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arXiv:2509.13528 · Physical Review Research

Evaluating the limits of Quantum Approximate Optimization Algorithm parameter transfer at high rounds on sparse Ising models with geometrically local cubic terms

Elijah Pelofske, Marek M. Rams, Andreas Bärtschi, Piotr Czarnik, Paolo Braccia, Łukasz Cincio, Stephan Eidenbenz

Published Mar 2, 2026• 1 citation
arXivHTMLPDF

Abstract

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 .

DOI
10.1103/p2lg-z4kn

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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 .
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.
Abstract We show that the quantum approximate optimization algorithm (QAOA) for higher-order, random coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from 16 up to 127 qubits for p = 1 up to p = 5, which allows for computationally efficient parameter transfer of QAOA angles. Matrix product state (MPS) simulation is used to compute noise-free QAOA performance. Hardware-compatible short-depth QAOA circuits are executed on ensembles of 100 higher-order Ising models on noisy IBM quantum superconducting processors with 16, 27, and 127 qubits using QAOA angles learned from a single 16-qubit instance using the JuliQAOA tool. We show that the best quantum processors find lower energy solutions up to p = 2 or p = 3, and find mean energies that are about a factor of two off from the noise-free distribution. We show that p = 1 QAOA energy landscapes remain very similar as the problem size increases using NISQ hardware gridsearches with up to a 414 qubit processor.
Quantum Approximate Optimization Algorithm (QAOA) enables solving combinatorial optimization problems on quantum computers by optimizing variational parameters for quantum circuits. We investigate a simplified approach that combines linear parameterization with parameter transferring, reducing the parameter space to just 4 dimensions regardless of the number of layers. This simplification draws inspiration from quantum annealing schedules providing both theoretical grounding and practical advantages. We compare this combined approach with standard QAOA and other parameter setting strategies such as INTERP and FOURIER, which require computationally demanding incremental layer-by-layer optimization. Notably, previously known methods like INTERP and FOURIER yield parameters that can be well fitted by linear functions, which supports our linearization strategy. Our analysis reveals that for the random Ising model, cost landscapes in this reduced parameter space demonstrate consistent structural patterns across different problem instances. Our experiments extend from classical simulation to actual quantum hardware implementation on IBM's Eagle processor, demonstrating the approach's viability on current NISQ devices. Furthermore, the numerical results indicate that parameter transferability primarily depends on the energy scale of problem instances, with normalization techniques improving transfer quality. Most of our numerical experiments are conducted on the random Ising model, while problem-dependence is also investigated across other models. A key advantage of parameter transferring is the complete elimination of instance-specific classical optimization overhead, as pre-trained parameters can be directly applied to other problem instances, reducing classical optimization costs by orders of magnitude for deeper circuits.
The Quantum Approximate Optimization Algorithm (QAOA) is a leading approach for combinatorial optimization on near-term quantum devices, yet its scalability is limited by the difficulty of optimizing 2p2p2p variational parameters for a large number ppp of layers. Recent empirical studies indicate that optimal QAOA angles exhibit concentration and transferability across problem sizes. Leveraging this observation, we propose a schedule-learning framework that transfers spectral-gap-informed adiabatic control strategies from small-scale instances to larger systems. Our method extracts the spectral gap profile of small problems and constructs a continuous schedule governed by ∂ts=κgq(s)\partial_t s = κg^q(s)∂t​s=κgq(s), where g(s)g(s)g(s) is the instantaneous gap and (κ,q)(κ, q)(κ,q) are global hyperparameters. Discretizing this schedule yields closed-form expressions for all QAOA angles, reducing the classical optimization task from 2p2p2p parameters to only 222, independent of circuit depth. This drastic parameter compression mitigates classical optimization overhead and reduces sensitivity to barren plateau phenomena. Numerical simulations on random QUBO and 3-regular MaxCut instances demonstrate that the learnt schedules transfer effectively to larger systems while achieving competitive approximation ratios. Our results suggest that gap-informed schedule transfers provide a scalable and parameter-efficient strategy for QAOA.
Quantum Approximate Optimization Algorithm (QAOA) is one of the most promising quantum algorithms for the Noisy Intermediate-Scale Quantum (NISQ) era. Quantifying the performance of QAOA in the near-term regime is of utmost importance. We perform a large-scale numerical study of the approximation ratios attainable by QAOA is the low-to medium-depth regime. To find good QAOA parameters we perform 990 million 10-qubit QAOA circuit evaluations. We find that the approximation ratio increases only marginally as the depth is increased, and the gains are offset by the increasing complexity of optimizing variational parameters. We observe a high variation in approximation ratios attained by QAOA, including high variations within the same class of problem instances. We observe that the difference in approximation ratios between problem instances increases as the similarity between instances decreases. We find that optimal QAOA parameters concentrate for instances in out benchmark, confirming the previous findings for a different class of problems.
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.