  We introduce several novel probabilistic quantum algorithms that overcome the
normal unitary restrictions in quantum machine learning by leveraging the
Linear Combination of Unitaries (LCU) method. Among our contributions are
quantum native implementations of Residual Networks (ResNet); demonstrating a
path to avoiding barren plateaus while maintaining the complexity of models
that are hard to simulate classically. Furthermore, by generalising to allow
control of the strength of residual connections, we show that the lower bound
of the LCU success probability can be set to any arbitrary desired value. We
also implement a quantum analogue of average pooling layers from convolutional
networks. Our empirical analysis demonstrates that the LCU success probability
remains stable for the MNIST database, unlocking a potential quadratic
advantage in terms of image size compared to classical techniques. Finally, we
propose a general framework for irreducible subspace projections for quantum
encoded data. Using this, we demonstrate a novel rotationally invariant
encoding for point cloud data via Schur-Weyl duality. We also show how this
framework can be used to parameterise and control the amount of symmetry in an
encoding; demonstrating improved classification performance for partially
permutation invariant encoded point cloud data when compared to non-invariant
or fully permutation invariant encodings. These new general algorithmic
frameworks are all constructed under the same LCU method, suggesting that even
more novel algorithms could be achieved by utilising the LCU technique.
