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.
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.
We propose a slack-free penalty formulation for constrained binary optimization that eliminates auxiliary slack variables and preserves the feasibility structure of the original problem. The proposed approach introduces a nonlinear custom penalty function to enforce inequality constraints directly in the objective function. To address the computational challenges associated with evaluating nonlinear penalties in variational quantum algorithms, we employ the finite-sampling method that avoids the exponential complexity required by exact expectation computation. Furthermore, we integrate the Conditional Value-at-Risk (CVaR) objective to improve optimization robustness and guide the search toward high-quality solutions. The proposed framework is evaluated on instances of the multi-dimensional knapsack problem, a classical benchmark in combinatorial optimization. We showcase that the proposed custom-penalty formulation combined with CVaR sampling achieves improved optimality gaps and more consistent performance compared with conventional slack-based QUBO formulations. We also tested our method on the Quantinuum H2 quantum hardware, demonstrating the possibilities of training small-sized VQE on noisy trapped-ion quantum computers. The results suggest that careful penalty design can play a critical role in enabling quantum and hybrid quantum-classical algorithms for constrained optimization problems that arise in operations research.
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.
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) 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 computing has emerged as a promising alternative for solving combinatorial optimization problems. The standard approach for encoding optimization problems on quantum processing units (QPUs) involves transforming them into their Quadratic Unconstrained Binary Optimization (QUBO) representation. However, encoding constraints of optimization problems, particularly inequality constraints, into QUBO requires additional variables, which results in more qubits. Considering the limited availability of qubits in NISQ machines, existing encoding methods fail to scale due to their reliance on large numbers of qubits. We propose a generalized exponential penalty framework for QUBO inequality constraints inspired by a class of exponential functions, which we call exponential penalization. This paper presents an encoding strategy for inequality constraints in combinatorial optimization problems, inspired by a class of exponential functions, which we call exponential penalization. The initial idea of using exponential penalties for QUBO formulation was introduced by Montanez-Barrera et al. by applying a specific exponential function to reduce qubit requirements. In this work, we extend that approach by conducting a comprehensive study on a broader class of exponential functions, analyzing their theoretical properties and empirical performance. Our experimental results demonstrate that an exponential penalization achieves 57%, 83% qubit number reduction for Bin Packing Problem (BPP) and Traveling Salesman Problem (TSP), respectively. And we demonstrate comparable solution quality to classical with a probability of 6% and 21% accuracy for BPP with 8 and TSP with 12 qubits, respectively.
Inequality constraints in combinatorial optimization are conventionally handled by introducing slack variables and quadratic penalties, which increase the qubit count and distort the optimization landscape. Slack-free formulations instead apply a nonlinear custom penalty function directly to the constraint, which are difficult to be implemented in quantum approximate optimization algorithm (QAOA). We show that the Walsh--Hadamard transform (WHT) removes this obstruction for any penalty function: a penalty applied to a linear constraint is exactly a weighted sum of Pauli-Z strings. We propose this framework as the Walsh custom penalty QAOA (WCP-QAOA). Truncating the expansion at Walsh degree k costs O(nk) Pauli terms and leaves the qubit count untouched. The truncation is the optimal degree-k approximation of the penalty in L2, and we give a condition on the discarded Walsh coefficients under which the truncated Hamiltonian still ranks every feasible solution below every infeasible one. Since evaluating a Walsh coefficient from its definition requires all 2n penalty values, we give an algorithm that returns any coefficient in sub-exponential time, so that the degree-k Hamiltonian is obtained in time polynomial in n for fixed k. We show that for the exponential penalty, there exists a closed product form of WHT, and hence a closed form expression to calculate the Walsh coefficients. We identify structural shortcuts that shorten the exact spectrum. We also tested WCP-QAOA in solving a 6-qubit knapsack problem.