Curvature-Aware Optimization of Noisy Variational Quantum Circuits via Weighted Projective Line Geometry

TL;DR

This paper introduces a geometric framework using weighted projective lines to model noise effects in variational quantum circuits, improving optimization stability and noise mitigation.

Contribution

It develops a differential-geometric approach with WPLs to characterize noise, enabling curvature-aware optimization and noise mitigation in quantum algorithms.

Findings

WPL geometry accurately captures noise-induced curvature deformation.

WPL-QNG stabilizes variational quantum eigensolver optimization.

Curvature-aware methods mitigate barren plateaus effectively.

Abstract

We develop a differential-geometric framework for variational quantum circuits in which noisy single- and multi-qubit parameter spaces are modeled by weighted projective lines (WPLs). Starting from the pure-state Bloch sphere CP1, we show that realistic hardware noise induces anisotropic contractions of the Bloch ball that can be represented by a pair of physically interpretable parameters (lambda_perp, lambda_parallel). These parameters determine a unique WPL metric g_WPL(a_over_b, b) whose scalar curvature is R = 2 / b^2, yielding a compact and channel-resolved geometric surrogate for the intrinsic information structure of noisy quantum circuits. We develop a tomography-to-geometry pipeline that extracts (lambda_perp, lambda_parallel) from hardware data and maps them to the WPL parameters (a_over_b, b, R). Experiments on IBM Quantum backends show that the resulting WPL geometries accurately capture anisotropic curvature deformation across calibration periods. Finally, we demonstrate that WPL-informed quantum natural gradients (WPL-QNG) provide stable optimization dynamics for noisy variational quantum eigensolvers and enable curvature-aware mitigation of barren plateaus.