Quantum annealing is a generic solver for combinatorial optimization problems that utilizes quantum fluctuations. Recently, there has been extensive research applying quantum annealers, which are hardware implementations of quantum annealing. Since quantum annealers can only handle quadratic unconstrained binary optimization problems, to solve constrained combinatorial optimization problems using quantum annealers, the constraints must be incorporated into the objective function. One such technique is the Ohzeki method, which employs a Hubbard-Stratonovich transformation to relax equality constraints, and its effectiveness for large-scale problems has been demonstrated numerically. This study applies the Ohzeki method to combinatorial optimization problems with inequality constraints. We show that inequality constraints can be relaxed into a similar objective function through statistical mechanics calculations similar to those for equality constraints. In addition, we evaluate the performance of this method in a typical inequality-constrained combinatorial optimization problem, the quadratic knapsack problem.
Quantum annealing is a generic solver for combinatorial optimization problems that utilizes quantum fluctuations. Recently, there has been extensive research applying quantum annealers, which are hardware implementations of quantum annealing. Since quantum annealers can only handle quadratic unconstrained binary optimization problems, to solve constrained combinatorial optimization problems using quantum annealers, the constraints must be incorporated into the objective function. One such technique is the Ohzeki method, which employs a Hubbard-Stratonovich transformation to relax equality constraints, and its effectiveness for large-scale problems has been demonstrated numerically. This study applies the Ohzeki method to combinatorial optimization problems with inequality constraints. We show that inequality constraints can be relaxed into a similar objective function through statistical mechanics calculations similar to those for equality constraints. In addition, we evaluate the performance of this method in a typical inequality-constrained combinatorial optimization problem, the quadratic knapsack problem.
Quantum annealing (QA) is used to solve combinatorial optimization problems (COPs). When COPs are implemented on quantum annealers, they are typically encoded as quadratic unconstrained binary optimization (QUBO) problems, but constraint encodings often increase the number of qubits and the embedding overhead. This issue is particularly important for COPs with inequality constraints, where standard slack-variable formulations introduce additional binary variables. Unbalanced penalization (UP) avoids slack variables, but the original UP formulation requires tuning two penalty coefficients and contains a squared residual term that can increase the number of quadratic couplings. In this paper, we propose the unbalanced penalization Ohzeki method (UPOM), which combines UP with the Ohzeki method for inequality-constrained COPs. UPOM replaces the two static penalty coefficients of original UP with an auxiliary-variable update and removes the squared residual term from the Hamiltonian used for sampling. We further propose the deep-unfolded unbalanced penalization Ohzeki method (DU-UPOM), which learns the step-size schedule of the UPOM update from training instances. Numerical experiments on random knapsack problems show that UPOM improves over original UP and that DU-UPOM reaches optimal solutions in fewer iterations than fixed-step UPOM and other baseline. These results demonstrate that the proposed framework reduces the tuning and embedding burdens of UP while making the Ohzeki method trainable for inequality constraints.
Quantum annealing is a model for quantum computing that is aimed at solving hard optimization problems by representing them as quadratic unconstrained binary optimization (QUBO) problems. Although many NP-hard problems can easily be formulated as binary-variable problems with a quadratic objective function, such formulations typically also include constraints, which are not allowed in a QUBO. Hence, such constraints are usually incorporated in the objective function as additive penalty terms. While there is substantial previous work on implementing linear equality constraints, the case of inequality constraints has not much been studied. In this paper, we propose a new approach for formulating and embedding inequality constraints as penalties and describe early implementation results.
We propose a new method for solving binary optimization problems under inequality constraints using a quantum annealer. To deal with inequality constraints, we often use slack variables, as in previous approaches. When we use slack variables, we usually conduct a binary expansion, which requires numerous physical qubits. Therefore, the problem of the current quantum annealer is limited to a small scale. In this study, we employ the alternating direction method of multipliers. This approach allows us to deal with various types using constraints in the current quantum annealer without slack variables. To test the performance of our algorithm, we use quadratic knapsack problems (QKPs). We compared the accuracy obtained by our method with a simulated annealer and the optimization and sampling mode of a D-Wave machine. As a result of our experiments, we found that the sampling mode shows the best accuracy. We also found that the computational time of our method is faster than that of the exact solver when we tackle various QKPs defined on dense graphs.
Quantum annealing is a global optimization algorithm that uses the quantum tunneling effect to speed-up the search for an optimal solution. Its current hardware implementation relies on D-Wave’s Quantum Processing Units, which are limited in terms of number of qubits and architecture while being restricted to solving quadratic unconstrained binary optimization (QUBO) problems. Consequently, previous applications of quantum annealing to real-life use cases have focused on problems that are either native QUBO or close to native QUBO. By contrast, in this paper we propose to tackle inequality constraints and non-quadratic terms. We demonstrate how to handle them with a realistic use case-a bus charging scheduling problem. First, we reformulate the original integer programming problem into a QUBO with the penalty method and directly solve it on a D-Wave machine. In a second approach, we dualize the problem by performing the Hubbard-Stratonovich transformation. The dual problem is solved indirectly by combining quantum annealing and adaptive classical gradient-descent optimizer. Whereas the penalty method is severely limited by the connectivity of the realistic device, we show experimentally that the indirect approach is able to solve problems of a larger size, offering thus a better scaling. Hence, the implementation of the Hubbard-Stratonovich transformation carried out in this paper on a scheduling use case suggests that this approach could be investigated further and applied to a variety of real-life integer programming problems under multiple constraints to lower the cost of mapping to QUBO, a key step towards the near-term practical application of quantum computing.
Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. However, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transformations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated Annealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total number of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.
Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. However, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transformations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated Annealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total number of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.