significantly taken into considerationthe exploitation probability, as optimizing exploitation prob-ability provides a trade-off between earlier convergence ofACO and effective escape from local optima. In 2008, LingWang et. al. extended their algorithm to solve fault detectionin chemical production process. [7] The authors had com-bined Support Vector Machine (SVM) with their proposedQACO to select fault features. In 2010, Panchi Li et. al. hadproposed a continuous quantum ant colony optimizationalgorithm where they have made each ant to be encodedwith a group of qubits to represent its own position. [6] Thealgorithm begins with selecting the local best path based onpheromone information as a heuristic followed by updatingeach of the ant's own qubits with the help of quantumrotation gate. In order to enable mutation and improvethe diversity of positions, some qubits have undergonemutations by quantum non-gates. The idea of continuous-time evolution was implemented by adding fitness functionvalue of the current ant position of the pheromone to updatethe heuristic information. The optimum position thus, isable to hold the greater fitness function value and fitnessfunction gradient value ensuring accelerated, guaranteedconvergence. Some other work in the domain of QACO wasthat of evacuation path optimization algorithm proposed byMin Liu et. al. in 2016. [8] In this paper, the advantage liesupon scalability of the method as it is suitable for multiplesource nodes to multiple destination nodes instead of asingle path between two locations. LASSICAL A NT C OLONY O PTIMIZATION FORSOLVING COMBINATORIAL OPTIMIZATION PROB - LEMS
The objective of a combinatorial optimization problem,associated with a set of problem instances, is to maxi-mize or minimize several parameters. [24] The solutionsof these intractable optimization problems incur exhaustiveor brute-force search, which is computationally hard. [20]Ant Colony Optimization (ACO) is a meta-heuristic processto solve computationally hard optimization problems, inwhich the idea is to allocate the computational resources
NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 3 to a set of relatively simple agents called artificial ants.Since, in our paper we have focused on proposing quantumalgorithm for simple ACO, where path searching behaviorof ants and pheromone updation rules are discussed, wewill have a brief introduction to simple ACO meta-heuristic.Simple ACO functions in two operating modes: forward(from nest towards the food) and backward (from food backto the nest). Forward ants build a solution by probabilis-tically choosing the next node to move to among those inthe adjacent positions with respect to current node. Thisprobabilistic choice is biased by pheromone trails previouslydeposited on the paths by other ants. Forward ants donot deposit any pheromone, which when associated withdeterministic backward moves, helps to eliminate loop for-mation.
Each ant builds a solution to the problem, where step-by-step decision making of each ant relies on the localinformation stored on the node itself or on the outgoing arcs.Search process begins with assignment of constant amountof pheromone ( τ ij = 1) to all arcs. [1] [2] An ant k locatingon node i computes the probability to choose j as the nextnode using pheromone trail τ ij as follows: p ijk = τ αij (cid:80) j ∈ Nki τ αij , if j ∈ N ki , if j / ∈ N ki (1) N ki = Neighbourhood of ant k when located on node i (excluding predecessor of node i )In ACO meta-heuristic, a problem-specific heuristic istaken into account for decision making. [4] [5] f ( j ) = arg { maxd = f easible k ( t )[ τ id ( t ) α .η βid ] } when r ≤ r j (cid:48) when r > r (2) f ( j ) = Constraint for transition function to move fromnode i to jτ id ( t ) = Pheromone trail at time tη id = problem specific heuristic information α = impact of heuristic information r = random number with uniform distribution in [0 , r = pre-specified parameter ranging from to , inclusive f easible k ( t ) = set of feasible nodes excluding alreadyvisited (predecessor) nodes by k -th ant before visiting node i , to prevent loop formation j (cid:48) = target point selected according to the followingprobability distribution. P kij ( t ) = [ τ ij ( t )] α .η βij (cid:80) d ∈ feasiblek ( t ) [ τ ij ( t )] α .η βij , if j ∈ f easible k ( t )0 , otherwise (3) Retracing step by step by the same path in backward modefirst begins with a scanning process for formed loop elimina-tion followed by deposition of ∆ τ k amount of pheromoneby each ant. An ant k traversing in backward mode throughthe arc ( i, j ) , will update the pheromone value as follows: τ (cid:48) ij = τ ij + ∆ τ k (4)This pheromone updation step ensures chances of forthcom-ing ants to trace the same path. Pheromone trail evaporation is an exploration mechanismto avoid quick convergence of all ants towards a local bestsolution, equivalently a suboptimal path. Process of decreas-ing the intensities of pheromone trails favours exploration ofdifferent paths during whole search space. The evaporationof pheromone trails can be expressed as follows: τ ∗ ij ← (1 − ρ ) .τ ij ∀ ( i, j ) ∈ E ( G ) (5) E ( G ) represents the set of all arcs of the graph, ρ ∈ (0 , is a parameter, τ ij ∗ is the updated pheromone level afterevaporation. [1] [2] [9] The pheromone evaporation process is interleaved with theprocess of pheromone deposition with ∆ τ k amount to allarcs. But at times, activating a local optimization procedureto implement centralized actions, is an important factor todecide utility of depositing additional pheromone to biasthe search process from a non-local perspective. [2] [1] [8]Evolution in simple ACO has taken place with differencein updation policy of pheromone deposition and evapora-tion. Ant-System(AS) update rule of simple ACO is replacedby Iteration-Best (IB) update rule in practice. IB update ruleintroduces emergence into a system while taking biased-ness towards good solutions into consideration obtainedthrough previous iterations. Another variant is Best-so-far(BS) update rule, which exhibits biasedness towards thebest solution available so far. [22] [23] Both of these policiessuffer from earlier convergence as the set of all sequencesof solution components that might be constructed by ACOalgorithm to produce feasible solutions are updated withthe set of all sequences of solution components obtainedthrough multiple solutions of previous iterations in caseof IB update and best solution obtained so far in caseof BS update. In order to avoid premature convergence,advanced ACO algorithms like Ant Colony System (ACS)and MAX MIN Ant System (MNAS) are used. UANTUM G ATES AND C IRCUITS
The smallest unit of information in quantum is representedas quantum bits or qubits. A qubit is thought to exist asa superposition of two pure states and . A state of asuperposed qubit can be expressed as follows: | ψ (cid:105) = α | (cid:105) + β | (cid:105) (6)Here, α and β are complex numbers with | α | and | β | representing probabilistic amplitudes of the superposed NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 4 qubit to be in | (cid:105) and | (cid:105) respectively. [11] Unlike classicalcomputing with irreversible gate logic, quantum computingperforms unitary evolution of quantum states and hencerelies on reversible logic of quantum gates. [11] A singlequbit gate is a kind of operator that acts on only onequbit at a time. These operators are described by × unitary matrices, where unitary matrix U has a property U † .U = U.U † = I . [11] [3] Such an operator, calledHadamard operator or Hadamard gate maps the basic state | (cid:105) to | (cid:105) + | (cid:105)√ and | (cid:105) to | (cid:105)−| (cid:105)√ . Hadamard gate is an 1-qubitversion of QFT (Quantum Fourier Transform). [11] H = 1 √ (cid:18) − (cid:19) ,H | (cid:105) = 1 √ (cid:18) − (cid:19) (cid:18) (cid:19) = 1 √ (cid:18) (cid:19) = 1 √ | (cid:105) + | (cid:105) ) (7)A quantum gate can act on N qubits simultaneously. Similarto the case of single qubit, the probability must be conservedwhen operating in multiple dimensions, and the operatorsare hence unitary. The simplest example is the well-knowntwo qubit Controlled N OT ( CN OT ) gate or Fenyman gate.Matrices are defined in the basis spanned by the two qubitstate vectors | (cid:105) ≡ [1000] T , | (cid:105) ≡ [0100] T , | (cid:105) ≡ [0010] T , | (cid:105) ≡ [0001] T , where the first qubit is the control qubit andthe second qubit is the target qubit. The CN OT gate flipsthe state of the target qubit conditioned on the control qubitbeing in state | (cid:105) . The action of the CN OT gate is given as | x (cid:105) | y (cid:105) → | x (cid:105) | y ⊕ x (cid:105) . [11] CN OT = Controlled-
N OT gate can be extended to Controlled-Controlled-
N OT ( C N OT ) gate, alternatively known asToffoli gate. It is universal reversible logic acting as a quan-tum operator with three input bits. If first two bits are set to1, it inverts the third bit. [11]Controlled phase (
CP HASE ) gate applies a Z -gate tothe target qubit conditioned on the control qubit being instate | (cid:105) . Z gate performs a π - rotation around the Z -axis. Z -gate is also referred as phase flip. [11] CP HASE = e iθ A Controlled-Controlled
N OT or ( C N OT ) gate gives theAND of two control qubits C and C . Chaining more thantwo Toffoli's together through AND operation among multi-ple control qubits c , c , ..., c n ( c .c .c ...c n ) and introducingfew ancilla qubits to store intermediate results, C n N OT gate can be implemented. In figure 2, a 4 qubit Toffoli( C N OT ) as MCT has been implemented with four controlqubits | c (cid:105) , | c (cid:105) , | c (cid:105) , | c (cid:105) and three ancilla qubits and onetarget qubit which will flip only when c .c .c .c = 1 . Afterapplying the final C n ( X ) , a reversible model of compu-tation has been implemented (compute-copy-uncompute)to clean up intermediate work qubits by undoing theircomputation and resulting ancilla qubits to | (cid:105) state. [17] Fig. 1: Basic quantum gates. (a) Controlled N OT gate (b)Controlled
P HASE gate (c) Controlled-Controlled
N OT gateFig. 2: Decomposition of a 4 qubit multi controlled tofolligate in
CN OT and C N OT gatesMeasuring a quantum state causes disturbance in quan-tum mechanical system by resulting in degeneration ofsuperposed quantum state and its convergence into classicalstate. A collection of measurement operators { M K } where K is a given measurement outcome, is not necessarilyunitary. Operators { M K } acting on Hilbert space of thegiven state satisfy completeness equation, (cid:80) K M + K M K = I .For a quantum state φ , the probability of obtaining themeasurement outcome m is P ( m ) = (cid:104) φ | M + m M m | φ (cid:105) andthe resulting quantum state is ( (cid:104) φ | M + m M m | φ (cid:105) ) − ∗ M m | φ (cid:105) .Completeness equation encodes the fact that measurementprobabilities over all the outcomes sum to unity.A quantum ORACLE is black box representation of aquantum circuit which acts as subroutine of a quantumalgorithm. Input to the ORACLE is a boolean function f ,such that f : { , } n → { , } n . Function f is said to bequeried via an ORACLE O f where, | x (cid:105) | q (cid:105) → | x (cid:105) | q ⊕ f ( x ) (cid:105) , (cid:96) x ∈ { , } n and q ∈ { , } m . The above mapping can beimplemented by an UNITARY circuit U f of the form: U f = (cid:88) x ∈{ , } n (cid:88) q ∈{ , } m | x (cid:105) (cid:104) x | X | q ⊕ f ( x ) (cid:105) (cid:104) q | (8)Effect of ORACLE needs to be determined in all basis states. NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 5
XISTING NOTION OF
QACO
AND THEIR LACKOF UNIVERSALITY
In ACO, complexity of exploring the possible paths fromfood source to nest and exploiting the whole search spaceincrease significantly with the increase in number of pathsand number of iterations to achieve better performance dueto sequential mode of execution of the algorithms. Quantumparallelization and quantum state entanglement can sub-stantially reduce the algorithmic complexity involved in ex-ploration of large solution space of optimization problems.A fault-tolerant quantum computer with error correctedqubits can encode number of paths simultaneously,which takes number of bits in its classical counterpart. Quantum-inspired Evolutionary Algorithms (QEA) fed byprobabilistic mechanism of quantum computation, havebeen applied in the existing research of QACO problems.The smallest information unit in QEA is Q-bit, defined as [ α, β ] T . [7] [30] α and β represent complex numbers to sat-isfy the normalization condition | α | + | β | = 1 . By a processof probabilistic observation, each Q-bit can be rendered intoone binary bit. A Q-bit representation, employing a Q-bit todescribe a probabilistic linear superposition can be extendedto a multi Q-bit system as shown in equation 9. Q = (cid:20) α β | α β | · · · | α m β m (cid:21) (9) Q = (cid:34) −√ √ | √ √ | −√ − (cid:35) (10)The above example in equation 10 represents a linearprobabilistic superposition of = 8 states as | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) and | (cid:105) , where its super-posed state can be described as: | ψ (cid:105) = √ | (cid:105) + √ | (cid:105) − √ | (cid:105) + √ | (cid:105) + √ | (cid:105) − √ | (cid:105) + √ | (cid:105) − √ | (cid:105) (11)A conventional binary solution is constructed through Q-bitobservation, where for a bit r i of a binary individual r , achosen value of random number η ∈ [0 , is compared with α i of Q-bit individual P . [7] The binary encoding process isas follows: (cid:40) r i = 0 , if | α i | > ηr i = 1 , if | α i | ≤ η (12)Generation step is followed by fitness evaluation step andits outcome is then processed through a quantum rotationgate R ( θ ) operating as follows: (cid:20) α i β i (cid:21) (cid:48) = R ( θ i ) (cid:20) α i β i (cid:21) = (cid:20) cosθ i − sinθ i sinθ i cosθ i (cid:21) (cid:20) α i β i (cid:21) (13)In order to converge to fitter states, quantum rotation gateis updated. [30] Rotating angle θ i has a huge significancein performance of Quantum-inspired EAs, where θ i can bedefined as follows: θ i = sign ( α i , β i )∆ θ i (14) sign ( α i , β i ) represents sign of rotating angle θ i to determinethe direction. The value of sign ( α i , β i ) and ∆ θ i are decidedby looking up into a table in order to compare performancebetween the solution provided by current individual andthe best solution obtained so far. [7] [30] The main disadvantage appearing during the process ofquantum state rotation lies in the dependency of using alookup table for fixing the quantum rotation angle. Fixedrotating angle causes negative impact on search speed foran adaptive network, hence can limit the universality ofsearch process by a significant extent. Enhancement of lo-cal searching ability and finding escape from local optimamight be of great challenge in an ineffective rotating an-gle updation strategy. Moreover, QEA is not a quantumalgorithm, rather it is an evolutionary algorithm inspiredby quantum. In the next three sections we have proposed,illustrated and analyzed a novel quantum algorithm forant colony optimization which is solely based on iterationdriven path selection and convergence to the path havingmaximum pheromone.
LGORITHM FOR Q UANTUM A NT C OLONY O PTIMIZATION
Our proposed Ant Colony Optimization algorithm
M N DAS is the quantum version of the very basic antcolony system, where the problem space comprises of sev-eral number of parallel paths existing between food sourceand Ant colony. We assume the number of ants startingfrom food source to ant colony in a given period of timeis evenly distributed with respect to time. Procedure beginswith quantum state preparation to encode all possible pathsencountered by ants while traversing from food sourceto colony. The encoded paths will undergo uniform su-perposition in order to be initially selected by the quan-tum ants with equal probability. Additionally,
M N DAS algorithm presumes that there is no pheromone alreadydeposited in any path prior to execution. An ORACLEfunction has been introduced for selecting paths and up-dating pheromone through multiple iterations. Generally,pheromone deposition and pheromone evaporation willtake place for selected and unselected paths respectivelyin each iteration. Pheromone updation is restricted for apath where convergence criterion has already been met.The ‘best-path’ obtained through a sufficient number ofiterations will automatically contain maximum amount ofdeposited pheromone. Once the solution is reached, ourprocedure will undergo a phase shift and an amplitudeamplification to identify the path from the initial superposi-tion of paths which further will be measured in classicalregisters. Figure 3 shows a detailed control flow of theexecution of
M N DAS algorithm.Procedure
M N DAS () begins with quantum andclassical register initialization for ant colony optimiza-tion problem. It encodes all the paths and pheromonein quantum registers through init Ant () procedure. Q [ p ] , Q [ p ] , ..., Q [ p x − ] are the x qubits for all encodedpaths to colony. A pheromone box with d number of qubits, Q [ ph ] , Q [ ph ] , ..., Q [ ph d − ] is introduced in init Ant () to NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 6
Fig. 3: Execution flow of
M N DAS algorithmkeep track of pheromone density distribution of each pathduring pheromone updation. Three additional qubits havebeen initialized with two ancillas Q [ a ] and Q [ a ] repre-senting temporary qureg contents and Q [ a target ] for targetqubit. init Ant () also sets number of iteration for conver-gence ( K ), total number of paths ( n ) and all the path weightsfrom food source to colony ( W ). A total of ( x + d + 3) entangled qubits are initialized with | (cid:105) of which first x qubits undergo quantum superposition with the help of x single qubit Hadamard gates.The initialization step along with problem encoding isfollowed by an iterative ORACLE . The
ORACLE consistsof procedure ant Execute () , which performs path selec-tion by picking up the indices of currently explored pathswith the help of M CT gates and update P heromone () to update the pheromone box. In order to encode a to-tal of paths with x = 4 and d = 4 to implementcorresponding ant Execute () for path index i (cid:54) = 11 ... , N OT gates ( X ) are used in respective positions to enable M CT ( C x N OT ) gate as shown in figure 4. Implementationphase of ant Execute () necessitates the decomposition of M CT gates into
CN OT and
T OF F OLI ( C N OT ) asshown in figure 2. In each iteration performed in procedure ant Execute () , ancilla qubits are raised for all the selectedpaths to be identified during update P heromone () . Un-compute task has been performed on path encoding qubits Algorithm 1: init Ant () Result:
Initialisation and parameter setting (cid:46)
Setting classical parameters K ← Constant to denote number of iterations; n ← Number of paths; W [ n ] ← Path weights; x ← (cid:100) log n (cid:101) Number of qubits for path encoding; Q [ p ..p x − ] ← Respective qubits for path encoding; Q [ a ] , Q [ a ] , Q [ a target ] ← Ancilla qubits; d ← Number of qubits to encode pheromonedistribution; Q [ ph ..ph d − ] ← Respective qubits to encodepheromone distribution; C [0 ..x − ← Number of classical registers formeasurement; (cid:46)
Initializing qubitsSet qureg Q [ p ..p x − , a , a , ph ..ph d − , a target ] as | (cid:105) ; (cid:46) Quantum superposition of x number of encodedpaths’ qubits. H ( Q [ p ..p x − ]) ; Algorithm 2: ant Execute ( t ) Result:
Possible path exploration by Quantum Ants while i in (0 , n − do (cid:46) Selecting paths corresponding to the iteration if t % W [ i ] == 0 thenwhile l in ( p , p x − ) doif i %2 == 0 then N OT ( Q [ l ]) ; endend (cid:46) Multi Controlled Toffoli implementation toraise ancilla for selected paths C x N OT ( Q [ p ..p x − ] , Q [ a ]) ; C x N OT ( Q [ p ..p x − ] , Q [ a ]) ; (cid:46) Reversible operation to achieve initial pathencoding while l in ( p , p x − ) doif i %2 == 0 then N OT ( Q [ l ]) ; endendendend N OT ( Q [ a ]) ; update P heromone () ; (cid:46) Resetting ancilla after pheromone box updation foreach path
RESET ( Q [ a ]) ; RESET ( Q [ a ]) ; NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 7 to get back their initial setting and moreover, the ancillaqubits are reset at the end of each iteration of algorithm 2.
Algorithm 3: update P heromone () Result:
Updating pheromone density based onselected paths (cid:46)
Avoiding pheromone deposition for the pathwhich has pheromone density as 111 ... 11. i.e.pheromone box for that path is full. C d N OT ( Q [ ph ..ph d − ] , Q [ a ]) ; pheromone Deposition () ; C d N OT ( Q [ ph ..ph d − ] , Q [ a ]) ; N OT ( Q [ ph ..ph d − )] ; (cid:46) Avoiding pheromone evaporation for the pathswhich has pheromone density as 000 ... 00. i.e.pheromone box for those paths are empty. C d N OT ( Q [ ph ..ph d − ] , Q [ a ]) ; N OT ( Q [ ph ..ph d − ]) ; pheromone Evaporation () ; Algorithm 4: pheromone Deposition () Result:
Increase in pheromone density of selectedpath by unitset r ← ph d − ; while m in ( r, ph ) do C N OT ( Q [ a ] , Q [ m ] , Q [ m + 1]) ; end CN OT ( Q [ a ] , Q [ m ]) ; Algorithm 5: pheromone Evaporation () Result:
Decrease in pheromone density for the pathsby unit excluding i ) Selected path , ii ) P aths having pheromone density of ... and iii ) P aths having pheromone density of ... set r ← ph d − ; CN OT ( Q [ a ] , Q [ ph ]) ; while m in ( ph , r ) do C N OT ( Q [ a ] , Q [ m ] , Q [ m + 1]) ; end Pheromone distribution among all paths is illustratedin the procedure update P heromone () . Figure 5 showsquantum ants perform pheromone Deposition () on theselected paths chosen earlier by procedure ant Execute () and pheromone Evaporation () for the unselected paths.The shorter paths are supposed to converge earlier in com-parison to comparatively longer paths through iterationsleading to global path convergence. Number of qubits cho-sen for encoding the pheromone box d causes variance inconvergence time of the ant colony optimization problem.For a qubit and a qubit pheromone box, the step-by-step global path convergence is described in figure 7 asexample. Number of qubits chosen to encode pheromonebox affects the overall performance of ACO. Often pre-mature convergence causes lack of adaptiveness into the Algorithm 6:
M N DAS () Result:
Global path convergence of Ant Colony (cid:46)
Qubits initialization init Ant () ; while t in (1 , K ) do (cid:46) Iteration for path selection and pheromoneupdation ant Execute ( t ) ; end (cid:46) Selecting best path C d N OT ( Q [ ph ..ph d − ] , Q [ a target ]) ; (cid:46) Phase shifting of target for best obtained path
CP HASE ( Q [ a target ] , Q [ P x − ] , π ) ; (cid:46) Quantum Amplitude Amplification
QAA ( Q [ p ..p x − ]) ; (cid:46) Measuring global path convergence for ant colonymeasure Q [ p ..p x − ] to C [0 ..x − ;system in case of link failures or path barriers. The realant colony behavior through natural synergy and groupintelligence allows the ants to choose the second optimalpath in case of any obstruction or unreachability in thebest path available so far. Our MNDAS QACO algorithmexhibits resemblance to real ant behavior with the helpof qubit expansion technique associated with pheromonebox updation. If number of qubits in pheromone box isincreased, it will undergo more number of iterations andthus will provide a tool for congestion-controlled trafficthrough controllable duration before global convergence. pheromone Deposition () and pheromone Evaporation () perform updation in pheromone box by maintaining thepush and pop sequence from an unique updation order.For a qubit pheromone box, the qureg content follows theunique pheromone updation order of − − − − − − − as shown in figure 7 (a), where with binary equivalent and with binary equivalent represent initialempty pheromone box and box with maximum pheromonerespectively. If d represents the total number of qubits toencode pheromone distribution, maximum ( (cid:98) log d (cid:99) +1 − )number of pheromone distribution states are available inour proposed QACO algorithm before ’best-path’ conver-gence as shown in figure 8. Lemma 1.
Second best shortest path will never converge asoptimal solution in
M N DAS () , if there already exists best pathwith minimum path weight.Proof. Each iteration of procedure ant Execute () performspath selection by applying modulo division arithmetic ofa specific iteration index ( K ) by each of the path weights.If that iteration index is a multiple of path weight, thenthe path ID of that corresponding path weight is selectedfor pheromone deposition. Say, the path with minimumweight is p min and the second best path is p min , where W min < W min ( W min , W min are path weights corre-sponding to paths p min and p min respectively). After asufficient number of iterations ( K threshold ), say, number oftimes pheromone has been deposited on paths p min and p min are d and d respectively. On the other hand, numberof times pheromone has been evaporated from paths p min NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 8
Fig. 4: Quantum circuit synthesis by MNDAS algorithm for 6 as a slected path in an anonymous iterationFig. 5: Quantum circuit for update P heromone () Fig. 6: Block diagram of update P heromone () with fanin and fan out of quantum gates showing control andtarget respectively for (a) pheromone Deposition () and (b) pheromone Evaporation () and p min are e and e respectively. Each deposition indi-cates selection and each evaporation indicates non-selectionof the corresponding path. Now if r and r denote thenumber of times paths p min and p min will be selectedin ant Execute () procedure and K (cid:48) denotes any randomiteration index ( K (cid:48) >> K threshold ), which is multiple of path weights of both p min and p min , K (cid:48) = r .W min (15) K (cid:48) = r .W min (16)From equations 15 and 16, we get ( W min < W min ) = ⇒ ( r > r )= ⇒ (( d > d ) ∧ ( e < e )) (17)Since, the shortest path with minimum path weight willalways undergo selection more than any other path presentin the graph (including the second best path), our algorithmguarantees convergence of the shortest path as the optimalsolution. Lemma 2.
If the shortest path is removed from the search spacebefore convergence, then our solution converges to the next bestpath after adequate iterations.Proof.
Unreachability in the best path can be mathematicallymapped as the path with infinite path weight (or a pathweight with very large magnitude). Since, our algorithmperforms pheromone box updation dynamically, it willadapt to the new changes which have taken place in thesystem. Due to being unselected in all the iterations takenplace after the previous shortest path has attained an infinitepath weight, the said path will eventually undergo severalevaporations and being at all (cid:48) s corresponding to its entryin the pheromone box. On the other hand, the secondoptimal path will have the chance to be selected maximumnumber of times during procedure ant Execute () . n = total number of paths present from food source to antcolony. P = set of all paths. NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 9
Fig. 7: Example of a qubit and qubit pheromone boxwith (a), (c) representing pheromone deposition on selectedpaths and (b), (d) representing pheromone evaporation fromunselected paths. p m = second optimal path (which currently is the best pathwith the minimum weight due to absence of the shortestpath). p i = any arbitrary path other than p m . W m , W i = path weights corresponding to p m and p i respec-tively. r m , r i = number of times paths p m and p i will be selectedrespectively. ∆ m , ∆ i = time taken for convergence in pheromone box by p m and p i respectively. ∃ p m ∈ P ∀ ( p i ∈ P ) ∧ (0 ≤ i ≤ n − W m < W i )= ⇒ ( r m > r i ) ∧ (∆ m < ∆ i )) (18)Whenever update P heromone () procedure is invoked,it typically puts a constraint on amount of pheromonedeposition based on the pheromone density in pheromonebox of the corresponding path. Paths with pheromone den-sity ... is supposed to contain maximum amount ofpheromone. In such cases, pheromone Deposition () pro-cedure restricts itself to increase the amount of deposited Fig. 8: Number of maximum pheromone states inpheromone box with respect to number of qubits.pheromone by changing the ancilla qubit Q [ a ] as furtherdeposition of pheromone will cause disturbance in con-vergence by resetting the pheromone density to ... . Onthe contrary, procedure pheromone Evaporation () doeswork only for the unselected paths of a particular iter-ation. The paths with pheromone density ... do noteven contain any pheromone. pheromone Evaporation () has to check an underflow condition by putting a bar-rier into pheromone to be evaporated from a path with-out any available pheromone deposited on it. There areother paths which might be unselected in i th iterationhaving pheromone density of ... . In order to pre-vent pheromone evaporation from an already convergentpath, pheromone Evaporation () also does not decrease thepheromone density of such paths as well since it will affectglobal convergence in QACO.After complete exploration of all possible paths by quan-tum ants, QACO will converge to optimal state with shortestpath chosen as output of procedure ant Execute () . Theindex value of the shortest path will be selected by usinga MCT gate with pheromone box Q [ ph , ph , ....ph d − ] ascontrol and Q [ a target ] as target. The MNDAS algorithm usesa controlled phase shift ( CP HASE ) gate performing a π rotation of the qubit in most significant position ( Q [ p x − ] )based on value of Q [ a target ] . Once phase is shifted for thebest path, procedure M N DAS () invokes Quantum Ampli-tude Amplification (QAA) technique in order to amplifythe probability density value of the path with minimumweight (shortest path from food source to colony). [3] QAAtechnique is followed by a measurement step where con-vergence of a quantum superposed state is mapped intoa classical register. The concern associated with most ofthe combinatorial optimization problems is convergence.Stochastic search procedures like classical ant colony opti-mization face challenges in achieving optimality in solutionobtained, as pheromone update often prevents an algorithmto reach optimal state. [26] [28] It is worth mentioningthat our proposed MNDAS QACO algorithm is well suitedin achieving convergence in value as well as convergencein solution. Our algorithm yields optimal solution atleastonce and optimality is preserved in the same solution withcourse of time-variant iterations; thus ensuring convergencein value and convergence in solution both. NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 10
Fig. 9: A example of Simple Ant colony optimization with 8different paths, encoded in a 3 qubit quantum system path id
W eight
21 18 16 11 5 2 11 14
TABLE 1: Cost corresponding to paths in a 3 qubit QACOas shown in Figure 9
ISULIZATION OF
MNDAS A
LGORITHM FOR
QACO
In real ant colonies, ants aim to find the shortest pathfrom a colony to food source. Since, ants deposit a certainamount of pheromone in its path from nest to food andwhile making the return trip, follow the same path markedpreviously along with depositing pheromone on the same,ants following the shorter path are expected to return earlier.The real key of our MNDAS algorithm follows the sameprinciple where the rate of deposition of pheromone hasbeen made faster on the shorter path in comparison tothe longer paths to induce pheromone evaporation effect.Pheromone evaporation takes place periodically by a certainamount at a constant rate which implies the existence offrequently visited paths only through pheromone deposi-tion as rarely visited paths by ants will undergo acceler-ated evaporation followed by no existence due to lack ofpheromone deposition. All ants starting their food searchingjourney can learn from the information left by previouslyvisited ants and can get guidance to follow the shorter pathdirected by maximum pheromone deposit.This foraging behavior of real ants can be mapped intoshortest path finding problem where a number of artifi-cial ants mimicking the data packets will build solutionsand exchange relevant information on the quality of thesolutions via a communication scheme which is expressedin our algorithm as update P heromone () consisting of pheromone Deposition () and pheromone Evaporation () quantumly. In order to elucidate our proposed M N DAS algorithm, we have considered a graph example with possible paths existing between food source and ant colonyas shown in figure 9.In init Ant () procedure, all the classical and quantumparameter settings have been performed. For a path countof n = 8 , the number of qubits representing path encodingis x = 3 . If we use a qubit pheromone box ( d = 4 ), theinitial quantum superposed state | ψ (cid:105) with ( x + d + 3 ) qubits can be denoted as equation 19, where all the qubits exceptthe path encoding qubits are initialized with | (cid:105) . | ψ (cid:105) = 1 / √ | (cid:105) + 1 / √ | (cid:105) +1 / √ | (cid:105) + 1 / √ | (cid:105) +1 / √ | (cid:105) + 1 / √ | (cid:105) +1 / √ | (cid:105) + 1 / √ | (cid:105) (19)Equation 19 also ensures uniform distribution of probabilitythrough quantum superposition to achieve equiprobableselection chances of all possible paths by quantum ants.The procedure ant Execute () performs multiple iterationsfor path selection and pheromone updation to achieve finalconvergence to best-path. Each iteration of ant Execute () will necessitate the pheromone box to undergo successivechanges in qubit states. A detailed transition showing thechanges in pheromone box content from the end of iteration K = 15 to the end of iteration K = 16 has been shownin figure 10. In each iteration, the path IDs selected forpheromone deposition and pheromone evaporation havebeen explicitly shown. For example as shown in figure 10during iteration K = 16 , path IDs and are selectedfor deposition as iteration index is a multiple of their pathweights and respectively. The path with IDs , , , and contain all s in pheromone box implying theirinapplicability for pheromone evaporation except the pathwith ID which is unselected in that iteration.The path encoding qubits get back to their ini-tial configuration due to uncompute operation speci-fied in ant Execute () . Moreover, the two ancilla qubits Q [ a ] and Q [ a ] will also be resetted at the end ofeach iteration. Q [ a target ] qubit is not involved in thewhole iterative procedure and hence, holds the ini-tial value. So, in each iteration only the state of thequbits Q [ P h , P h , P h , P h ] representing pheromone boxwill be updated. Figure 11 represents the status ofqubit states of pheromone box in multiple iterations for K = 0 , , , , , , , , , , , , , , . In itera-tion K = 21 , the qubit register for pheromone box containsall s corresponding to path ID . The path with ID denotes the shortest path corresponding to the minimumpath weight as shown in figure 11.The function ant Execute () is the input to the ORACLEwhich yields the shortest path as output of the ORACLE.Controlled phase shift gate performs a π -rotation of pathindex of the selected shortest path enabling the amplitudeof the selected path to be phase shifted by π to undergoQuantum Amplitude Amplification (QAA). QAA amplifiesthe probability amplitude of the selected path as shownin figure 12 by applying phase inversion followed by per-forming inversion about mean operation on target qubit Q [ a target ] . M N DAS
QACO algorithm is free from earlierconvergence to local optima and lack of universality ofsearch space with the help of adaptive quantum algorithms,4 and 5.
ESULT AND ANALYSIS OF
MNDAS Q
UANTUM
ACO
ALGORITHM
We have implemented
M N DAS
QACO algorithm usingIBM QISKIT. The algorithm has been executed on both NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 11
Fig. 10: Change in qubit states of the pheromone box during a single iteration for K = 16 . Qubit a target is not shown as itis unaffected during quantum iteration. NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 12
Fig. 11: Qubit state status of pheromone box for the iteration K = 0 , , , , , , , , , , , , , , Fig. 12: QAA amplifies the probability amplitude of theselected path.Fig. 13: Convergence to the shortest path in a -path ACOqubit IBMQ ( ibmq melbourne ) and QASM simulator for ≤ paths and higher number of paths respectively. Since, RESET operation is currently not supported, we haveimplemented each step of iteration separately by initializingqubits with the output state of previous iteration. IBMQ er-ror threshold values, ζ corresponding to a single qubit quan-tum gate U and CN OT gate are depicted as . e − ≤ ζ U ≤ . e − and . e − ≤ ζ CNOT ≤ . e − respectively with average qubit frequency of . GHzapproximately. The Ant colony specified in figure 9 showsa simple network with paths with path ids , , ... , .Encoding such paths in our proposed M N DAS () al-gorithm requires qubits. Among all possible path costs,minimum is which in turn, is associated with path ID asshown in figure 12. The outcome of a particular executionof our algorithm on qubits and number of iterationsas K = 200 is shown in figure 13 where, the path id undergoes amplitude amplification after being selectedas shortest path with probabilistic amplitude ( p selected ) of . where p ( ∀ x ∈ n ∧ x (cid:54) = selected ) << p selected .Another example is taken into consideration with possible paths and number of qubits to encode all possiblepaths with path weight as minimum cost (shortest path)and path id as shown in figure 14 for the distribution of NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 13 path id
W eight
12 9 24 131 17 99 11 100 path id
W eight
24 31 64 79 73 6 67 101
TABLE 2: Cost corresponding to paths in a 4 qubit QACOFig. 14: Convergence to the shortest path in a -path ACOpath weights as specified as table 2. The expected optimization time of well-known MAX-MINAnt System (MMAS) for single destination shortest pathon a graph G with m number of vertices is O ( m + mρ ) ,where ρ is evaporation factor. [15] In our proposed QACOalgorithm M N DAS () , a single quantum superposed stateis prepared to encode all possible paths, thus explorationof all the possible n paths can be incorporated in a singleiteration in O (1) time. Since procedure M N DAS () has cho-sen K number of iterations to be performed for global pathconvergence, the total complexity of ant Execute () through pheromone U pdation () requires O ( K.n ) time complexity.The QAA performed in M N DAS algorithm incurs a com-plexity of O ( √ n ) . So, overall complexity of our proposednovel quantum ACO is O ( K.n + √ n ) ∈ O ( n ) , as K is con-stant. Thus, our quantum algorithm provides polynomialspeedup over its classical counterpart. ONCLUSION
In this paper, we have proposed a novel quantum al-gorithm for ant colony optimization to solve computa-tionally hard combinatorial optimization problems. Ouralgorithm MNDAS QACO (Mrityunjay-Nivedita-Debdeep-Amlan-Subhansu Quantum Ant Colony Optimization) ap-proaches a novel quantum algorithm to be run on a quan-tum hardware instead of quantum-inspired evolutionaryACO algorithms available so far. Our approach can be mod-elled as quantum module for a variety of NP-Hard problemsnamely Travelling Salesman Problem (TSP), Vehicle RoutingProblem and Network Routing Problem.MNDAS QACO is an adaptive quantum algorithm en-suring reliability in obtaining the shortest path through pheromone U pdate () quantum module. We have also builtup a fault prevention mechanism through structural con-straints applied over pheromone deposition and pheromoneevaporation to achieve unaffected global convergence ofACO problems. Our future work will incorporate quantumgate cost optimization through fault tolerant logic synthesisof quantum circuits to reduce the gate cost and improve theoverall efficiency of our algorithm. In order to physicallyimplement the multi-qubit quantum gates like MCT, Toffolispecified in our algorithms, a considerable number of SWAPgates have to be introduced which in turn, will incur ahuge cost over head. This might be our extended researcharea to work in order to reduce circuit complexity and gateoverhead through optimized synthesis of quantum physicalcircuits. Moreover, we will also focus on enhancing thisquantum ACO algorithm for other complex variants of antcolony system. A CKNOWLEDGMENT
The authors would like to thank Prof. (Dr.) SubhansuBandyopadhyay, former Professor, Department of Com-puter Science and Engineering, University of Calcutta, fordiscussions on this subject matter. R EFERENCES [1] M. Dorigo, M. Birattari and T. Stutzle,
Ant colony optimization , IEEEComputational Intelligence Magazine, vol. 1, no. 4, pp. 28-39, Nov.2006.[2] A. Colorni, M. Dorigo and V. Maniezzo,
Distributed optimization byant colonies , Proceedings of ECAL91- European Conf. on ArtificialLife, vol. 1, pp. 134-142, 1991.[3] Lov K. Grover,
A fast quantum mechanical algorithm for databasesearch , Proceedings of the twenty-eighth annual ACM symposiumon Theory of Computing, pp. 212-219, July 1996.[4] M. Dorigo, V. Maniezzo and A. Colorni,
Positive feedback as asearch strategy , Tech. Report 91-016, Dipartimentodi Elettronica,Politecnico di Milano, Italy, 1991.[5] M. Dorigo, V. Maniezzo and A. Colorni,
Ant system: optimizationby a colony of cooperating agents , IEEE Trans. on Systems, Man andCybernetics part B, vol. 26, pp. 29-41, 1996.[6] Panchi Li, Kaoping Song and Erlong Yang,
Quantum Ant ColonyOptimization with Application , Sixth International Conference onNatural Computation (ICNC), 2010.[7] Ling Wang, Qun Niu and Minrei Fei,
A novel quantum ant colony op-timization algorithm and its application to fault diagnosis , Transactionsof the Institute of Measurement and Control, vol. 30, pp. 313-329,2008.[8] Min Liu, Feng Zhang, Yunlong Ma, Hemanshu Roy Pota andWeiming Shen
Evacuation path optimization based on quantum antcolony algorithm , Advanced Engineering Informatics, vol. 30, pp.259-267, 2016.[9] M. Dorigo, V. Maniezzo and A. Colorni,
A cooperative learningapproach to the traveling salesman problem , IEEE Trans. EvolutionaryComput., vol. 1, pp. 53-66 ,1997.[10] M. Reimann, K. Doerner and R. F. Hartl,
D-ants: savings based antsdivide and conquer the vehicle routing problems , Comput. Oper. Res.,vol. 31, pp. 563-591, 2004.[11] N. B. Lovett,
Application of quantum walks on graph structures toquantum computing , Ph.D. Dissertation, School of Physics & As-tronomy, University of Leeds, 2011.[12] B. Doerr, F. Neumann, D. Sudholt, C. Witt,
Runtime analysis of the1-ANT Ant Colony Optimizer , Theoretical Computer Science vol.412 (17) pp. 1629 - 1644, 2011.[13] W.J. Gutjahr, G. Sebastiani,
Runtime analysis of ant colony optimiza-tion with best-so-far reinforcement , Methodology and Computing inApplied Probability 10, pp. 409–433, 2008.[14] F. Neumann, C. Witt,
Runtime analysis of a simple ant colony opti-mization algorithm , Algorithmica vol. 54 (2) pp. 243–255, 2009.
NOVEL QUANTUM ALGORITHM FOR ANT COLONY OPTIMIZATION, VOL. 11, NO. 8, MARCH 2020 14 [15] T. St ¨utzle, H.H. Hoos,
MAX–MIN ant system , Journal of FutureGeneration Computer Systems vol. 16 pp. 889–914, 2000.[16] D. Sudholt,
Crossover is provably essential for the Ising model on trees ,in: Proceedings of the 7th Genetic and Evolutionary ComputationConference (GECCO ’05), ACM Press, pp. 1161–1167, 2005.[17] P. Gupta, A. Agrawal and N. K. Jha,
An algorithm for synthesisof reversible logic circuits , IEEE Trans. Computer-Aided Design ofIntegrated Circuits and Systems, vol. 25, pp 2317-2330, 2006.[18] M. F. Demiral,
Ant Colony Optimization for a Variety of ClassicAssignment Problems , International Turkish World Engineering andScience Congress, Antalya, December 2017.[19] P. Zhao, P. Zhao and X. Zhang,
A new ant colony optimization for theknapsack problem , 2006 7th International Conference on Computer-Aided Industrial Design and Conceptual Design, Hangzhou, pp.1-3, 2006.[20] C. Blum,
Theoretical and Practical Aspects of Ant Colony Optimiza-tion , Dissertations in Artificial Intelligence, Vol. 282, AkademischeVerlagsgesellschaft Aka GmbH, Berlin, Germany, 2004.[21] J Yang, X Shi, M Marchese and Y Liang,
An ant colony optimizationmethod for generalized TSP problem , Progress in Natural Science Vol.18 (11), pp. 1417-1422, 10 November 2008.[22] C. Blum and M. Dorigo,
The hyper-cube framework for ant colonyoptimization , IEEE Trans. Systems, Man, Cybernet.-Part B, vol. 34(2), pp. 1161-1172, 2004.[23] C. Blum and M. Dorigo,
Search bias in ant colony optimization: on therole of competition-balanced systems , IEEE Trans. Evol. Comput., vol.9 (2), pp. 159-174, 2005.[24] C. Blum, A. Roli,
Metaheuristics in combinatorial optimization:overview and conceptual comparison , ACM Comput. Surveys, vol.35 (3), pp. 268-308, 2003.[25] A Puris, R Bello, Y Trujillo, A Nowe, Y Mart´ınez,
Two-Stage ACO toSolve the Job Shop Scheduling Problem , Progress in Pattern Recogni-tion, Image Analysis and Applications. CIARP 2007. Lecture Notesin Computer Science, vol. 4756. Springer, Berlin, Heidelberg, 2007.[26] W. J. Gutjahr,
ACO algorithms with guaranteed convergence to theoptimal solution , Inform. Process. Lett., vol. 82 (3), pp. 145-153, 2002.[27] J. H. Holland,
Adaption in Natural and Artificial Systems , The Uni-versity of Michigan Press, Ann Harbor, MI, 1975.[28] H. H. Hoos and T. St ¨utzle,
Stochastic Local Search: Foundations andApplications , Elsevier, Amsterdam, The Netherlands, 2004.[29] L. Kallel, B. Naudts and A. Rogers (Eds.), emphTheoretical As-pects of Evolutionary Computing, Natural Computing Series,Springer, Berlin, Germany, 2001.[30] L. Wang, Q. Niu and M Fei,
A Novel Quantum Ant Colony Opti-mization Algorithm , Lecture Notes in Computer Science book series(LNCS), vol. 4688, Bio-Inspired Computational Intelligence andApplications, pp 277-286, 2007
Mrityunjay Ghosh is currently associated withUniversity of Calcutta and leading research anddevelopment of industry first quantum lab ofeastern India. Previously, he was head of quan-tum technologies of an Israel based companyand worked as Assistant Professor of differentstate level Universities. He has 11 years of ex-perience in industry, research and academia.His research areas of interest include reversiblecomputing, quantum algorithm, quantum circuitsimulation and synthesis. Presently, he is lead-ing a project on development of a quantum based cryptographic suiteusing quantum random number generator.
Nivedita Dey has full time association with thefirst quantum research and development lab ofKolkata, India. Prior to this, Nivedita had workedfor an Israel based company with a designa-tion of Quantum Software Researcher and alsomade her contribution in academia as AssistantProfessor in Computer Science. Her research in-terests include quantum algorithm design, quan-tum circuit synthesis and quantum informationprocessing. She is presently involved as one ofthe key persons in development of a quantumbased cryptographic suite and in designing its use cases in varieddomain of industry and academia.
Debdeep Mitra is a post graduate in ComputerScience & Engineering. Debdeep had worked inthe research and development wing of an Israelbased company. He has teaching experience ofabout 2 years. His research interests includequantum algorithm, quantum circuit simulationand quantum machine learning. He was involvedin a project on generation of a pure random num-bers using quantum mechanical system and isfocussed to contribute in quantum computationarea for development of society and mankind atlarge.