Near-Optimal Parameter Tuning of Level-1 QAOA for Ising Models

V Vijendran; Dax Enshan Koh; Eunok Bae; Hyukjoon Kwon; Ping Koy Lam; Syed M Assad

doi:10.22331/q-2026-07-15-2158

Near-Optimal Parameter Tuning of Level-1 QAOA for Ising Models

V Vijendran^1,2,3, Dax Enshan Koh^2,4,5,6, Eunok Bae^7,8, Hyukjoon Kwon⁷, Ping Koy Lam^2,1,3, and Syed M Assad^2,3

¹Centre for Quantum Technologies (CQT), National University of Singapore, Singapore 117543, Republic of Singapore
²Quantum Innovation Centre (Q.InC), Agency for Science, Technology and Research (A*STAR), 2 Fusionopolis Way, Innovis #08-03, Singapore 138634, Republic of Singapore
³Centre for Quantum Computation and Communication Technologies (CQC2T), Department of Quantum Science and Technology, Research School of Physics, Australian National University, Acton 2601, Australia
⁴Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Republic of Singapore
⁵Science, Mathematics and Technology Cluster, Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372, Republic of Singapore
⁶Engineering Cluster, Singapore Institute of Technology, 1 Punggol Coast Road, Singapore 828608, Republic of Singapore
⁷School of Computational Sciences, Korea Institute for Advanced Study (KIAS), Seoul 02455, Korea
⁸Electronics and Telecommunications Research Institute (ETRI), Daejeon 34129, Korea

Published:	2026-07-15, volume 10, page 2158
Editor:	Alessio Benavoli
Eprint:	arXiv:2501.16419v4
Doi:	https://doi.org/10.22331/q-2026-07-15-2158
Citation:	Quantum 10, 2158 (2026).

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Abstract

The Quantum Approximate Optimisation Algorithm (QAOA) tackles combinatorial optimisation problems by encoding their solutions into the ground state of an Ising Hamiltonian prepared by a $p$-level parameterised circuit, with the angles tuned classically. Parameter optimisation is widely regarded as a central bottleneck, even for the shallowest circuits. Focusing on QAOA at $p=1$ (QAOA$_1$), we show that tuning the two angles $(\gamma, \beta)$ for weighted Ising models is not a black-box search but a structured signal-processing problem. We prove that the QAOA$_1$ expectation value is a partial Fourier series in $\gamma$ whose frequencies are determined explicitly by the problem's couplings and fields, giving instance-wise bandwidth bounds and, via the Nyquist–Shannon theorem, the sampling resolution needed to avoid the aliasing that causes coarse-grid searches to return spurious optima. We then eliminate the mixer angle analytically, computing $\beta^*(\gamma)$ in closed form to reduce the search to one dimension, and apply a subdivision algorithm that locates the globally optimal $\gamma$ in polynomial time with a certificate of optimality when the weights are commensurable and bounded. For regular weighted graphs, we further prove the conventional wisdom that the globally optimal $\gamma^* \in \mathbb{R}^+$ concentrates near zero and coincides with the first local optimum, giving a rigorous account of the empirical success of small-angle initialisation and allowing gradient descent to replace exhaustive line searches. Validated within Recursive QAOA (RQAOA) on weighted instances of 128 and 256 qubits, our method consistently outperforms both coarsely optimised RQAOA and semidefinite programming.

Featured image: A weighted Ising/QUBO instance defines a problem Hamiltonian whose spectrum sets the frequencies accessible to the QAOA$_1$ landscape; since the shallow circuit expresses only a subset of these, the expectation value is a truncated Fourier series. We compute its bandwidth analytically, use it to set a Nyquist-safe sampling scale, and eliminate $\beta$ in closed form, reducing the problem to a one-dimensional optimisation over $\gamma$ solved with a certificate of global optimality – with the optimum concentrating near $\gamma \approx 0$ on average.

Popular summary

Quantum computers offer a promising approach to combinatorial optimisation – problems that arise everywhere from logistics and finance to drug discovery, where the goal is to find the best configuration among an astronomically large number of possibilities. One of the most studied algorithms for this purpose is the Quantum Approximate Optimisation Algorithm (QAOA), which encodes a problem into a quantum Hamiltonian and searches for the spin configuration that minimises a cost function. Even at its shallowest useful depth — just one layer, called QAOA$_1$ – reliable tuning of its two free parameters, a problem angle $\gamma$ and a mixer angle $\beta$, has remained surprisingly elusive in practice.

The difficulty lies hidden in the structure of the cost landscape. As the parameters are varied, the output of QAOA$_1$ does not change smoothly – it oscillates, and the rate of oscillation grows with the weights and connectivity of the problem being solved. For large, dense, or heavily weighted problems, these oscillations become so rapid that standard coarse searches over the parameters miss them entirely, landing on spurious minima rather than the true optimum. This explains a widely observed but poorly understood failure mode: naive parameter optimisation, even with local refinement, can produce dramatically suboptimal results – not because the underlying problem is fundamentally hard to optimise, but because the search is being conducted at the wrong resolution.

Our main contribution is to replace this heuristic search with a principled, mathematically grounded procedure. We show that the oscillation pattern of QAOA$_1$ can be fully characterised in advance, directly from the coefficients of the problem itself, allowing us to calculate exactly how finely the parameters need to be sampled to avoid missing the true optimum. We then eliminate one of the two parameters entirely: for any fixed value of $\gamma$, the best choice of $\beta$ can be computed directly, reducing the original two-dimensional search to a much simpler one-dimensional problem over $\gamma$ alone. We solve this remaining problem using a systematic search procedure that progressively narrows down the range of possible values, ruling out regions that provably cannot contain the best solution – producing not merely a good parameter setting, but a guarantee that nothing better was missed.

For large regular weighted problems, we further prove that the best choice of $\gamma$ typically lies close to zero and coincides with the first local optimum encountered when searching outward from zero. This provides a rigorous theoretical explanation for why simple strategies that start near zero tend to work well in practice, and allows a fast local search to replace an exhaustive one.

We validate the full approach within Recursive QAOA (RQAOA), where accurate parameter tuning consistently enables RQAOA$_1$ to outperform both coarsely tuned variants and classical semidefinite programming across dense, weighted benchmark instances with 128 and 256 qubits.

► BibTeX data

@article{V2026nearoptimal,
  doi = {10.22331/q-2026-07-15-2158},
  url = {https://doi.org/10.22331/q-2026-07-15-2158},
  title = {Near-{O}ptimal {P}arameter {T}uning of {L}evel-1 {QAOA} for {I}sing {M}odels},
  author = {V Vijendran and Koh, Dax Enshan and Bae, Eunok and Kwon, Hyukjoon and Lam, Ping Koy and Assad, Syed M},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2158},
  month = jul,
  year = {2026}
}

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Cited by

[1] V Vijendran, Dax Enshan Koh, Ping Koy Lam, and Syed M Assad, "Classical and Quantum Heuristics for the Binary Paint Shop Problem", arXiv:2509.15294, (2025).

[2] Malick A. Gaye, Omar Shehab, Paraj Titum, and Gregory Quiroz, "Quantum optimization with classical chaos", Quantum Science and Technology 11 2, 025039 (2026).

[3] Eunok Bae, Hyukjoon Kwon, V. Vijendran, and Soojoon Lee, "Modified recursive QAOA for exact MAX-CUT solutions on bipartite graphs: closing the gap beyond QAOA limit", Journal of Physics A Mathematical General 58 48, 485304 (2025).

[4] Filip B. Maciejewski, Stuart Hadfield, Oscar Wallis, George Pennington, Sebastian Brandhofer, Stefan Woerner, Daniel J. Egger, and Davide Venturelli, "Quantum Approximate Optimization via Noise-Directed Adaptive Warm-Starting", arXiv:2607.09368, (2026).

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