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Quantum Multiple Rotation Averaging

Shuteng Wang, Natacha Kuete Meli, Michael Möller, Vladislav Golyanik

TL;DR

This work introduces IQARS, the first end-to-end framework that solves multiple rotation averaging (MRA) on Ising machines by phrasing it as a sequence of local QUBO subproblems. It preserves the non-Euclidean $SO(3)$ geometry through Rodrigues-based exponential maps and a Taylor-linearized update in tangent space, while discretizing updates with a binary encoding suitable for quantum annealing. A posterior analysis protocol augments annealer outputs by Boltzmann-weighted aggregation over low-energy samples, improving solution fidelity. Empirical results on synthetic and real data show approximately 10–15% residual reduction over strong classical methods under modest hardware limits, highlighting a viable pathway for quantum-enhanced 3D vision as hardware scales.

Abstract

Multiple rotation averaging (MRA) is a fundamental optimization problem in 3D vision and robotics that aims to recover globally consistent absolute rotations from noisy relative measurements. Established classical methods, such as L1-IRLS and Shonan, face limitations including local minima susceptibility and reliance on convex relaxations that fail to preserve the exact manifold geometry, leading to reduced accuracy in high-noise scenarios. We introduce IQARS (Iterative Quantum Annealing for Rotation Synchronization), the first algorithm that reformulates MRA as a sequence of local quadratic non-convex sub-problems executable on quantum annealers after binarization, to leverage inherent hardware advantages. IQARS removes convex relaxation dependence and better preserves non-Euclidean rotation manifold geometry while leveraging quantum tunneling and parallelism for efficient solution space exploration. We evaluate IQARS's performance on synthetic and real-world datasets. While current annealers remain in their nascent phase and only support solving problems of limited scale with constrained performance, we observed that IQARS on D-Wave annealers can already achieve ca. 12% higher accuracy than Shonan, i.e., the best-performing classical method evaluated empirically.

Quantum Multiple Rotation Averaging

TL;DR

This work introduces IQARS, the first end-to-end framework that solves multiple rotation averaging (MRA) on Ising machines by phrasing it as a sequence of local QUBO subproblems. It preserves the non-Euclidean geometry through Rodrigues-based exponential maps and a Taylor-linearized update in tangent space, while discretizing updates with a binary encoding suitable for quantum annealing. A posterior analysis protocol augments annealer outputs by Boltzmann-weighted aggregation over low-energy samples, improving solution fidelity. Empirical results on synthetic and real data show approximately 10–15% residual reduction over strong classical methods under modest hardware limits, highlighting a viable pathway for quantum-enhanced 3D vision as hardware scales.

Abstract

Multiple rotation averaging (MRA) is a fundamental optimization problem in 3D vision and robotics that aims to recover globally consistent absolute rotations from noisy relative measurements. Established classical methods, such as L1-IRLS and Shonan, face limitations including local minima susceptibility and reliance on convex relaxations that fail to preserve the exact manifold geometry, leading to reduced accuracy in high-noise scenarios. We introduce IQARS (Iterative Quantum Annealing for Rotation Synchronization), the first algorithm that reformulates MRA as a sequence of local quadratic non-convex sub-problems executable on quantum annealers after binarization, to leverage inherent hardware advantages. IQARS removes convex relaxation dependence and better preserves non-Euclidean rotation manifold geometry while leveraging quantum tunneling and parallelism for efficient solution space exploration. We evaluate IQARS's performance on synthetic and real-world datasets. While current annealers remain in their nascent phase and only support solving problems of limited scale with constrained performance, we observed that IQARS on D-Wave annealers can already achieve ca. 12% higher accuracy than Shonan, i.e., the best-performing classical method evaluated empirically.
Paper Structure (28 sections, 4 theorems, 40 equations, 13 figures, 4 tables, 2 algorithms)

This paper contains 28 sections, 4 theorems, 40 equations, 13 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Review: Adiabatic Quantum Computing for Solving QUBO Problems
  4. Quantum MRA
  5. Taylor Linearization on $\text{SO}(3)$ Manifold
  6. Binary Encoding of Iterative Update
  7. Required Hardware Resource Analysis
  8. Algorithmic Protocol of IQARS
  9. Solution Refinement via Posterior Analysis
  10. Experiments
  11. Discussion and Conclusion
  12. Review: MRA
  13. Review: Quantum System Evolution during Annealing
  14. Review: Adiabatic Theorem
  15. Proof of Proposition 1
  16. ...and 13 more sections

Key Result

Proposition 1

Problem RA_final can be formulated as a quadratic optimization problem in matrix form: with $R = \left[ \right]^\top \in \mathbb{R}^{9N\times 1}$, where "$\mathrm{vec}(\cdot)$" denotes the vectorization of a matrix by stacking its columns, and ${Q} \in \mathbb{R}^{ 9N \times 9N}$ is the cost matrix:

Figures (13)

  • Figure 1: IQARS formulates MRA as QUBO sub-problems executable via quantum annealing to efficiently search for high-quality solutions. Estimated rotations are acquired from QUBO solutions once converged. Annealer's image is taken from hsu2013dwave.
  • Figure 2: Schematic local search behavior visualization under varying penalty strengths $\alpha$.
  • Figure 3: IQARS convergence behavior for synchronizing $N{=}10$ synthetic random noiseless rotations.
  • Figure 4: Visualization of IQARS's hyperparameter choice on the convergence behavior (MRA residual on log scale on y-axis).
  • Figure 5: Benchmark results of IQARS against other solvers for MRA on a synthetic noisy dataset across noise levels $\sigma$; ${R_{i}}_{GT}$ represents the ground truth of $R_i$.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • Remark 2
  • Remark 3
  • Proposition 3
  • proof
  • Remark 4
  • Theorem 1