Pattern or Not? QAOA Parameter Heuristics and Potentials of Parsimony

TL;DR

This study systematically interrogates how classical parameters influence QAOA performance on NP-complete problems (MaxCut, VertexCover, Max3SAT) using large-scale simulations and a simple sequential parameter initialisation. It demonstrates that optimal parameters often deviate from previously proposed patterns, especially at higher depths, and that the energy landscape becomes increasingly gamma-invariant as beta→0, reducing sensitivity to gamma. Among parameter strategies, iterative fixed-parameter expansion is a fast, robust baseline, while linear ramps and fully optimised schedules can outperform it at larger depths under suitable initialisation. The findings suggest practical pathways to robust QAOA deployment on NISQ devices, including instance-aware initialisation and depth-aware optimisation schedules, while highlighting that slower convergence can yield better long-depth performance. Overall, the work provides nuanced insights into when and why QAOA parameter patterns hold, guiding more resilient parametrisation strategies for near-term quantum optimization.

Abstract

Structured variational quantum algorithms such as the Quantum Approximate Optimisation Algorithm (QAOA) have emerged as leading candidates for exploiting advantages of near-term quantum hardware. They interlace classical computation, in particular optimisation of variational parameters, with quantum-specific routines, and combine problem-specific advantages -- sometimes even provable -- with adaptability to the constraints of noisy, intermediate-scale quantum (NISQ) devices. While circuit depth can be parametrically increased and is known to improve performance in an ideal (noiseless) setting, on realistic hardware greater depth exacerbates noise: The overall quality of results depends critically on both, variational parameters and circuit depth. Although identifying optimal parameters is NP-hard, prior work has suggested that they may exhibit regular, predictable patterns for increasingly deep circuits and depending on the studied class of problems. In this work, we systematically investigate the role of classical parameters in QAOA performance through extensive numerical simulations and suggest a simple, yet effective heuristic scheme to find good parameters for low-depth circuits. Our results demonstrate that: (i) optimal parameters often deviate substantially from expected patterns; (ii) QAOA performance becomes progressively less sensitive to specific parameter choices as depth increases; and (iii) iterative component-wise fixing performs on par with, and at shallow depth may even outperform, several established parameter-selection strategies. We identify conditions under which structured parameter patterns emerge, and when deviations from the patterns warrant further consideration. These insights for low-depth circuits may inform more robust pathways to harnessing QAOA in realistic quantum compute scenarios.

Figures (26)

• Figure 1: Proposed sequential parameter initialisation method.
• Figure 2: Symmetries specific to MaxCut (lhs) and general QAOA (rhs) illustrated by a single 16-vertex, 3-regular instance of MaxCut and a single 16 qubit Max3SAT instance with $\alpha=4.\overline{3}$. Top rows: Overall optimisation landscape $F_{p}(\gamma, \beta)$ with $\gamma,\beta\in[-\pi,\pi]$ and symmetry axes marked in orange. Bottom rows: Restriction to the subset $\gamma\in[-\frac{\pi}{2},\frac{\pi}{2}]$, $\beta\in[-\frac{\pi}{4},\frac{\pi}{4}]$ for MaxCut and $\gamma\in[-\pi,\pi]$, $\beta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ in general that captures all information. The landscapes are obtained by the sequential method.
• Figure 3: Average residual energy $\bar{r}$ for 40 3-regular MaxCut instances of sizes 10 to 16, with $\gamma,\beta$ set to sequentially fixed parameters (top row), optimised parameters using COBYLA starting from the sequential parameters (second row), $\text{LR}_{-\beta}\xspace$ parameters with $\Delta_{\beta}=-0.3 , \Delta_{\gamma}=0.6$ (third row), and $\text{LR}_{+\beta}\xspace$ parameters with $\Delta_{\beta}=0.3 , \Delta_{\gamma}=0.6$ (bottom row).
• Figure 4: Standard deviation of the residual energy $\bar{r}$ in \ref{['fig:avg_parameter_scan_maxcut_symmetry']} over 40 MaxCut instances.
• Figure 5: Average and standard deviation of the approximation quality for 40 3-regular MaxCut instances of sizes 10 to 16 when fixing parameters at each layer $p$ according to the considered methods.