Improving the efficiency of QAOA using efficient parameter transfer initialization and targeted-single-layer regularized optimization with minimal performance degradation

TL;DR

This study addresses the inefficiency of optimizing QAOA for MaxCut by combining parameter transfer initialization from small, donor graphs with Trotterized Quantum Annealing and targeted-single-layer optimization. By transferring optimized parameters to larger, similar-graph instances and optimizing a single QAOA layer, the approach achieves substantial speedups while maintaining near-optimal performance in unweighted graphs; weighted graphs show more variable behavior, which can be mitigated with regularization. L2 (ridge) regularization is demonstrated to smooth the optimization landscape and reduce instances where full optimization outperforms selective optimization. Overall, the work provides a practical framework for efficiently deploying QAOA on diverse graph families with quantifiable trade-offs between computation time and solution quality.

Abstract

Quantum approximate optimization algorithm (QAOA) have promising applications in combinatorial optimization problems (COPs). We investigated the MaxCut problem in three different families of graphs using QAOA ansats with parameter transfer initialization followed by targeted single layer optimization. For 3 regular (3R), Erdos Renyi (ER), and Barabasi Albert (BA) graphs, the parameter transfer approach achieved mean approximation ratios of 0.9443 for targeted-single layer optimization as compared to 0.9551 of full optimization. It represents 98.88 percent optimal performance, with 8.06 times computational speedup in unweighted graphs. But, in weighted graph families, optimal performance is relatively low (less than 90 percent) for higher nodes graph, suggesting parameter transfer followed by targeted-single-layer optimization is not ideal for weighted graph families, however, we find that for some weighted families (weighted 3-regular) this approach works perfectly. In 8.92 percent test cases, targeted single layer optimization outperformed the full optimization, indicating that complex parameter landscape can trap full optimization in sub-optimal local minima. To mitigate this inconsistency, ridge (L2) regularization is used to smoothen the solution landscape, which helps the optimizer to find better optimum parameters during full optimization and reduces these inconsistent test cases from 8.92 percent to 3.81 percent. This work demonstrates that efficient parameter initialization and targeted-single-layer optimization can improve the efficiency of QAOA with minimal performance degradation.

Figures (13)

• Figure 1: Schematic diagram demonstrating working of QAOA with parameters transfer initialization from a simple instance $G_1$($n_d$ nodes) to more complex instance $G_2$($n_a\geq n_d$) followed by targeted-single layer ($k^{th}$ layer) optimization with $L_2$ regularization.
• Figure 2: Parameters solution space: (a)for $10n-u3R$ and (b) for $16n-u3R$ graph using $p=1$ QAOA ansatz.
• Figure 3: Clustering of parameters: (a) Plot of optimal $\gamma^{*}_i$ as a function of increasing layers $p$ for similar instances of u3R family.(b) Similar plot for $\beta^{*}_{i}$.
• Figure 4: $L_2$ Regularization: Approximation ratios for $20n-u3R$ graph using various optimization techniques following parameters transfer initialization from $8n-u3R$ graph: (a) representing complex landscape trapping the full optimization, causing $r_s\geq r_f$, when any one of layers $5\ \rm{or}\ 7$ is selectively optimized. This problem is mitigated in (b) using $L_2$-Regularization which makes the parameter solution space smooth and improve the full optimization accuracy ($r_f$). Here $r_f,r_s$, and $r_n$ are approximation ratios of acceptor graph ($20n-u3R$), obtained by full, targeted-selective, and no optimizations respectively. Here regularization strength is taken as, $\lambda = 0.0001$
• Figure 5: Optimal target layer: Figure representing heat-maps to find the probability $P(p_k:r_k>r_i)$ of targeted layer $p_k$ providing best optimum cut ($r_k>r_i$) for (a) u3R, (b)Unweighted Barabási–Albert (uBA) graph, (c)Unweighted Erdős–Rényi (uER) graph, and (d) w3R graph families.