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The use of vectorial parameterization to create geometrical representations in computational models has a large number of applications. One particular application is the calculation of the 3D rotational motion of rigid bodies, that could be used for the spatial location estimation from objects. Provided the algebraic nature of this problem, it could benefit from Quantum Computing, in particular several vectors could be superposed to be transformed with a single operation, providing a quantum processing advantage. In this article, we propose an implementation of a Quantum Computing algorithm to compute Euler-Rodrigues Parameters to model rigid body rotations to transform arbitrary functions, rotating multiple vectors in superposition. We developed this algorithm using Qiskit, taking into account the limitations imposed by the current Noisy Intermediate Scale Quantum (NISQ) devices, such as the reduced number of qubits available and the limited coherence time.

Boosting is an ensemble learning method that converts a weak learner into a strong learner in the PAC learning framework. Freund and Schapire designed the Godel prize-winning algorithm named AdaBoost that can boost learners, which output binary hypotheses. Recently, Arunachalam and Maity presented the first quantum boosting algorithm with similar theoretical guarantees. Their algorithm, which we refer to as QAdaBoost henceforth, is a quantum adaptation of AdaBoost and only works for the binary hypothesis case. QAdaBoost is quadratically faster than AdaBoost in terms of the VC-dimension of the hypothesis class of the weak learner but polynomially worse in the bias of the weak learner. Izdebski et al. posed an open question on whether we can boost quantum weak learners that output non-binary hypothesis. In this work, we address this open question by developing the QRealBoost algorithm which was motivated by the classical RealBoost algorithm. The main technical challenge was to provide provable guarantees for convergence, generalization bounds, and quantum speedup, given that quantum subroutines are noisy and probabilistic. We prove that QRealBoost retains the quadratic speedup of QAdaBoost over AdaBoost and further achieves a polynomial speedup over QAdaBoost in terms of both the bias of the learner and the time taken by the learner to learn the target concept class. Finally, we perform empirical evaluations on QRealBoost and report encouraging observations on quantum simulators by benchmarking the convergence performance of QRealBoost against QAdaBoost, AdaBoost, and RealBoost on a subset of the MNIST dataset and Breast Cancer Wisconsin dataset., Comment: 24 pages, 2 figures, 1 table