Iterative Interpolation Schedules for Quantum Approximate Optimization Algorithm

TL;DR

The paper tackles the challenge of scaling QAOA parameter optimization to high depth, where there are $2p$ parameters to tune as the number of layers $p$ grows. It introduces Iterative Interpolation (II), an approach that expresses QAOA schedules as smooth functions in an orthonormal basis, optimizing a small set of coefficients and progressively increasing both depth $p$ and the number of active coefficients $C$. Compared with prior Fourier-based interpolation, II yields faster convergence and higher-quality schedules across SK, LABS, and portfolio optimization, enabling $p$ values well into the thousands for LABS and revealing a mild, polynomial-depth growth for SK but exponential scaling for LABS. This framework opens the door to exploring high-depth QAOA regimes and provides practical insights into problem-dependent scaling, with potential extensions to other continuous-time quantum optimization settings.

Abstract

Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum optimization heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA solves optimization problems using a parameterized circuit with $p$ layers, with higher $p$ leading to better solutions. Existing methods require optimizing $2p$ independent parameters which is challenging for large $p$. In this work, we present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and the number of coefficients until convergence, our approach enables construction of high-quality schedules for large $p$. We demonstrate our method achieves better performance with fewer optimization steps than current approaches on three problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. As an application of our technique, we observe a mild growth of QAOA depth sufficient to solve SK model exactly, a result of independent interest.

Figures (7)

• Figure 1: As $p$ grows, QAOA TTS decreases despite increasing circuit depth. Here TTS is the total number of QAOA layers that must be executed in expectation to solve the LABS problem exactly. Data reproduced from Ref. shaydulin2023evidence.
• Figure 2: Building blocks of Iterative Interpolation. a) The parameter schedules vary smoothly with QAOA layer index. b) The coefficients of parameter schedules in the Chebyshev basis decay rapidly, showing that the first few modes have the largest contribution. c) QAOA performance with parameters reconstructed using only the first 12 coefficients is similar to that with the original schedule.
• Figure 3: Performance of II in comparison to the Fourier method. Across different problem sizes considered, II performs significantly better when compared to the Fourier method. a) SK model: II achieves $50\%$ overlap with the ground state using $3.5\times$ fewer total layers. b) Portfolio optimization: II consistently reaches approximation ratios $\geq 0.9$ with fewer total layers than Fourier. c) LABS: II achieves better approximation ratios ($\simeq 0.95$) while Fourier plateaus below $0.74$, despite using substantially fewer total layers. The values annotated in Figs. a) and b) correspond to the median values across different seeds for each problem instance.
• Figure 4: II obtains the optimal values for LABS. a) The comparison across different QAOA parameter schedules for solving the LABS problem. The II schedule attains the optimal MFs for the $N$ values considered, while the Fourier and Linear schedules fail to do so. b) The II schedule corresponding to $N=25$, a $p=1005$ and AR$=0.965$. c) The impact of the basis function choice on performance. Although the choice of basis can be optimized for each instance, the overall performance across different $N$ remains comparable for both choices.
• Figure 5: The depth $p$ required to achieve a fixed overlap with the optimal state for the SK model (a) and LABS (b). For the SK model, we fit the depth to a polynomial function in $N$ and report a high goodness-of-fit. However, we are not able to make rigorous claims about scaling. For LABS, the depth appears to grow exponentially as a function of $N$ for the LABS problem (b). For the SK model (a), the growth of $p$ appears slower.