Interpolation-based coordinate descent method for parameterized quantum circuits

Abstract

Parameterized quantum circuits (PQCs) are fundamental to many hybrid quantum-classical algorithms. However, existing structure-based optimizers for the training of PQCs, such as Rotosolve and sequential minimal optimization, rely on heuristic node selection or ignore statistical noise, which limits their robustness and accuracy. To address this issue, we propose an interpolation-based coordinate descent (ICD) method as a unified framework for all structure-based optimizers. ICD approximates the cost function through interpolation, recovers its trigonometric structure, and performs global one-dimensional updates on individual parameters. Unlike previous methods, ICD derives optimal interpolation nodes that minimize statistical errors from measurements. For the common case of r equidistant frequencies, we prove that equidistant nodes with spacing 2π/(2r + 1) jointly minimize the mean squared error of Fourier coefficient estimates, the condition number of the interpolation matrix, and the average variance of the approximated cost function. Numerical experiments confirm the superior robustness and efficiency of ICD over gradient-based methods. Parameterized quantum circuits are a common tool in variational quantum algorithms and quantum machine learning. The authors design an interpolation-based coordinate descent method that reconstructs the cost landscape from a few circuit runs and achieves more efficient training than standard gradient and coordinate descent methods in our numerical tests.