Transfer learning of optimal QAOA parameters in combinatorial optimization

TL;DR

This work tackles the challenge of selecting QAOA parameters for combinatorial optimization by introducing transfer learning (TL) to reuse optimized parameters $\{\gamma_i,\beta_i\}_{i=0}^{p-1}$ across different problem instances. Using COBYLA to optimize on small instances ($N_q\leq 20$) with $p=10$, they evaluate transfer to larger instances ($N_q\leq 42$) and across COPs such as TSP, BPP, KP, PO, MIS, and MaxCut, finding that BPP provides the strongest generalization. The study shows that TL can maintain a ground-state success probability above the quadratic speedup over random guessing for several COPs and sizes, and demonstrates practical TL on multiple quantum hardware platforms as well as cross-platform TL to D-Wave, highlighting potential for reducing classical optimization overhead in near-term quantum computing. These findings imply that certain QAOA parameter landscapes exhibit cross-COP and cross-platform structure, motivating future HUBO extensions and broader TL strategies for quantum optimization.

Abstract

Solving combinatorial optimization problems (COPs) is a promising application of quantum computation, with the Quantum Approximate Optimization Algorithm (QAOA) being one of the most studied quantum algorithms for solving them. However, multiple factors make the parameter search of the QAOA a hard optimization problem. In this work, we study transfer learning (TL), a methodology to reuse pre-trained QAOA parameters of one problem instance into different COP instances. This methodology can be used to alleviate the necessity of classical optimization to find good parameters for individual problems. To this end, we select small cases of the traveling salesman problem (TSP), the bin packing problem (BPP), the knapsack problem (KP), the weighted maximum cut (MaxCut) problem, the maximal independent set (MIS) problem, and portfolio optimization (PO), and find optimal $β$ and $γ$ parameters for p layers. We compare how well the parameters found for one problem adapt to the others. Among the different problems, BPP is the one that produces the best transferable parameters, maintaining the probability of finding the optimal solution above a quadratic speedup over random guessing for problem sizes up to 42 qubits and p = 10 layers. Using the BPP parameters, we perform experiments on IonQ Harmony and Aria, Rigetti Aspen-M-3, and IBM Brisbane of MIS instances for up to 18 qubits. The results indicate that IonQ Aria yields the best overlap with the ideal probability distribution. Additionally, we show that cross-platform TL is possible using the D-Wave Advantage quantum annealer with the parameters found for BPP. We show an improvement in performance compared to the default protocols for MIS with up to 170 qubits. Our results suggest that there are QAOA parameters that generalize well for different COPs and annealing protocols.

Figures (10)

• Figure 1: Example of the TL methodology for transferring the parameters from BPP to MIS. (a) Quantum annealing initialization of the QAOA parameters for $p=10$ layers using the BPP, (b) self-optimization step using QAOA, (c) final $\beta$ and $\gamma$ parameters transferred to the MIS problem.
• Figure 2: D-Wave modified schedule example. The circles represent the modified point in the realtion between $s_i$ and $t_i$.
• Figure 3: Example for the QAOA parameter optimization of the BPP. (a) Quantum annealing initialization of the QAOA parameters for $p=10$ layers, and (b) final $\beta_i$ and $\gamma_i$ for $i=0, ..., p-1$ angles for the BPP with 3 items (12 qubits).
• Figure 4: (a)-(b)Classical optimization of the cost function value at each iteration for a random 12 qubit (3 items) BPP problem and 12 nodes MaxCut problem. (a) BPP (b) MaxCut. The inset plot shows the final parameters found for each problem. The maximum number of iterations is 2400. (c) MaxCut cost function versus the number of iterations for the classical optimization for problem sizes 10 and 14 qubits. Different lines represent the different seeds.
• Figure 5: Comparison between TL (solid line) and self-optimization (dashed line) for different COPs (see legend). Here, each marker represents the mean value over 5 random cases. The dashed lines with small markers represent the problems optimized using the procedure in Sec. \ref{['Sec:QUBO']}, and the solid lines with big markers represent the results of applying TL from the BPP. The quadratic speedup over random guessing (black dotted line) is presented as a guiding line.