In variational quantum algorithms, constraints are usually added to the problem objective via penalty terms. Already for linear inequality constraints, this procedure requires additional slack qubits. These extra qubits tend to blow up the search space and complicate the parameter landscapes to be navigated by the classical optimizers. In this work, we explore approaches to model linear inequalities for quantum algorithms without these drawbacks. More concretely, our main suggestion is to omit the slack qubits completely and evaluate the inequality classically during parameter tuning. We test our methods on QAOA as well as on Trotterized adiabatic evolution, and present empirical results. As a benchmark problem, we consider different instances of the multi-knapsack problem. Our results show that removing the slack bits from the circuit Hamiltonian and considering them only for the expectation value yields better solution quality than the standard approach. The tests have been carried out using problem sizes up to 26 qubits. Our methods can in principle be applied to any problem with linear inequality constraints, and are suitable for variational as well as digitized versions of adiabatic quantum computing.
In variational quantum algorithms, constraints are usually added to the problem objective via penalty terms. Already for linear inequality constraints, this procedure requires additional slack qubits. These extra qubits tend to blow up the search space and complicate the parameter landscapes to be navigated by the classical optimizers. In this work, we explore approaches to model linear inequalities for quantum algorithms without these drawbacks. More concretely, our main suggestion is to omit the slack qubits completely and evaluate the inequality classically during parameter tuning. We test our methods on QAOA as well as on Trotterized adiabatic evolution, and present empirical results. As a benchmark problem, we consider different instances of the multi-knapsack problem. Our results show that removing the slack bits from the circuit Hamiltonian and considering them only for the expectation value yields better solution quality than the standard approach. The tests have been carried out using problem sizes up to 26 qubits. Our methods can in principle be applied to any problem with linear inequality constraints, and are suitable for variational as well as digitized versions of adiabatic quantum computing.
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.
Quantum optimization is emerging as a prominent candidate for exploiting the capabilities of near-term quantum devices. Many application-relevant optimization tasks require the inclusion of inequality constraints, usually handled by enlarging the Hilbert space through the addition of slack variables. This approach, however, requires significant additional resources especially when considering multiple constraints. Here, we study an alternative direct implementation of these constraints within the quantum approximate optimization algorithm, achieved using qudit-sum gates, and compare it to the slack variable method generalized to qudits. We benchmark these approaches on three paradigmatic optimization problems. We find that the direct implementation of the inequality penalties vastly outperforms the slack variables method, especially when studying real-world inspired problems with many constraints. Within the direct penalty implementation, a linear energy penalty for unfeasible states outperforms other investigated functional forms, such as the canonical quadratic penalty. The proposed approach may thus be an enabling step for approaching realistic industry-scale and fundamental science problems with large numbers of inequality constraints.
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.
Traditional methods for handling (inequality) constraints in the quantum approximate optimization (QAOA) typically rely on penalty terms and slack variables, which increase problem complexity and expand the search space. More sophisticated mixer-based QAOA variants restrict the search within the feasible assignments but often suffer from prohibitive circuit complexity. This paper presents a penalty-free formalism for incorporating inequality constraints into the cost function of QAOA using an oracle-based subroutine that evaluates constraint satisfaction in an additional register, subsequently called indicator function QAOA (IF-QAOA). Applied to the knapsack problem, we demonstrate in numerical simulations the superior performance of IF-QAOA over conventional penalty-based approaches. Using advanced QAOA simulation techniques, we find that IF-QAOA achieves significantly higher solution quality and a faster time to solution in 82% of our benchmark cases, even though circuit depth is approximately three times larger. Analysis of the scaling behavior shows favorable scaling of IF-QAOA compared to penalty-based methods. Also, benchmarks against the recently developed quantum tree generator QAOA for knapsack problems (P. Christiansen , ) demonstrated higher solution quality for circuits of similar depth. Additionally, this paper introduces a method for approximating the indicator function when the number of ancillary qubits is limited or the constraint function is noninteger. With a specialized simulation algorithm based on projective measurements, we empirically demonstrate that this formalism can encode general inequality constraints using a fixed number of ancillary qubits.
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.
Constraint handling remains a key bottleneck in quantum combinatorial optimization. While slack-variable-based encodings are straightforward, they significantly increase qubit counts and circuit depth, challenging the scalability of quantum solvers. In this work, we investigate a suite of Lagrangian-based optimization techniques including dual ascent, bundle methods, cutting plane approaches, and augmented Lagrangian formulations for solving constrained combinatorial problems on quantum simulators and hardware. Our framework is applied to three representative NP-hard problems: the Travelling Salesman Problem (TSP), the Multi-Dimensional Knapsack Problem (MDKP), and the Maximum Independent Set (MIS). We demonstrate that MDKP and TSP, with their inequality-based or degree-constrained structures, allow for slack-free reformulations, leading to significant qubit savings without compromising performance. In contrast, MIS does not inherently benefit from slack elimination but still gains in feasibility and objective quality from principled Lagrangian updates. We benchmark these methods across classically hard instances, analyzing trade-offs in qubit usage, feasibility, and optimality gaps. Our results highlight the flexibility of Lagrangian formulations as a scalable alternative to naive QUBO penalization, even when qubit savings are not always achievable. This work provides practical insights for deploying constraint-aware quantum optimization pipelines, with applications in logistics, network design, and resource allocation.