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The success of modern deep learning hinges on the ability to train neuralnetworks at scale. Through clever reuse of intermediate information,backpropagation facilitates training through gradient computation at a totalcost roughly proportional to running the function, rather than incurring anadditional factor proportional to the number of parameters - which can now bein the trillions. Naively, one expects that quantum measurement collapseentirely rules out the reuse of quantum information as in backpropagation. Butrecent developments in shadow tomography, which assumes access to multiplecopies of a quantum state, have challenged that notion. Here, we investigatewhether parameterized quantum models can train as efficiently as classicalneural networks. We show that achieving backpropagation scaling is impossiblewithout access to multiple copies of a state. With this added ability, weintroduce an algorithm with foundations in shadow tomography that matchesbackpropagation scaling in quantum resources while reducing classical auxiliarycomputational costs to open problems in shadow tomography. These resultshighlight the nuance of reusing quantum information for practical purposes andclarify the unique difficulties in training large quantum models, which couldalter the course of quantum machine learning.

On quantum backpropagation, information reuse, and cheating measurement collapse