Quantum Approximate Optimisation Algorithms (QAOA) hold immense promise for tackling complex optimisation problems, but their performance is often hampered by the challenges of optimising numerous parameters. Phuong‑Nam Nguyen, from Viettel High Technology Industries Corporation, alongside colleagues, present a novel framework called LOTUS (Layer‑Ordered Temporally Unified Schedules) to address this issue. LOTUS reimagines the optimisation process, transforming a chaotic, high‑dimensional search into a more manageable, low‑dimensional dynamical system through a Hybrid Fourier‑Autoregressive (HFA) mapping. This approach not only delivers consistently superior results—exceeding the performance of established optimisers such as L‑BFGS‑B and COBYLA—but also significantly reduces computational demands by requiring fewer optimisation iterations than conventional methods. The research represents a substantial step towards realising the full potential of QAOA for real‑world applications.

The study pinpointed pathological behaviours within standard QAOA optimisation landscapes, specifically discrete piecewise‑constant transitions in parameter schedules and a tendency for parameters to swap roles between layers due to inherent permutation symmetry. Experiments revealed these issues were amplified when scaling the number of qubits or increasing circuit depth, leading to unstable parameters and a performance gap compared with functionally‑defined schedules.

To overcome these challenges, the authors engineered a Hybrid Fourier‑Autoregressive (HFA) mapping, replacing independent layer‑wise angles with a unified parameterisation. This innovative approach enforces global temporal coherence within the QAOA circuit while preserving local flexibility, effectively breaking the permutation symmetry that hinders traditional optimisers.

Key findings include:

• Dimensionality collapse: LOTUS reduces optimisation complexity to O(1) relative to circuit depth, enabling the training of deeper circuits without the exponential performance degradation typical of classical optimisers.

• Performance gains: Benchmarking shows LOTUS consistently outperforms standard optimisers, achieving up to 27.2 % improvement in expectation values versus L‑BFGS‑B and 20.8 % improvement over COBYLA.

• Iteration reduction: LOTUS requires over 90 % fewer iterations than algorithms such as Powell or SLSQP, and more than 86 % fewer than TNC and L‑BFGS‑B, dramatically lowering computational cost.

• Depth transferability: Schedules optimised at lower depths serve as effective initialisations for deeper circuits, further accelerating training.

These results demonstrate that replacing independent layer‑wise parameters with an HFA mapping enforces global temporal coherence while retaining local flexibility. Across a range of test cases, LOTUS achieved a remarkable 27.2 % improvement in expectation values compared with L‑BFGS‑B and over 26 % improvement versus both TNC and SLSQP. Convergence was reached with 93.3 % fewer iterations than the Powell method.

The reduction in computational cost is critical for tackling larger, more complex problems that are currently intractable for conventional optimisation techniques. By collapsing the search space to a low‑dimensional dynamical system, LOTUS provides a robust pathway for scaling QAOA toward utility‑scale quantum advantage, offering a superior balance between solution quality and efficiency.

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Additional Information

Paper: “LOTUS: Layer‑ordered Temporally Unified Schedules For Quantum Approximate Optimization Algorithms”

arXiv: https://arxiv.org/abs/2601.07851

The authors note that the current work applies the HFA ansatz exclusively to the MaxCut problem; future research will explore its efficacy on other NP‑hard challenges such as MaxSAT, TSP, and QUBO.