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> given a unitary matrix, could a neural network predict a number of gates needed to simulate that matrix as a QC and thereby give us a measure of complexity

That's really interesting. I'm curious—did you explore whether the predictivity of the neural network was influenced by any hidden subgroup structure in the unitary matrix? Seems like the matrix symmetries could play a significant role in determining the gate complexity.

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I didn’t include this in my thesis but from what I remember looking at hundreds if not thousands of matrices and their QC solutions some symmetries would immediately make it so that way less gates would be needed but it could also be deceptive and completely depended on which gate set you have available. If you take the matrix for the quantum furier transform for example with a gate set of phase gates and a hadamard gate then for the 100% solution you n hadamard gates and n! phase gates for an n qubit circuit even though the matrix is highly symmetrical. If your gate set was Clifford + toffli you would be able to do it with n*log(n). And then depending on how close you wanna approximate it you could get away with even less. But I have not gone into further analysis on which symmetries would have which effect on which gate set and whether it would influence predictivity. But it would be fun to investigate for sure!

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