Variational Quantum Algorithms (VQAs) are promising approaches for solving combinatorial optimization problems on near-term quantum devices. However, the role played by quantum resources such as entanglement and coherence during the optimization process remains poorly understood. In this work, we investigate the evolution of these resources in two VQAs--the Quantum Approximate Optimization Algorithm (QAOA) and the Feedback-based Algorithm for Quantum Optimization (FALQON)--applied to the Minimum Vertex Cover problem. We employ a variational resource-quantification framework based on the Bures metric and Uhlmann's theorem, in which an auxiliary quantum circuit is optimized to identify the closest resource-free states. This approach allows us to quantify Bures entanglement and Bures coherence of the algorithmic states without requiring full quantum state tomography. For the considered problem instance, QAOA exhibits a pronounced accumulation of entanglement in its initial layers, followed by a progressive reduction as the algorithm converges toward the optimal solution. Its coherence likewise decreases overall with increasing circuit depth, although a finite residual value persists due to the degeneracy of the optimal-solution subspace. In contrast, FALQON exhibits a substantially lower entanglement peak, reached at larger circuit depth, followed by a decay as convergence proceeds. Its coherence decreases smoothly and monotonically together with the optimization error. These results reveal distinct quantum-resource trajectories associated with the two optimization strategies and illustrate how entanglement and coherence evolve as variational quantum states approach solutions of a classical combinatorial optimization problem.
Variational Quantum Algorithms (VQAs) are promising approaches for solving combinatorial optimization problems on near-term quantum devices. However, the role played by quantum resources such as entanglement and coherence during the optimization process remains poorly understood. In this work, we investigate the evolution of these resources in two VQAs--the Quantum Approximate Optimization Algorithm (QAOA) and the Feedback-based Algorithm for Quantum Optimization (FALQON)--applied to the Minimum Vertex Cover problem. We employ a variational resource-quantification framework based on the Bures metric and Uhlmann's theorem, in which an auxiliary quantum circuit is optimized to identify the closest resource-free states. This approach allows us to quantify Bures entanglement and Bures coherence of the algorithmic states without requiring full quantum state tomography. For the considered problem instance, QAOA exhibits a pronounced accumulation of entanglement in its initial layers, followed by a progressive reduction as the algorithm converges toward the optimal solution. Its coherence likewise decreases overall with increasing circuit depth, although a finite residual value persists due to the degeneracy of the optimal-solution subspace. In contrast, FALQON exhibits a substantially lower entanglement peak, reached at larger circuit depth, followed by a decay as convergence proceeds. Its coherence decreases smoothly and monotonically together with the optimization error. These results reveal distinct quantum-resource trajectories associated with the two optimization strategies and illustrate how entanglement and coherence evolve as variational quantum states approach solutions of a classical combinatorial optimization problem.
Variational quantum algorithms (VQAs) for combinatorial optimization routinely employ entangling gates as a default design choice, yet the role of entanglement, in its amount and structure, remains poorly understood. This gap is particularly consequential for problems governed by diagonal Hamiltonians, whose ground states are classical product states and therefore require no entanglement in principle, raising the fundamental question of whether and how entangling gates help or hinder the variational search. We investigate this question for MaxCut by introducing two complementary control mechanisms that provide smooth, monotonic control over hardware-efficient ansatz (HEA) entanglement as quantified by the Meyer-Wallach measure Q, and by benchmarking against QAOA as a problem-structured reference. Tracking the entanglement trajectory Q(t) throughout VQA training reveals that when the ansatz grants the optimizer indirect control over entanglement through its parameters, it consistently drives entanglement down. In line with this tendency, a fully separable ansatz outperforms all entangled hardware-efficient configurations, establishing a monotonic relationship: less problem-agnostic entanglement yields better performance. In contrast, QAOA, whose entanglement is structurally derived from the problem Hamiltonian, maintains high entanglement yet achieves competitive solution quality, demonstrating that entanglement structure, not merely quantity, determines its utility. These findings suggest that HEAs for diagonal Hamiltonians are inappropriate and that variational approaches to combinatorial optimization should prioritize problem-structured circuit designs.
The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.
Combinatorial optimization is a promising application for near-term quantum computers, however, identifying performant algorithms suited to noisy quantum hardware remains as an important goal to potentially realizing quantum computational advantages. To address this we present a variational quantum algorithm for solving combinatorial optimization problems with linear-depth circuits. Our algorithm uses an ansatz composed of Hamiltonian generators designed to control each term in the target combinatorial function, along with parameter updates following a modified version of quantum imaginary time evolution. We evaluate this ansatz in numerical simulations that target solutions to the MAXCUT problem. The state evolution is shown to closely mimic imaginary time evolution, and its optimal-solution convergence is further improved using adaptive transformations of the classical Hamiltonian spectrum. With these innovations, the algorithm consistently converges to optimal solutions, with interesting highly-entangled dynamics along the way. We further demonstrate the success of this approach by performing optimization of a truncated version of our ansatz with up to 32 qubits in a trapped-ion quantum computer, using measurements from the quantum computer to train the ansatz and prepare optimal solutions with high fidelity in the majority of cases we consider. The success of these large-scale quantum circuit optimizations, in the presence of realistic hardware constraints and without error mitigation, mark significant progress on the path towards solving combinatorial problems using quantum computers. We conclude our performant and resource-minimal approach is a promising candidate for potential quantum computational advantages.
Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparameterization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA) -- a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems -- MAX-CUT and MAX-2-SAT. For MAX-CUT we observe that overparametriation is both sufficient and (statistically) necessary for attaining exact solutions, as confirmed numerically for up to 20 qubits. In fact, for MAX-CUT on 2-regular graphs we show the necessity to be exact, based on the analytically found optimal depth. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.
Abstract Variational Quantum Algorithms (VQAs) offer a promising approach to solving optimization problems on quantum computers. In this paper, we investigate the use of VQAs to solve the Learning With Errors (LWE) problem, a crucial challenge in post-quantum cryptography. We propose a VQA-based algorithm and a Quantum Approximate Optimization Algorithm (QAOA) for solving the Learning Parity with Noise (LPN) problem, a special case of LWE. We examine the sample complexity of combinatorial optimization methods for LPN and analyze the performance of these two variational quantum algorithms in terms of quantum resources. For the VQA algorithm, both the number of qubits and the circuit depth scale polynomially with the secret key length n . In the case of QAOA, the number of qubits matches the secret key length n , but the CNOT gate count increases exponentially with n . However, numerical simulations of the VQA-based algorithm show that choosing optimal ansatzes and optimization methods can significantly enhance success rates.
The role of entanglement in quantum optimization remains actively debated. To address this question, we focus on the fixed-parameter expanding-depth regime of the quantum approximate optimization algorithm (QAOA), where a compact two-parameter schedule is trained once on small instances and then applied as the problem size and circuit depth increase. To probe this regime beyond full state-vector simulation, we perform approximate matrix product state simulations for up to 50 qubits and 100 layers and quantify entanglement by the bond dimension. We observe an entangle--disentangle profile, with the peak bond dimension decreasing with depth and eventually saturating. This observation motivates an extreme approximation: projecting the state onto the product-state manifold (bond dimension one) after every two-qubit interaction. Based on this approximation, we introduce BOND-1, a quantum-inspired classical solver. Despite the drastic simplification, BOND-1 achieves cut ratios above 0.95 relative to the best known values on standard GSet MaxCut benchmarks with up to 20000 variables, and in some cases it matches those values. It achieves these results without per-instance optimization and has linear memory cost, while per-instance tuning can provide further improvement. These results show that, in this regime, a substantial fraction of the optimization power of QAOA survives even in the complete absence of entanglement. Our conclusions, however, are specific to this setting and do not imply that entanglement is unnecessary for quantum optimization in general.