Out of the Loop: Structural Approximation of Optimisation Landscapes and non-Iterative Quantum Optimisation

Tom Krüger; Wolfgang Mauerer

doi:10.22331/q-2025-11-06-1903

Out of the Loop: Structural Approximation of Optimisation Landscapes and non-Iterative Quantum Optimisation

¹Technical University of Applied Sciences Regensburg, Regensburg, Germany
²FI CODE, Universität der Bundeswehr München, Munich, Germany
³Siemens AG, Technology, Munich, Germany

Published:	2025-11-06, volume 9, page 1903
Editor:	Marco Cerezo
Eprint:	arXiv:2408.06493v3
Doi:	https://doi.org/10.22331/q-2025-11-06-1903
Citation:	Quantum 9, 1903 (2025).

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Abstract

The Quantum Approximate Optimisation Algorithm (QAOA) is a widely studied quantum-classical iterative heuristic for combinatorial optimisation. While QAOA targets problems in complexity class NP, the classical optimisation procedure required in every iteration is itself known to be NP-hard. Still, advantage over classical approaches is suspected for certain scenarios, but nature and origin of its computational power are not yet satisfactorily understood.
By introducing means of efficiently and accurately approximating the QAOA optimisation landscape from solution space structures, we derive a new algorithmic variant of unit-depth QAOA for two-level Hamiltonians (including all problems in NP): Instead of performing an iterative quantum-classical computation for each input instance, our non-iterative method is based on a quantum circuit that is instance-independent, but problem-specific. It matches or outperforms unit-depth QAOA for key combinatorial problems, despite reduced computational effort.
Our approach is based on proving a long-standing conjecture regarding instance-independent structures in QAOA. By ensuring generality, we link existing empirical observations on QAOA parameter clustering to established approaches in theoretical computer science, and provide a sound foundation for understanding the link between structural properties of solution spaces and quantum optimisation.

Featured image: Overview of our main foundational contributions. Left: Previously conjectured parameter clustering in QAOA optimisation landscapes \(F_{1}^{(i)}(\beta, \gamma)\) (derived from a combinatorial optimisation objective function \(c^{(i)}(\vec{z})\) of a decision problem instance) suggests that shared macroscopic similarities exist between instances \(i\). Averaging over these reveals that the expected landscape \(E(F_{1}(\beta, \gamma))\) is a common structure at the problem-global level.
Right: Shared macroscopic features exist across all instances, based on structural properties manifest in the solution spaces. Aggregation (which may be possible analytically, but can always be performed using empirical sampling) leads to a macroscopic description of a problem-global solution space, from which the expected optimisation landscape can be efficiently approximated as \(\tilde{E}(F_{1}(\beta, \gamma))\). Most importantly, we prove that the approximation has a bounded difference to the underlying exact quantity. The approximation approach (ingredients indicated by a blue background) and its consequences are subject of this paper.

Popular summary

Variational quantum algorithms (VQA) became extremely popular in the current era of quantum computing. Here a parameterised circuit ansatz gets classically optimized for each instance to be solved. In order to harness the biggest quantum advantage the classical parameter optimization costs have to be reduces as much as possible. Based on a sound theoretical underpinning proving a long standing empirical conjecture we derive a new non-iterative variant of the unit-depth version of the most common VQA – the Quantum Approximate Optimisation Algorithm (QAOA) – that works without instance specific parameter optimization.

In this work, we provide a sound theoretical explanation and proof of the longstanding empirical conjecture of the existence of problem wide structures that cause instance optimal QAOA parameters to cluster. More specifically, we derive an error bounded structural approximation theorem that shows how the expected unit-depth QAOA parameter optimisation landscape can be precisely approximated by structural information about the problem solution spaces. For problems with constant sized solution spaces our approximation is even exact. We further provide an extensive set of examples building on each other to introduce possible applications of our approximation method. After that we show that parameter optimisation on this approximate landscape yields as set of problem wide close to optimal parameters, resulting in a non iterative version of QAOA that matches standard QAOA.

We thus have two main contributions to build upon: The landscape approximation theorem, that opens up research on QAOA to the vast body of research in theoretical computer science on the structure of solution spaces. And second, a strong theoretical underpinning for non-iterative QAOA methods.

► BibTeX data

@article{Kruger2025outofloop,
  doi = {10.22331/q-2025-11-06-1903},
  url = {https://doi.org/10.22331/q-2025-11-06-1903},
  title = {Out of the {L}oop: {S}tructural {A}pproximation of {O}ptimisation {L}andscapes and non-{I}terative {Q}uantum {O}ptimisation},
  author = {Kr{\"{u}}ger, Tom and Mauerer, Wolfgang},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1903},
  month = nov,
  year = {2025}
}

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[2] Wolfgang Mauerer and Manuel Schönberger, Proceedings of the 3rd workshop on Quantum Computing and Quantum-Inspired Technology for Data-Intensive Systems and Applications 36 (2026) ISBN:9798400727030.

[3] Vincent Eichenseher, Maja Franz, Christian Wolff, and Wolfgang Mauerer, "Going off Pattern? QAOA Parameter Heuristics and Potentials of Parsimony", arXiv:2510.08153, (2025).

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