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.
We formulate MRA as a global synchronization problem on the Lie group \( SO(3) \). Given noisy relative rotations \( \tilde{R}_{ij} \in SO(3) \) between camera pairs \( (i,j) \), the goal is to recover globally consistent absolute rotations \( \{R_i\}_{i=1}^{N} \), where \( N \) denotes the number of cameras. Using the squared Frobenius (chordal) distance, the MRA objective is \[ \min_{R_1,\dots,R_N \in SO(3)} \sum_{(i,j)} \| \tilde{R}_{ij} R_i - R_j \|_F^2 , \] where \( \|\cdot\|_F \) denotes the Frobenius norm. Instead of solving this non-convex problem directly on the manifold, we perform optimization in the tangent space of \( SO(3) \) using the exponential map. Each rotation matrix is parameterized as \( R_i = \exp(\mathcal{R}(v_i)) \), where \( v_i \in \mathbb{R}^3 \) is a Lie algebra vector in \( \mathfrak{so}(3) \), and \( \mathcal{R}(\cdot) : \mathbb{R}^3 \rightarrow \mathfrak{so}(3) \) denotes the skew-symmetric operator that maps a vector to its corresponding matrix representation. At iteration \( k \), the nonlinear rotation mapping is locally approximated using a first-order Taylor expansion in the tangent space. This yields a quadratic objective in the update variable \( \Delta v \in \mathbb{R}^{3N} \): \[ \min_{\Delta v} \; \Delta v^\top \hat{Q}\,\Delta v + \hat{c}^\top \Delta v , \] where \( \hat{Q} \) is the local quadratic coefficient matrix and \( \hat{c} \) is the linear term obtained from the linearization. To enable execution on quantum annealers, the continuous update vector \( \Delta v \) is discretized within a bounded search window \( \|\Delta v\|_\infty \). Binary encoding transforms the discretized variables into a vector \( q \in \{0,1\}^{3Nm} \), where \( m \) is the number of qubits used per dimension for discretization. This results in a quadratic unconstrained binary optimization (QUBO) problem \[ \min_{q \in \{0,1\}^{3Nm}} \; q^\top \tilde{Q}\,q + \tilde{c}^\top q . \] The proposed IQARS algorithm solves these local QUBO subproblems sequentially on a quantum annealer and updates the tangent variables via \( v^{k+1} = v^k + \Delta v^{k+1} \).
We analyze the convergence behavior of IQARS during iterative optimization executed on quantum annealing hardware. At iteration \( k \), the algorithm constructs a local QUBO approximation of the rotation-averaging objective; \( R(v^k) \) represents the corresponding set of rotations obtained through the exponential map. Convergence is evaluated using four complementary metrics: (1) the sampled QUBO energy spectrum \( \{E_m\} \) returned by the annealer, where \( E_m \) denotes the \( m \)-th lowest energy; (2) neighboring energy gaps \( E_{m+1} - E_m \); (3) the rotation-synchronization residual \( \| \tilde{R}_{ij} R_i - R_j \|_F \), and (4) the rotation-update magnitude \( \| R(v^{k+1}) - R(v^k) \|_F \). Empirically, IQARS exhibits stable convergence behavior on annealing hardware. The synchronization residual decreases consistently across iterations while the neighboring energy gaps remain well separated, indicating stable local optimization dynamics despite hardware noise and coarse discretization. These observations support the effectiveness of the iterative quantum-local search strategy: sequential QUBO refinement combined with exponential-map updates maintains the \( SO(3) \) manifold structure while enabling annealing-based exploration of the non-convex MRA objective.
We evaluate IQARS on synthetic rotation-averaging problems with controlled measurement noise. Starting from ground-truth camera rotations, noisy relative rotations are generated by applying random perturbations on the SO(3) manifold. We compare IQARS with classical rotation-averaging solvers with synchronization residual averaged over all rotation measurements. As shown in the figure below, IQARS consistently produces lower residuals across a wide range of noise levels, indicating improved robustness to noisy relative-rotation observations.
We integrate our proposed IQARS algorithm into a global SfM workflow to replace the conventional rotation synchronization module. Specifically, IQARS is used to synchronize camera rotations prior to global pose estimation and dense reconstruction. Experiments on standard multi-view benchmarks demonstrate that the recovered camera orientations enable accurate 3D geometry reconstruction, producing visually consistent meshes and stable camera alignment even under measurement noise. These results highlight that quantum-annealing-based rotation synchronization can serve as a practical component within modern SfM systems, bridging quantum optimization techniques with real-world 3D vision pipelines.
@inproceedings{Wang2026quantum,
title={Quantum Multiple Rotation Averaging},
author={Wang, Shuteng and Kuete Meli, Natacha and Möller, Michael and Golyanik, Vladislav},
booktitle={International Conference on 3D Vision},
year={2026}
}