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 (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.
Abstract Solving combinatorial optimization problems of the kind that can be codified by quadratic unconstrained binary optimization (QUBO) is a promising application of quantum computation. Some problems of this class suitable for practical applications such as the traveling salesman problem (TSP), the bin packing problem (BPP), or the knapsack problem (KP) have inequality constraints that require a particular cost function encoding. The common approach is the use of slack variables to represent the inequality constraints in the cost function. However, the use of slack variables considerably increases the number of qubits and operations required to solve these problems using quantum devices. In this work, we present an alternative method that does not require extra slack variables and consists of using an unbalanced penalization function to represent the inequality constraints in the QUBO. This function is characterized by larger penalization when the inequality constraint is not achieved than when it is. We evaluate our approach on the TSP, BPP, and KP, successfully encoding the optimal solution of the original optimization problem near the ground state cost Hamiltonian. Additionally, we employ D -Wave Advantage and D -Wave hybrid solvers to solve the BPP, surpassing the performance of the slack variables approach by achieving solutions for up to 29 items, whereas the slack variables approach only handles up to 11 items. This new approach can be used to solve combinatorial problems with inequality constraints with a reduced number of resources compared to the slack variables approach using quantum annealing or variational quantum algorithms.
Solving NP-hard constrained combinatorial optimization problems using quantum algorithms remains a challenging yet promising avenue toward quantum advantage. Variational Quantum Algorithms (VQAs), such as the Variational Quantum Eigensolver (VQE), typically require constrained problems to be reformulated as unconstrained ones using penalty methods. A common approach introduces slack variables and quadratic penalties in the QUBO formulation to handle inequality constraints. However, this leads to increased qubit requirements and often distorts the optimization landscape, making it harder to find high-quality feasible solutions. To address these issues, we explore a slack-free formulation that directly encodes inequality constraints using custom penalty functions, specifically the exponential function and the Heaviside step function. These steplike penalties suppress infeasible solutions without introducing additional qubits or requiring finely tuned weights. Inspired by recent developments in quantum annealing and thresholdbased constraint handling in gate-based algorithms, we implement and evaluate our approach on the Multiple Knapsack Problem (MKP). Experimental results show that the step-based formulation significantly improves feasibility and optimality rates compared to unbalanced penalization, while reducing overall qubit overhead.
Combinatorial optimization problems are one of the target applications of current quantum technology, mainly because of their industrial relevance, the difficulty of solving large instances of them classically, and their equivalence to Ising Hamiltonians using the quadratic unconstrained binary optimization (QUBO) formulation. Many of these applications have inequality constraints, usually encoded as penalization terms in the QUBO formulation using additional variables known as slack variables. The slack variables have two disadvantages: (i) these variables extend the search space of optimal and suboptimal solutions, and (ii) the variables add extra qubits and connections to the quantum algorithm. Recently, a new method known as unbalanced penalization has been presented to avoid using slack variables. This method offers a trade-off between additional slack variables to ensure that the optimal solution is given by the ground state of the Ising Hamiltonian, and using an unbalanced heuristic function to penalize the region where the inequality constraint is violated with the only certainty that the optimal solution will be in the vicinity of the ground state. This work tests the unbalanced penalization method using real quantum hardware on D-Wave Advantage for the traveling salesman problem (TSP). The results show that the unbalanced penalization method outperforms the solutions found using slack variables and sets a new record for the largest TSP solved with quantum technology.
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 computing approaches, such as Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA), have emerged as promising candidates for solving combinatorial optimization problems. Since these approaches typically work with Quadratic Unconstrained Binary Optimization (QUBO) problems, constraints, when present, are commonly encoded as penalty terms. Conventional approaches to incorporate inequality constraints into QUBO formulations additionally require the introduction of slack variables, increasing the required number of qubits. The Knapsack and Bin Packing problems are optimization problems with such inequality constraints. We show that the number of slack variables required to formulate the Knapsack problem as a QUBO problem can be reduced using certain properties specific to this problem. Moreover, we present two separate strategies for the Knapsack and Bin Packing problems that avoid using the costly slack variables altogether. Instead, both strategies solve several QUBO problems, with varying penalty constants, such that at least one of them is a good representation of the given optimization problem. Numerical simulations of QAOA, as well as experiments on a D-Wave quantum annealer, show that these slack-less approaches can improve performance compared to the conventional slack variable method (in terms of success probability), even when the latter is allowed to vary penalty constants too.
Quantum annealing (QA) has attracted research interest as a sampler and combinatorial optimization problem (COP) solver. A recently proposed sampling-based solver for QA significantly reduces the required number of qubits, being capable of large COPs. In relation to this, a trainable sampling-based COP solver has been proposed that optimizes its internal parameters from a dataset by using a deep learning technique called deep unfolding. Although learning the internal parameters accelerates the convergence speed, the sampler in the trainable solver is restricted to using a classical sampler owing to the training cost. In this study, to utilize QA in the trainable solver, we propose classical-quantum transfer learning, where parameters are trained classically, and the trained parameters are used in the solver with QA. The results of numerical experiments demonstrate that the trainable quantum COP solver using classical-quantum transfer learning improves convergence speed and execution time over the original solver.