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

Gunhee Cho, Jessie Wang, Angela Yue

TL;DR

This work introduces a curvature-aware framework for noisy variational quantum circuits by modeling the noisy single-qubit parameter space as a weighted projective line (WPL) with constant curvature R=2/b^2 and anisotropy encoded in a/b, derived from two principal Bloch-ball contractions (\lambda_{\perp},\lambda_{\parallel}). It provides a tomography-to-WPL pipeline to obtain (a/b,b,R) from minimal idle-channel tomography, validates the approach on IBM hardware, and develops WPL-based quantum natural gradients (WPL-QNG) that stabilize optimization and mitigate barren plateaus under drift. The framework extends to multi-qubit settings via product-orbifold geometry, yielding a block-diagonal QFIM that can be efficiently inverted with Moore–Penrose preconditioning and curvature-aware step sizes. Empirically, WPL curvature remains stable across calibration windows, while anisotropy captures device-specific decoherence signatures; in VQE tasks, WPL-QNG achieves superior convergence and drift robustness compared to Euclidean GD and Bloch-sphere QNG. Overall, curvature becomes a measurable resource that links hardware tomography, differential geometry, and efficient, noise-aware quantum optimization for NISQ devices.

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.

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

TL;DR

This work introduces a curvature-aware framework for noisy variational quantum circuits by modeling the noisy single-qubit parameter space as a weighted projective line (WPL) with constant curvature R=2/b^2 and anisotropy encoded in a/b, derived from two principal Bloch-ball contractions (\lambda_{\perp},\lambda_{\parallel}). It provides a tomography-to-WPL pipeline to obtain (a/b,b,R) from minimal idle-channel tomography, validates the approach on IBM hardware, and develops WPL-based quantum natural gradients (WPL-QNG) that stabilize optimization and mitigate barren plateaus under drift. The framework extends to multi-qubit settings via product-orbifold geometry, yielding a block-diagonal QFIM that can be efficiently inverted with Moore–Penrose preconditioning and curvature-aware step sizes. Empirically, WPL curvature remains stable across calibration windows, while anisotropy captures device-specific decoherence signatures; in VQE tasks, WPL-QNG achieves superior convergence and drift robustness compared to Euclidean GD and Bloch-sphere QNG. Overall, curvature becomes a measurable resource that links hardware tomography, differential geometry, and efficient, noise-aware quantum optimization for NISQ devices.

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.

Paper Structure

This paper contains 66 sections, 8 theorems, 111 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation: noise deforms variational geometry
  3. From Bloch sphere symmetry breaking to weighted projective lines
  4. Contributions
  5. Organization of the paper
  6. Single-Qubit Geometry: From Bloch Sphere to WPL
  7. Pure-state Bloch sphere and quantum Fisher metric
  8. Definition of the weighted projective line P(a,b;kappa)
  9. WPL metric and proof that R=2/b^2
  10. Orbifold structure and cone singularities
  11. Consequences for quantum natural gradient
  12. From Channels to Geometry: Map (lambda⊥,lambda∥)→(a/b,b,R)
  13. Axis-aligned CPTP channels and Bloch maps
  14. Principal contractions and identifiability
  15. Model-to-data identification of WPL parameters
  16. ...and 51 more sections

Key Result

Proposition 2.1

On the smooth locus of $\mathbb{P}(a,b;\kappa)$, the scalar curvature of the WPL metric is

Figures (3)

  • Figure 1: VQE convergence under noisy channels. (a) WPL-QNG exhibits stable and fast reduction of energy error under static noise, whereas Bloch-QNG oscillates and Euclid-GD converges slowly. (b) Under synthetic drift, curvature tracking and WPL-based preconditioning stabilize convergence relative to static geometry.
  • Figure A1: Drift-robustness diagnostics for WPL-QNG.
  • Figure A2: Ablation studies for WPL-QNG under noisy VQE.

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 3.1: Transversal symmetry of phase-covariant channels
  • proof
  • Remark 3.2
  • Proposition 3.3: Axis alignment from degenerate singular values
  • ...and 14 more