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arXiv:2606.05311

Setting angles in quantum approximate optimization at utility-scale

Maosheng Guo, Joel Jurado Diaz, Anurag Ramesh, Conrad J. Haupt, Alberto Baiardi, Dimitrios Athanasakos, M. Emre Sahin, Oscar Wallis, George Pennington, Christian Arenz, Sebastian Brandhofer, Georgios Korpas, Ieva Čepaitė, J. A. Montañez-Barrera, Jakub Marecek, Davide Venturelli, Stephan Eidenbenz, David E. Bernal Neira, Daniel J. Egger

quant-phSubmitted Jun 3, 2026• Announced Jun 5, 2026• 0 citations
v1 · Jun 3, 2026v2 · Sep 23, 2026
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Abstract

The quantum approximate optimization algorithm (QAOA) is a powerful heuristic that seeks to solve combinatorial optimization problems using quantum hardware and classical optimization in tandem. Various methods exist to train the parameterized quantum circuits that serve as an ansatz in QAOA. However, which method works best to identify optimal angles for a given problem instance remains poorly understood, especially at utility-scale, i.e., $100$ qubits or more. In this work, we address this challenge through utility-scale benchmarks from which we distill operational guidance for QAOA practitioners. First, we investigate approximation techniques, such as matrix product states and Pauli propagation, to find optimal angles. Second, we train QAOA on small-scale representative problems and transfer the angles to larger ones. We then validate the results on quantum hardware for utility-scale problem instances that can be meaningfully executed. In this way, we identify insights for QAOA angle setting strategies that work best for problems at the utility scale, including as a function of resource cost for the search. Crucially, the operational implications we draw from our benchmarks will help quantum optimization practitioners execute QAOA end-to-end pipelines efficiently on current and future hardware.

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The Quantum Approximate Optimization Algorithm (QAOA) and its variants are leading algorithms to help solve challenging combinatorial optimization problems. QAOA maps these problems to Ising Hamiltonians such that their ground state is the desired solution. Candidate solutions are then drawn from a variational quantum circuit. The performance of QAOA hinges on a set of variational angles. However, finding good angles is an instance-specific and computationally expensive non-convex classical optimization task. Here, we learn the mapping from the problem instance to high-quality QAOA angles. Our learned models replace the per-instance optimization, that we call the solver, with a single forward pass that is several orders of magnitude faster than the solver with a near identical quality. Our models further generalize to unseen topologies such as Barabási-Albert and 2D grid graphs. Experiments on quantum hardware confirm that QAOA samples with angles produced by our open-source models are indistinguishable from solver angles. Crucially, models sensitive to the topology of the problem instance increase the quality of the QAOA angles. Our work amortizes a major bottleneck in QAOA by an up-front data generation and training cost. Importantly, this cost can be amortized over time in settings where many similar problems are optimized on a regular basis, as is often the case in applications for industry.

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Abstract As combinatorial optimization is one of the main quantum computing applications, many methods based on parameterized quantum circuits are being developed. In general, a set of parameters are being tweaked to optimize a cost function out of the quantum circuit output. One of these algorithms, the Quantum Approximate Optimization Algorithm stands out as a promising approach to tackling combinatorial problems. However, finding the appropriate parameters is a difficult task. Although QAOA exhibits concentration properties, they can depend on instances characteristics that may not be easy to identify, but may nonetheless offer useful information to find good parameters. In this work, we study unsupervised Machine Learning approaches for setting these parameters without optimization. We perform clustering with the angle values but also instances encodings (using instance features or the output of a variational graph autoencoder), and compare different approaches. These angle-finding strategies can be used to reduce calls to quantum circuits when leveraging QAOA as a subroutine. We showcase them within Recursive-QAOA up to depth 3 where the number of QAOA parameters used per iteration is limited to 3, achieving a median approximation ratio of 0.94 for MaxCut over 200 Erdős-Rényi graphs. We obtain similar performances to the case where we extensively optimize the angles, hence saving numerous circuit calls.

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The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters, which is usually done using a variational optimization loop. Parameter transfer, or parameter concentration, is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable nonvariational learning method for QAOA angle finding. In this work, we systematically study QAOA parameter transferability from small problem sizes (16 and 27 decision variables) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher-order terms using the Julia based QAOA simulation tool to perform classical angle finding for up to 49 QAOA layers ( p ) . Parameter transfer of the fixed angles is validated using a combination of full state vector, projected entangled pair states, matrix product state, and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to other (unseen) problem instances does not in general provide monotonically improving performance as a function of p —there are many cases where the performance temporarily decreases as a function of p —but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the 100 + qubit instances. We also sample the hardware-compatible Ising models using the ensemble of transfer-learned QAOA parameters on several superconducting qubit IBM quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM noisy intermediate-scale quantum processors up to p = 5 on , up to p = 9 on , and up to p = 10 on .

arXiv:2509.13528•Physical Review Research•Mar 2, 2026•1 citation
The quantum approximate optimization algorithm (QAOA) is a powerful heuristic that seeks to solve combinatorial optimization problems using quantum hardware and classical optimization in tandem. Various methods exist to train the parameterized quantum circuits that serve as an ansatz in QAOA. However, which method works best to identify optimal angles for a given problem instance remains poorly understood, especially at utility-scale, i.e., 100100100 qubits or more. In this work, we address this challenge through utility-scale benchmarks from which we distill operational guidance for QAOA practitioners. First, we investigate approximation techniques, such as matrix product states and Pauli propagation, to find optimal angles. Second, we train QAOA on small-scale representative problems and transfer the angles to larger ones. We then validate the results on quantum hardware for utility-scale problem instances that can be meaningfully executed. In this way, we identify insights for QAOA angle setting strategies that work best for problems at the utility scale, including as a function of resource cost for the search. Crucially, the operational implications we draw from our benchmarks will help quantum optimization practitioners execute QAOA end-to-end pipelines efficiently on current and future hardware.
The Quantum Approximate Optimization Algorithm (QAOA) and its variants are leading algorithms to help solve challenging combinatorial optimization problems. QAOA maps these problems to Ising Hamiltonians such that their ground state is the desired solution. Candidate solutions are then drawn from a variational quantum circuit. The performance of QAOA hinges on a set of variational angles. However, finding good angles is an instance-specific and computationally expensive non-convex classical optimization task. Here, we learn the mapping from the problem instance to high-quality QAOA angles. Our learned models replace the per-instance optimization, that we call the solver, with a single forward pass that is several orders of magnitude faster than the solver with a near identical quality. Our models further generalize to unseen topologies such as Barabási-Albert and 2D grid graphs. Experiments on quantum hardware confirm that QAOA samples with angles produced by our open-source models are indistinguishable from solver angles. Crucially, models sensitive to the topology of the problem instance increase the quality of the QAOA angles. Our work amortizes a major bottleneck in QAOA by an up-front data generation and training cost. Importantly, this cost can be amortized over time in settings where many similar problems are optimized on a regular basis, as is often the case in applications for industry.
The main object of study of this work is the Quantum Approximate Optimization Algorithm (QAOA) —the best-known heuristic quantum algorithm for addressing combinatorial optimization problems. Despite its popularity, QAOA poses major challenges. Namely, its performance remains poorly understood, especially in the low depth regime most relevant to near-term quantum computing. Besides, the algorithm is of variational nature, relying on several hyperparameters to be classically trained. In this work, we contribute to addressing these issues by developing techniques to understand the performance of QAOA at low depth, and deducing from these pretrained hyperparameters, thereby eliminating or considerably reducing the need for a classical optimization loop. We first consider applying QAOA to constraint satisfaction problems with the goal of maximizing the algorithm’s approximation ratio (fraction of satisfied constraints). As a particular example, we consider QAOA applied to MAXCUT on non-regular graphs, showing how analytic techniques provide good guess hyperparameters in this setting. Next, we analyze the performance of QAOA on a well-studied constraint satisfaction problem: random k-SAT. In this case, the performance is measured by the probability of finding an assignment satisfying all constraints (“success probability”). To estimate this quantity, we develop new techniques, mapping quantum expectation values of the QAOA circuits to classical spin models. We further show how these techniques can be generalized to a larger family of random constraint satisfaction problems. We conclude the work by two feasibility studies inspired by real-world use cases encountered in collaborations with industrial partners: peptide sampling, and solving flow problems. These problems are naturally phrased by encoding combinatorial variables in qudits, a relatively unexplored setting in the existing literature. We present partial analytic and numerical results for optimal QAOA hyperparameters and the scaling of its success probability in this framework, underlining potential specific difficulties when using QAOA with qudits.
The quantum approximate optimization algorithm (QAOA) is a quantum heuristic for combinatorial optimization that has been demonstrated to scale better than state-of-the-art classical solvers for some problems. For a given problem instance, QAOA performance depends crucially on the choice of the parameters. While average-case optimal parameters are available in many cases, meaningful performance gains can be obtained by fine-tuning these parameters for a given instance. This task is especially challenging, however, when the number of circuit executions (shots) is limited. In this work, we develop an end-to-end protocol that combines multiple parameter settings and fine-tuning techniques. We use large-scale numerical experiments to optimize the protocol for the shot-limited setting and observe that optimizers with the simplest internal model (linear) perform best. We implement the optimized pipeline on a trapped-ion processor using up to 32 qubits and 5 QAOA layers, and we demonstrate that the pipeline is robust to small amounts of hardware noise. To the best of our knowledge, these are the largest demonstrations of QAOA parameter fine-tuning on a trapped-ion processor in terms of two-qubit gate count.
The quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems.Despite its promise for near-term quantum applications, not much is currently understood about the QAOA's performance beyond its lowestdepth variant.An essential but missing ingredient for understanding and deploying the QAOA is a constructive approach to carry out the outer-loop classical optimization.We provide an in-depth study of the performance of the QAOA on MaxCut problems by developing an efficient parameter-optimization procedure and revealing its ability to exploit nonadiabatic operations.Building on observed patterns in optimal parameters, we propose heuristic strategies for initializing optimizations to find quasioptimal p-level QAOA parameters in O½polyðpÞ time, whereas the standard strategy of random initialization requires 2 OðpÞ optimization runs to achieve similar performance.We then benchmark the QAOA and compare it with quantum annealing, especially on difficult instances where adiabatic quantum annealing fails due to small spectral gaps.The comparison reveals that the QAOA can learn via optimization to utilize nonadiabatic mechanisms to circumvent the challenges associated with vanishing spectral gaps.Finally, we provide a realistic resource analysis on the experimental implementation of the QAOA.When quantum fluctuations in measurements are accounted for, we illustrate that optimization is important only for problem sizes beyond numerical simulations but accessible on near-term devices.We propose a feasible implementation of large MaxCut problems with a few hundred vertices in a system of 2D neutral atoms, reaching the regime to challenge the best classical algorithms.
Abstract As combinatorial optimization is one of the main quantum computing applications, many methods based on parameterized quantum circuits are being developed. In general, a set of parameters are being tweaked to optimize a cost function out of the quantum circuit output. One of these algorithms, the Quantum Approximate Optimization Algorithm stands out as a promising approach to tackling combinatorial problems. However, finding the appropriate parameters is a difficult task. Although QAOA exhibits concentration properties, they can depend on instances characteristics that may not be easy to identify, but may nonetheless offer useful information to find good parameters. In this work, we study unsupervised Machine Learning approaches for setting these parameters without optimization. We perform clustering with the angle values but also instances encodings (using instance features or the output of a variational graph autoencoder), and compare different approaches. These angle-finding strategies can be used to reduce calls to quantum circuits when leveraging QAOA as a subroutine. We showcase them within Recursive-QAOA up to depth 3 where the number of QAOA parameters used per iteration is limited to 3, achieving a median approximation ratio of 0.94 for MaxCut over 200 Erdős-Rényi graphs. We obtain similar performances to the case where we extensively optimize the angles, hence saving numerous circuit calls.
The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters, which is usually done using a variational optimization loop. Parameter transfer, or parameter concentration, is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable nonvariational learning method for QAOA angle finding. In this work, we systematically study QAOA parameter transferability from small problem sizes (16 and 27 decision variables) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher-order terms using the Julia based QAOA simulation tool to perform classical angle finding for up to 49 QAOA layers ( p ) . Parameter transfer of the fixed angles is validated using a combination of full state vector, projected entangled pair states, matrix product state, and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to other (unseen) problem instances does not in general provide monotonically improving performance as a function of p —there are many cases where the performance temporarily decreases as a function of p —but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the 100 + qubit instances. We also sample the hardware-compatible Ising models using the ensemble of transfer-learned QAOA parameters on several superconducting qubit IBM quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM noisy intermediate-scale quantum processors up to p = 5 on , up to p = 9 on , and up to p = 10 on .