Applying analog quantum computing to solve optimisation problems: case studies on Max-Cut and CVRP

It is widely believed that the real parallel computation achieved by quantum computers has an enormous computing potential. In order to expand its applicable field, we have investigated the fusion of quantum and neural computations. As a first step of implementing learning function on quantum computers, we have proposed a novel quantum associative memory (QuAM) by considering an analogy between neural associative network and qubit network. The memorizing procedure of the QuAM is realized with a Hamiltonian derived from qubit-qubit interactions, and the retrieving procedure is based on the adiabatic Hamiltonian evolution. The memory capacity of the QuAM has been nominally estimated as 2N−1 where N is a number of qubits, but its retrieve property has not been discussed in our previous study. This paper proposes a retrieving process for the QuAM and evaluates its performance in detail. The results indicate that the average of the retrieving probability is over 50% even when the qubit network memorizes 2N−1 patterns and thus the QuAM is successfully implemented.

Quantum computing is gaining popularity across a wide range of scientific disciplines due to its potential to solve long-standing computational problems that are considered intractable with classical computers. One promising area where quantum computing has potential is in the speed-up of NP-hard optimisation problems that are common in industrial areas such as logistics and finance. Newcomers to the field of quantum computing who are interested in using this technology to solve optimisation problems do not have an easily accessible source of information on the current capabilities of quantum computers and algorithms. This paper aims to provide a comprehensive overview of the theory of quantum optimisation techniques and their practical application, focusing on their near-term potential for noisy intermediate scale quantum devices. The paper starts by drawing parallels between classical and quantum optimisation problems, highlighting their conceptual similarities and differences. Two main paradigms for quantum hardware are then discussed: analogue and gate-based quantum computers. While analog devices such as quantum annealers are effective for some optimisation problems, they have limitations and cannot be used for universal quantum computation. In contrast, gate-based quantum computers offer the potential for universal quantum computation, but they face challenges with hardware limitations and accurate gate implementation. The paper provides a detailed mathematical discussion with references to key works in the field, as well as a more practical discussion with relevant examples. The most popular techniques for quantum optimisation on gate-based quantum computers, the quantum approximate optimisation algorithm and the quantum alternating operator ansatz framework, are discussed in detail. However, it is still unclear whether these techniques will yield quantum advantage, even with advancements in hardware and noise reduction. The paper concludes with a discussion of the challenges facing quantum optimisation techniques and the need for further research and development to identify new, effective methods for achieving quantum advantage.

Particle swarm optimization (PSO) is adopted to solve capacitated vehicle routing problems (CVRPs). A nonlinear S decrease curve is employed to adjust the inertia weight of PSO to balance global and local searches. Meanwhile, local search and path relink are also involved to improve the quality of solutions. Experiments were conducted on Auger at problems in the OR Library. Experimental results will were compared with those from other studies. Experimental results prove the algorithm designed in this article can effectively solve capacitated type vehicle routing problems.

Combining genetic local search into a multi-population Imperialist Competitive Algorithm for the Capacitated Vehicle Routing Problem

In measurement-based quantum computation (MBQC), a special highly-entangled state (called a resource state) allows for universal quantum computation driven by single-qubit measurements and post-measurement corrections. Physical realisations of this model have been achieved in various physical systems for low numbers of qubits. The large number of qubits necessary to construct the resource state constitutes one of the main down sides to MBQC. However, in some instances it is possible to extend the resource state on the fly, meaning that not every qubit must be realised in the devices simultaneously. We consider the question of the minimal number of physical qubits that must be present in a system to directly implement a given measurement pattern. For measurement patterns with n inputs, n outputs and m total qubits which have flow, we show that only min(n + 1, m) qubits are required, while the number of required qubits can be as high as m-2 for measurement patterns with only gflow. We discuss the implications of removing the Clifford part of a measurement pattern, using well-established transformation rules for Pauli measurements, for the presence of flow versus gflow, and hence the effect on the minimum number of physical qubits required to directly realise the measurement pattern.

Design and development of a hybrid artificial bee colony algorithm for the environmental vehicle routing problem

Capacitated vehicle routing problem is a vehicle routing problem that has constraints in the form of vehicle capacity. It is a matter of optimization to find the optimal route for some vehicles with a specific capacity and homogeneous vehicles that serve some agents with several known requests before the distribution process takes place. LPG distribution is the application of capacitated vehicle routing problems. This research aims to determine the LPG distribution route based on the Clarke and Wright Algorithm and LINGO. The capacitated vehicle routing problem model will be solved in two phases. The first stage completes the capacitated vehicle routing problem model using the Clarke and Wright algorithm. The second stage is completing the results of the first stage in the form of LINGO-based sub tours to obtain the optimal solution. The solution of the capacitated vehicle routing problem in distribution LPG using the Clarke and Wright algorithm has a total distance of 151.94 km. The solution of the capacitated vehicle routing problem in distribution LPG using LINGO has a total distance of 161.59 km.

eCommerce, postal and logistics’ planners require to solve large-scale capacitated vehicle routing problems (CVRPs) on a daily basis. CVRP problems are NP-Hard and cannot be easily solved for large problem instances. Given their complexity, we propose a methodology to reduce the size of CVRP problems that can be later solved with state-of-the-art optimization solvers. Our method is an efficient version of clustering that considers the constraints of the original problem to transform it into a more tractable version. We call this approach Constrained Clustering Capacitated Vehicle Routing Solver (CC-CVRS) because it produces a soft-clustered vehicle routing problem with reduced decision variables. We demonstrate how this method reduces the computational complexity associated with the solution of the original CVRP and how the computed solution can be transformed back into the original space. Extensive numerical experiments show that our method allows to solve very large CVRP instances within seconds with optimality gaps of less than 16%. Therefore, our method has the following benefits: it can compute improved solutions with small optimality gaps in near real-time, and it can be used as a warm-up solver to compute an improved solution that can be used as an initial solution guess by an exact solver.

Advances in Quantum Machine Learning and Deep Learning for Image Classification: A Survey

Vehicle Routing Problem (VRP) is a classical NP-hard problem in combinational optimization with great practicality. A branch of VRP is the Capacitated Vehicle Routing Problem (CVRP), where vehicles have capacity constraints. In this paper, a Hybrid Chaotic Particle Swarm Optimization (HCPSO) is proposed to solve CVRP. To make this algorithm successful, three main components play an important role. Firstly, we introduce a new mutual mapping method to encode and decode between decimal numbers and integer solutions. Secondly, the ergodicity and sensitivity on initial conditions of chaos theory are utilized to achieve chaotic initialization and renewing. Thirdly, various local search strategies like neighbor change strategy, move strategy are employed to improve the local search ability. In the end, benchmarks have been tested. It is clearly shown that for small and medium sized CVRP, the algorithm is efficient and effective and the results are better than those of some recent papers.

Capacitated vehicle routing problem is extension of vehicle routing Problem where the purpose is to determine a set of routes to fulfil all the transportation requests with a given fleet of vehicles in such a way that the travelling cost comes out to be a minimum. In this paper, the demands of the customers will be very well known in advance, but they are known imprecisely. To deal with such kind of impreciseness, interval type-2 fuzzy number (IT2FN) can be used. Therefore, a capacitated vehicle routing problem with interval type-2 fuzzy number has been modelled. A procedure to solve the model has been introduced using Clark and Wright algorithm.

We consider the asymmetric capacitated vehicle routing problem (CVRP), a particular case of the standard asymmetric vehicle routing problem in which only the vehicle capacity constraints are imposed. CVRP is known to be NP-hard and finds practical applications in distribution and scheduling. We describe two new bounding procedures for CVRP, based on the so-called additive approach. Each procedure computes a sequence of nondecreasing lower bounds, obtained by solving different relaxations of CVRP. Effective implementations of the procedures are also outlined which considerably reduce the computational effort. The two procedures are combined into an overall bounding algorithm. A branch-and-bound exact algorithm is then proposed, whose performance is enhanced by means of reduction procedures, dominance criteria, and feasibility checks. Extensive computational results on both real-world and random test problems are presented, showing that the proposed approach favorably compares with previous algorithms from the literature.

The Capacitated Vehicle Routing Problem (CVRP) presents significant challenges in shipping route optimization and logistics management. These challenges include balancing vehicle capacity, minimizing travel distance, and efficiently grouping delivery points, all of which are crucial for enhancing operational efficiency and reducing costs. This research aims to apply a combination of the Sweep and Nearest Neighbor algorithms to address the CVRP, seeking to improve route efficiency and manage vehicle capacity effectively. The Sweep algorithm is employed to cluster pickup points based on their polar angle from the depot, facilitating efficient grouping and optimal vehicle capacity management. Within each cluster, the Nearest Neighbor algorithm is implemented to optimize the sequence of visits, minimizing total travel distance by sequentially selecting the next closest point. The Haversine Distance is used to calculate the distances between points, ensuring geographical accuracy compared to the Euclidean method. Experimental results demonstrate that this hybrid approach yields shorter routes. Quantitative analysis shows a significant reduction 13% in total travel distance when using this combination of algorithms, highlighting its effectiveness in solving the CVRP. This research demonstrates that combining the Sweep and Nearest Neighbor algorithms provides an efficient solution to the CVRP, improving route optimization and vehicle capacity management. The findings contribute valuable insights to logistics management, with practical implications for enhancing shipping route efficiency.

The capacitated vehicle routing problem (CVRP) is regarded as an NP-hard problem. Moreover, the CVRP is described as a model that can be used in many applications such as transport, logistics, and distribution. The exact algorithms can find exact optimal solutions on the small-sized problem instances; however, for large-sized instances it is difficult to find the exact optimal solutions in polynomial time. This reason motivated the researchers to present heuristic/metaheuristic algorithms to solve large-sized problem instances within a reasonable computational time. One of the good algorithms that deal with the CVRP is the ant colony optimization (ACO) algorithm. Several ACO algorithms have been suggested in the literature, such as the ant system (AS) algorithm, ant colony system (ACS) algorithm, and so on. On the other hand, ACO is designed to solve the path problem that finds the best way. However, this algorithm still lacks exploratory mechanisms, which results in premature convergence and stagnation issues. Therefore, we propose to develop an enhanced ACS (EACS) algorithm for solving the CVRP based on subpaths. In our proposed algorithm, we propose to utilize the K-nearest neighbour (KNN) algorithm for finding the best initial solution and then enhance the diversity mechanism of the proposed algorithm by avoiding the generation of the same solution using subpaths. This uses the diversity of the generated solution to find a better solution with a shorter route in a reasonable amount of computational time. Furthermore, we propose to apply the three-opt algorithm to the completed subtour and the k-opt algorithm to the subpath gained from the experience of the subpath. Finally, to verify the effectiveness of the proposed EACS algorithm, the algorithm is tested on some CVRP instances and is compared with one of the state-of-the-art methods, namely, the enhanced simulated annealing algorithm. The comparative study showed a better performance of our EACS compared to the enhanced simulated annealing algorithm.

The ability to simulate a fermionic system on a quantum computer is expected to revolutionize chemical engineering, materials design, nuclear physics, to name a few. Thus, optimizing the simulation circuits is of significance in harnessing the power of quantum computers. Here, we address this problem in two aspects. In the fault-tolerant regime, we optimize the Rz and T gate counts along with the ancilla qubit counts required, assuming the use of a product-formula algorithm for implementation. We obtain a savings ratio of two in the gate counts and a savings ratio of eleven in the number of ancilla qubits required over the state of the art. In the pre-fault tolerant regime, we optimize the two-qubit gate counts, assuming the use of the variational quantum eigensolver (VQE) approach. Specific to the latter, we present a framework that enables bootstrapping the VQE progression towards the convergence of the ground-state energy of the fermionic system. This framework, based on perturbation theory, is capable of improving the energy estimate at each cycle of the VQE progression, by about a factor of three closer to the known ground-state energy compared to the standard VQE approach in the test-bed, classically-accessible system of the water molecule. The improved energy estimate in turn results in a commensurate level of savings of quantum resources, such as the number of qubits and quantum gates, required to be within a pre-specified tolerance from the known ground-state energy. We also explore a suite of generalized transformations of fermion to qubit operators and show that resource-requirement savings of up to more than 20%, in small instances, is possible.