Optimization deals with problem of finding the minimum of a given cost function (the relationship between the total cost of production and the quantity of a product produced) and in the combinatorial optimisation problems the cost function depends on a large number of variables and hard problems are those for which the computational time is not bound by any polynomial in the problem size [1]. To solve hard optimization problems is a challenging task. Several techniques have been developed to get to the solution(s) of such problems. Here we will discuss about some of the problems and also the techniques that have already been implemented.
Optimization deals with problem of finding the minimum of a given cost function (the relationship between the total cost of production and the quantity of a product produced) and in the combinatorial optimisation problems the cost function depends on a large number of variables and hard problems are those for which the computational time is not bound by any polynomial in the problem size [1]. To solve hard optimization problems is a challenging task. Several techniques have been developed to get to the solution(s) of such problems. Here we will discuss about some of the problems and also the techniques that have already been implemented.
Quantum annealing is analogous to simulated annealing with a tunneling mechanism substituting for thermal activation. Its performance has been tested in numerical simulation with mixed conclusions. There is a class of optimization problems for which the efficiency can be studied analytically using techniques based on the statistical mechanics of spin glasses. Solving hard combinatorial problems by temperature annealing is a classic method in computer science.1) The problem is formulated in terms of a cost function, which one can identify as an energy. The dynamics are then a combination of a systematic descent in energy perturbed by a ’noise’, which is turned off gradually, inducing
Critical decision-making issues in science, engineering, and industry are based on combinatorial optimization; however, its application is inherently limited by the NP-hard nature of the problem. A specialized paradigm of analogue quantum computing, quantum annealing (QA), has been proposed to solve these problems by encoding optimization problems into physical energy landscapes and solving them by quantum tunnelling systematically through exploration of solution space. This is a critical review that summarizes the current applications of quantum annealing to combinatorial optimization and includes a theoretical background, hardware designs, algorithm implementation strategies, encoding and embedding schemes, protocols to benchmark quantum annealing, areas of implementation, and links with the quantum algorithms implementation with gate-based hardware and classical solvers. We develop a unified framework, relating adiabatic quantum dynamics, Ising and QUBO models, stoquastic and non-stoquastic Hamiltonians, and diabatic transitions to modern flux-qubit annealers (Chimera, Pegasus, Zephyr topologies), and emergent architectures (Lechner-Hauke-Zoller systems, Rydberg atom platforms), and hybrids of quantum and classical computation. Through our analysis, we find that overhead in embedding and encoding is the largest determinant of the scalability and performance (this is not just the number of qubits). Minor embeddings also usually have a physical qubit count per logical variable of between 5 and 12 qubits, which limits effective problem capacity by 80-92% and, due to chain-breaking errors, compromises the quality of solutions.
Abstract Quantum annealing has been applied to combinatorial optimization problems in recent years. In this paper we study the possibility to use quantum annealing for solving the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from a conveyor belt onto pallets. The problem is NP-hard and can be solved using linear programming approaches. We developed two QUBO (quadratic unconstrained binary optimization) objective functions based on a bin stack-up solution and a pallet stack-up solution for this problem suitable for a quantum annealer. The number of variables was minimized to increase the performance and their dependence on the number of bins and pallets was discussed. The performances of both methods were studied for various small problem sizes on a D-Wave quantum annealer. We found that only tiny instances could be solved and looked at the terms of the QUBO-formulations, which cause the quantum annealer to fail for larger problem sizes. Furthermore we compare the results to the performance of a classic computer using the same QUBO-formulations.
In planning and scheduling, large-scale optimization problems often have complex limits and a lot of factors, which makes them hard to answer with standard methods. Quantum annealing has become a hopeful way to deal with these problems because it can quickly look into a lot of possible solutions. This essay looks into how quantum annealing might be used to solve optimization problems in transportation and scheduling, with a focus on how it could be used for route planning, inventory management, and arranging workers. To do these optimization jobs, quantum annealing is used. This is a quantum computing method that gets the global minimum of a cost function. We use the D-Wave quantum annealer in this method. It turns the problem's limits and objective function into a quadratic unconstrained binary optimization (QUBO) model. The quantum annealer then works on this model to find the best answers. The suggested method combines traditional preparation methods with quantum annealing to get the best results for big schedule and transportation issues. We use quantum annealing to solve difficult multi-objective optimization problems, like reducing shipping time, cost, and resource use all at the same time. Preliminary results show that quantum annealing can make solutions much better and use computers much more efficiently than traditional methods for optimizing supplies and schedules. Quantum annealing works especially well with problems that have a lot of dimensions. It can provide scalable answers that sometimes work better than standard methods.
Quantum annealing, or quantum stochastic optimization, is a classical randomized algorithm which provides good heuristics for the solution of hard optimization problems. The algorithm, suggested by the behaviour of quantum systems, is an example of proficuous cross contamination between classical and quantum computer science. In this survey paper we illustrate how hard combinatorial problems are tackled by quantum computation and present some examples of the heuristics provided by quantum annealing. We also present preliminary results about the application of quantum dissipation (as an alternative to imaginary time evolution) to the task of driving a quantum system toward its state of lowest energy.
Quantum annealing is a promising tool for solving optimization problems, similar in some ways to the traditional (classical) simulated annealing of Kirkpatrick et al. Simulated annealing takes advantage of thermal fluctuations in order to explore the optimization landscape of the problem at hand, whereas quantum annealing employs quantum fluctuations. Intriguingly, quantum annealing has been proved to be more effective than its classical counterpart in many applications. We illustrate the theory and the practical implementation of both classical and quantum annealing – highlighting the crucial differences between these two methods – by means of results recently obtained in experiments, in simple toy-models, and more challenging combinatorial optimization problems (namely, Random Ising model and Travelling Salesman Problem). The techniques used to implement quantum and classical annealing are either deterministic evolutions, for the simplest models, or Monte Carlo approaches, for harder optimization tasks. We discuss the pro and cons of these approaches and their possible connections to the landscape of the problem addressed.