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LOTUS: Layer-ordered Temporally Unified Schedules For Quantum Approximate Optimization Algorithms

Phuong-Nam Nguyen

TL;DR

QAOA often suffers from high-dimensional, layer-wise independent parameter optimization that leads to barren plateaus and permutation-symmetry ambiguities as depth grows. LOTUS replaces the conventional $2p$ independent angles with a low-dimensional Hybrid Fourier–Autoregressive (HFA) generator to produce temporally coherent schedules, effectively collapsing the optimization dimensionality to $O(1)$ in depth and breaking permutation symmetry. The approach yields improved solution quality and reduced training costs on weighted MaxCut benchmarks and demonstrates depth-transfer capabilities, enabling scalable training of deeper circuits. This framework provides a practical path toward utility-scale quantum advantage by combining global spectral structure with local adaptability in QAOA schedules.

Abstract

In this paper, we introduce LOTUS (Layer-Ordered Temporally-Unified Schedules), which is a framework that restructures QAOA from a high-dimensional, chaotic search into a low-dimensional dynamical system. By replacing independent layer-wise angles with a Hybrid Fourier-Autoregressive (HFA) mapping, LOTUS enforces global temporal coherence while maintaining local flexibility. LOTUS consistently outperforms standard optimizers, achieving up to a $27.2\%$ improvement in expectation values over L-BFGS-B and $20.8\%$ compared with COBYLA. Besides, our proposed method drastically reduces computational costs, requiring over $90\%$ fewer iterations than methods like Powell or SLSQP.

LOTUS: Layer-ordered Temporally Unified Schedules For Quantum Approximate Optimization Algorithms

TL;DR

QAOA often suffers from high-dimensional, layer-wise independent parameter optimization that leads to barren plateaus and permutation-symmetry ambiguities as depth grows. LOTUS replaces the conventional independent angles with a low-dimensional Hybrid Fourier–Autoregressive (HFA) generator to produce temporally coherent schedules, effectively collapsing the optimization dimensionality to in depth and breaking permutation symmetry. The approach yields improved solution quality and reduced training costs on weighted MaxCut benchmarks and demonstrates depth-transfer capabilities, enabling scalable training of deeper circuits. This framework provides a practical path toward utility-scale quantum advantage by combining global spectral structure with local adaptability in QAOA schedules.

Abstract

In this paper, we introduce LOTUS (Layer-Ordered Temporally-Unified Schedules), which is a framework that restructures QAOA from a high-dimensional, chaotic search into a low-dimensional dynamical system. By replacing independent layer-wise angles with a Hybrid Fourier-Autoregressive (HFA) mapping, LOTUS enforces global temporal coherence while maintaining local flexibility. LOTUS consistently outperforms standard optimizers, achieving up to a improvement in expectation values over L-BFGS-B and compared with COBYLA. Besides, our proposed method drastically reduces computational costs, requiring over fewer iterations than methods like Powell or SLSQP.
Paper Structure (38 sections, 3 theorems, 9 equations, 5 figures, 2 algorithms)

This paper contains 38 sections, 3 theorems, 9 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Backgrounds:
  3. Key observations:
  4. Contributions:
  5. Results:
  6. Methods
  7. Framework overview
  8. Framework details
  9. From discrete pathologies to continuous dynamics:
  10. The hybrid Fourier-autoregressive (HFA) ansatz:
  11. Optimization space:
  12. Algorithms:
  13. Theoretical analysis
  14. Definitions
  15. Key theoretical Results
  16. ...and 23 more sections

Key Result

Theorem 3.1

For any depth $p$, the ratio of the HFA parameter space to the standard space vanishes as $p \to \infty$:

Figures (5)

  • Figure 1: Performance benchmarking of the LOTUS framework against traditional optimizers (L-BFGS-B, SLSQP, Powell, COBYLA, and TNC) for weighted MaxCut, showing significant percentage improvements in both expectation values (solution quality) and total iteration counts (computational efficiency)
  • Figure 2: Comparative results between LOTUS and other optimizers
  • Figure 3: Preliminary results of QAOA training with existing optimizers
  • Figure 4: Experimental results of LOTUS using $8$ qubits and $24$-layered QAOA
  • Figure 5: Experimental results of LOTUS using $12$ qubits and $24$-layered QAOA

Theorems & Definitions (7)

  • Definition 3.1: Circuit Depth and Layer Indexing
  • Definition 3.2: Standard QAOA Parameter Space
  • Definition 3.3: LOTUS HFA Ansatz
  • Definition 3.4: QAOA Cost Function
  • Theorem 3.1
  • Theorem 3.2: Layer-wise Lipschitz continuity
  • Theorem 3.3: Depth Transfer Invariance