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arXiv:2603.00260

Constrained Quantum Optimization at Utility Scale: Application to the Knapsack Problem

Naeimeh Mohseni, Julien-Pierre Houle, Ibrahim Shehzad, Giorgio Cortiana, Corey O'Meara, Adam Bene Watts

quant-phSubmitted Feb 27, 2026• Announced Mar 3, 2026• 0 citations
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Abstract

Constrained combinatorial optimization problems are challenging for quantum computing, particularly at utility-relevant scales and on near-term hardware. At the same time, these problems are of practical significance in industry; for example, the Unit Commitment (UC) problem in energy systems involves complex operational constraints. To address this challenge, we apply copula-QAOA (cop-QAOA), a hardware-efficient approach for constrained optimization to a single-period UC that can be reduced to a one-dimensional knapsack. Cop-QAOA biases the quantum state toward feasible solutions using constant-depth mixers and appropriately biased initial states. We implement our benchmark on problem instances that are confirmed to be hard for classical solvers such as Gurobi. Our results show that cop-QAOA often finds solutions better than a lazy greedy baseline and very close to, and in some instances surpasses, those obtained by Gurobi, with only a few QAOA rounds. This work presents the largest successful demonstration of the knapsack problem on IBM Quantum hardware using up to 150 qubits, and more generally, the largest demonstration of constrained combinatorial optimization where constraints are enforced via shallow mixers.

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Constrained combinatorial optimization problems are challenging for quantum computing, particularly at utility-relevant scales and on near-term hardware. At the same time, these problems are of practical significance in industry; for example, the Unit Commitment (UC) problem in energy systems involves complex operational constraints. To address this challenge, we apply copula-QAOA (cop-QAOA), a hardware-efficient approach for constrained optimization to a single-period UC that can be reduced to a one-dimensional knapsack. Cop-QAOA biases the quantum state toward feasible solutions using constant-depth mixers and appropriately biased initial states. We implement our benchmark on problem instances that are confirmed to be hard for classical solvers such as Gurobi. Our results show that cop-QAOA often finds solutions better than a lazy greedy baseline and very close to, and in some instances surpasses, those obtained by Gurobi, with only a few QAOA rounds. This work presents the largest successful demonstration of the knapsack problem on IBM Quantum hardware using up to 150 qubits, and more generally, the largest demonstration of constrained combinatorial optimization where constraints are enforced via shallow mixers.
Combinatorial optimization is a promising area for achieving quantum speedup. The quantum approximate optimization algorithm (QAOA) is designed to search for low-energy states of the Ising model, which correspond to near-optimal solutions of combinatorial optimization problems (COPs). However, effectively dealing with constraints of COPs remains a significant challenge. Existing methods, such as tailoring mixing operators, are typically limited to specific constraint types, like one-hot constraints. To address these limitations, we introduce a method for engineering a compressed space that represents the feasible solution space with fewer qubits than the original. Our approach includes a scalable technique for determining the unitary transformation between the compressed and original spaces on gate-based quantum computers. We then propose compressed space QAOA, which seeks near-optimal solutions within this reduced space, while utilizing the Ising model formulated in the original Hilbert space. Experimental results on a quantum simulator demonstrate the effectiveness of our method in solving various constrained COPs.
This paper introduces two techniques that make the standard Quantum Approximate Optimization Algorithm (QAOA) more suitable for constrained optimization problems. The first technique describes how to use the outcome of a prior greedy classical algorithm to define an initial quantum state and mixing operation to adjust the quantum optimization algorithm to explore the possible answers around this initial greedy solution. The second technique is used to nudge the quantum exploration to avoid the local minima around the greedy solutions. To analyze the benefits of these two techniques we run the quantum algorithm on known hard instances of the Knapsack Problem using unit depth quantum circuits. The results show that the adjusted quantum optimization heuristics typically perform better than various classical heuristics.
Combinatorial optimization problems play a central role in computer science and real-world decision-making, as they aim to find the optimal solution from a large set of possibilities. Among these, the Knapsack Problem is a fundamental example with many practical applications. The Quantum Approximate Optimization Algorithm (QAOA) is a quantum-classical hybrid approach designed to address such problems. However, existing simulation approaches for QAOA applied to the Knapsack Problem do not scale well because they fail to exploit the problem's structural properties. To overcome these limitations, this study exploits the entangling characteristics of circuits in the Knapsack Problem and develops an optimized solver tailored to this setting. The key technical contributions include an entanglement-free circuit simulation method that reduces complexity from exponential to linear for non-entangling circuits, enabling 24,000x speedups for QAOA ansatz circuit simulations, and a two-phase parameter selection strategy that enhances the efficiency of finding optimal Knapsack solutions. Our solver extends the scalability of QAOA simulations from the previous limit of approximately 10 qubits to 100 qubits within a system memory of 256 GB. It achieves performance gains of over 100,000x for solving Knapsack Problems while maintaining high solution quality.
Combinatorial optimization problems play a vital role in computer science and engineering, with applications in logistics, network design, finance, and resource allocation. However, many of these problems are NP-hard, making them computationally expensive to solve using classical optimization techniques, especially for large-scale instances. In recent years, quantum computing has emerged as a promising paradigm capable of addressing these limitations by leveraging quantum mechanical principles such as superposition and entanglement. This study investigates the application of quantum computing for solving combinatorial optimization problems, with a focus on algorithm design, complexity analysis, and real-world applicability. The research primarily employs the Quantum Approximate Optimization Algorithm (QAOA) and Grover's search algorithm to solve representative problems such as Max-Cut and the Knapsack problem. The performance of these algorithms is evaluated using key metrics including accuracy, execution time, and approximation ratio, and is compared with classical optimization techniques. The results demonstrate that QAOA is capable of generating near-optimal solutions, achieving accuracy levels of up to 93% for small problem instances. However, the study also highlights the challenges associated with current quantum systems, including noise, limited qubit availability, and increased execution time. Comparative analysis reveals that while classical methods remain more efficient for small-scale problems, quantum algorithms show significant potential for scalability and handling complex optimization tasks. The findings suggest that hybrid quantum-classical approaches offer a practical solution in the current Noisy Intermediate-Scale Quantum (NISQ) era. The study concludes that although quantum computing is still evolving, it holds strong potential for transforming combinatorial optimization in the future.
This paper proposes a novel combination of constraint encoding methods for the Quantum Approximate Optimization Ansatz (QAOA). Real-world optimization problems typically consist of multiple types of constraints. To solve these optimization problems with quantum methods, normally, all constraints are added as quadratic penalty terms to the objective, which expands the search space and increases problem complexity. This work proposes a general workflow that extracts and encodes specific constraints directly into the circuit of QAOA: One-hot constraints are enforced throughXYX YXY-mixers that restrict the search space to the feasible sub-space naturally. Inequality constraints are implemented through oracle-based Indicator Functions (IF). This paper focuses on the numerical benchmarks of the combined approach for solving the Multi-Knapsack (MKS) and the Prosumer Problem (PP), a modification of the MKS in the domain of electricity optimization. To this end, we introduce computational techniques that efficiently simulate the two presented constraint architectures. SinceXYX YXY-mixers restrict the search space, specific state vector entries are always zero and can be omitted from the simulation, saving valuable memory and computing resources. We benchmark the combined method against the established QUBO formulation, yielding a better solution quality and probability of sampling the optimal solution. Despite more complex circuits, the time-to-solution is more than an order of magnitude faster compared to the baseline methods and exhibits more favorable scaling properties.
This paper introduces a novel approach to the Quantum Approximate Optimization Algorithm (QAOA), specifically tailored to the knapsack problem. We combine the recently proposed quantum tree generator as an efficient state preparation circuit for all feasible solutions to the knapsack problem with the framework of Grover-mixer QAOA to form the first representative of Amplitude Amplification-mixer QAOA (AAMQAOA). On hard benchmark sets with up to 20 knapsack items, we demonstrate our method's improved performance over the current state-of-the-art Copula-QAOA. However, for larger instance sizes, both approaches fail to deliver better outcomes than greedily packing items in descending value-to-weight ratio, at least for the considered circuit depths. For sufficiently high circuit depths, however, we can prove that AAM-QAOA will eventually be able to sample the optimal solution.