Scaling QAOA: transferring optimal adiabatic schedules from small-scale to large-scale variational circuits

TL;DR

This work addresses the scalability bottleneck of QAOA by learning a spectral-gap informed adiabatic schedule from small instances and transferring it to larger problems. The schedule is mapped to a discrete QAOA circuit via a trotterized approximation, collapsing the optimization over $2p$ angles to just two global hyperparameters $\kappa$ and $q$ with derivatives governed by $\dfrac{ds}{dt}=\kappa g^q(s)$. By normalizing problem coefficients and fitting gap profiles (mean/median) with Bézier curves, the authors obtain closed-form angle expressions and demonstrate effective transfer to larger instances for MaxCut and random QUBOs, often outperforming vanilla QAOA at comparable depths. The approach reduces classical overhead, mitigates barren plateaus, and offers a principled inductive strategy for variational quantum algorithms in near-term devices.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading approach for combinatorial optimization on near-term quantum devices, yet its scalability is limited by the difficulty of optimizing \(2p\) variational parameters for a large number \(p\) of layers. Recent empirical studies indicate that optimal QAOA angles exhibit concentration and transferability across problem sizes. Leveraging this observation, we propose a schedule-learning framework that transfers spectral-gap-informed adiabatic control strategies from small-scale instances to larger systems. Our method extracts the spectral gap profile of small problems and constructs a continuous schedule governed by \(\partial_t s = κg^q(s)\), where \(g(s)\) is the instantaneous gap and \((κ, q)\) are global hyperparameters. Discretizing this schedule yields closed-form expressions for all QAOA angles, reducing the classical optimization task from \(2p\) parameters to only \(2\), independent of circuit depth. This drastic parameter compression mitigates classical optimization overhead and reduces sensitivity to barren plateau phenomena. Numerical simulations on random QUBO and 3-regular MaxCut instances demonstrate that the learnt schedules transfer effectively to larger systems while achieving competitive approximation ratios. Our results suggest that gap-informed schedule transfers provide a scalable and parameter-efficient strategy for QAOA.

Figures (6)

• Figure 1: Sampled instantaneous gaps for $n=10$ qubits on 1000 QUBOs with coefficients randomly drawn in [-1,1] (left) and the final gap distribution (right).
• Figure 2: Bézier fitting of the mean (degree 3) and median (degree 7) sampled gaps obtained in the learning phase.
• Figure 3: Experimental results of our heuristic and QAOA on 100 instances of 3-regular unweighted MaxCut problems of size $n=20$. Showing the approximation ratios (upper) and optimized hyperparameters obtained via classical optimization (lower). The learning part of our heuristic is done on random QUBOs of size $n=10$ and the classical optimization is done for $2$ (our heuristic) and $2p$ (QAOA) hyperparameters.
• Figure 4: Experimental results of our heuristic and QAOA on 100 instances of 3-regular weighted MaxCut problems of size $n=20$. Showing the approximation ratios (upper) and optimized hyperparameters obtained via classical optimization (lower). The learning part of our heuristic is done on random QUBOs of size $n=10$ and the classical optimization is done for $2$ (our heuristic) and $2p$ (QAOA) hyperparameters.
• Figure 5: Experimental results of our heuristic and QAOA on 100 instances of random QUBO problems of size $n=20$. Showing the approximation ratios (upper) and optimized hyperparameters obtained via classical optimization (lower). The learning part of our heuristic is done on random QUBOs of size $n=10$ and the classical optimization is done for $2$ (our heuristic) and $2p$ (QAOA) hyperparameters.