Quantum Levenberg--Marquardt Algorithm for optimization in Bundle Adjustment

TL;DR

This paper introduces a quantum algorithm to solve the bundle adjustment optimization problem more efficiently, demonstrating potential speedups and improved convergence in simulated quantum environments for computer vision tasks.

Contribution

It develops a quantum Levenberg--Marquardt algorithm that reduces complexity in solving normal equations, advancing quantum computing applications in computer vision optimization.

Findings

Quantum algorithm reduces complexity with respect to number of points.

Improved convergence rate observed in toy-model experiments.

Analysis of success probability on current quantum hardware.

Abstract

In this paper we develop a quantum optimization algorithm and use it to solve the bundle adjustment problem with a simulated quantum computer. Bundle adjustment is the process of optimizing camera poses and sensor properties to best reconstruct the three-dimensional structure and viewing parameters. This problem is often solved using some implementation of the Levenberg--Marquardt algorithm. In this case we implement a quantum algorithm for solving the linear system of normal equations that calculates the optimization step in Levenberg--Marquardt. This procedure is the current bottleneck in the algorithmic complexity of bundle adjustment. The proposed quantum algorithm dramatically reduces the complexity of this operation with respect to the number of points. We investigate 9 configurations of a toy-model for bundle adjustment, limited to 10 points and 2 cameras. This optimization problem is solved both by using the sparse Levenberg-Marquardt algorithm and our quantum implementation. The resulting solutions are presented, showing an improved rate of convergence, together with an analysis of the theoretical speed up and the probability of running the algorithm successfully on a current quantum computer. The presented quantum algorithm is a seminal implementation of using quantum computing algorithms in order to solve complex optimization problems in computer vision, in particular bundle adjustment, which offers several avenues of further investigations.