Preconditioning Natural and Second Order Gradient Descent in Quantum Optimization: A Performance Benchmark

TL;DR

This paper investigates how preconditioning and second-order information affect optimization of variational quantum circuits in QAOA for MaxCut. It benchmarks quasi-Newton and quantum natural gradient methods on shallow QAOA with Bayesian hyperparameter tuning, and introduces SP-BFGS to stabilize Hessian updates under gradient noise. The results show DFP and SP-BFGS offer the best convergence and robustness among quasi-Newton methods, while QNG variants converge quickly at small scales but suffer from QFIM noise at larger sizes; stochastic second-order methods require careful tuning and may not beat first-order baselines. The work provides practical guidance on optimizer choice for VQAs and highlights avenues for improving QFIM-based preconditioning and secant-penalization strategies.

Abstract

The optimization of parametric quantum circuits is technically hindered by three major obstacles: the non-convex nature of the objective function, noisy gradient evaluations, and the presence of barren plateaus. As a result, the selection of classical optimizer becomes a critical factor in assessing and exploiting quantum-classical applications. One promising approach to tackle these challenges involves incorporating curvature information into the parameter update. The most prominent methods in this field are quasi-Newton and quantum natural gradient methods, which can facilitate faster convergence compared to first-order approaches. Second order methods however exhibit a significant trade-off between computational cost and accuracy, as well as heightened sensitivity to noise. This study evaluates the performance of three families of optimizers on synthetically generated MaxCut problems on a shallow QAOA algorithm. To address noise sensitivity and iteration cost, we demonstrate that incorporating secant-penalization in the BFGS update rule (SP-BFGS) yields improved outcomes for QAOA optimization problems, introducing a novel approach to stabilizing BFGS updates against gradient noise.

Figures (35)

• Figure 1: Example of iterative Bayesian process using a Gaussian kernel. Objective function of the form $f(x) =(x - 2)^2 + 1$. In practice the objective function is not known, it is only displayed for illustration. Edge acquisition policy and GP + simple Matèrn kernel
• Figure 2: 3 nodes problem (left) and 7 node problem (right) PCA analysis in a grid exploration. Increasing the problem size drastically increases rugosity and local minima, central symmetry due to symmetries in parametrization of Pauli gates.
• Figure 3: For 300 grid discretization, examples of two randomly picked directions in 5 nodes problem App. \ref{['sec:problem_def']}, defining two random parameterized lines. Two random directions define a plan in the hyper-space in which the objective function is defined, parameters $\alpha$ and $\beta$ are unique parameterization of each random lines.
• Figure 4: Each method normalized on their averages. We find three families of convergence behaviors: first are the skewd distributions: BFGS, SR1 and NCG. Well-behaved leptokurtic distributions (peaked) with SP-BFGS, DFP, QNG(block and diagonal), qBroyden and m-QNG. Last behavior are platykurtic distributions (flat) with qBang.
• Figure 5: Convergence analysis, quasi-Newton types methods for benchmark problem, top figure problem with 3 nodes, low figure problem with 5 nodes of seed=32, 50 experiments over randomly sampled initial conditions. 512 shots, 60 iterations