Existing methods for learning 3D representations are deep neural networks trained and tested on classical hardware. Quantum machine learning architectures, despite their theoretically predicted advantages in terms of speed and the representational capacity, have so far not been considered for this problem nor for tasks involving 3D data in general. This paper thus introduces the first quantum auto-encoder for 3D point clouds. Our 3D-QAE approach is fully quantum, i.e. all its data processing components are designed for quantum hardware. It is trained on collections of 3D point clouds to produce their compressed representations. Along with finding a suitable architecture, the core challenges in designing such a fully quantum model include 3D data normalisation and parameter optimisation, and we propose solutions for both these tasks. Experiments on simulated gate-based quantum hardware demonstrate that our method outperforms simple classical baselines, paving the way for a new research direction in 3D computer vision. The source code is available at https://4dqv.mpi-inf.mpg.de/QAE3D/.

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