Unflattening by Flattening

Working with quantum Fourier models, the feature map a.k.a. encoding is one of the most interesting aspects as the frequencies, and therefore the accesible domain of functions such a model can approximate, is mainly set by this stage. In our latest paper, we step back a little and look on how the distribution of input data that is encoded by the feature map, affect the training, specifically the output variance. That might sound a bit strange first, but the algebraic input purity and the circuit's dynamical Lie algebra are proven to have an effect on the output purity and hence the variance of the loss and hence the trainability. We show that for some circuit families there is indeed an interesting effect which causes the loss variance to decay and thus render such circuits untrainable in case of clustered input data. When the same data is then preprocessed such that it approaches a uniform distribution, the exact same model becomes trainable again. This turns out to not just be a constructed theory example but can also be observed in training. Although we only show this effect mathematically for a single data encoding circuit, we numerically see this effect also happening in data re-uploading circuits.

Feel free to checkout our paper on Arxiv.

I'm very happy that we got accepted at QTML26 with an extended abstract of this work.