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Neural Networks (NNs) are essential components of intelligent systems because they produce efficient solutions to problems of overwhelming complexity for conventional computing methods. There are lots of papers showing that NNs can approximate a wide variety of functions, but comparatively very few discuss their limitations. To illustrate the role played by information coding in NN computations we define and study sensitive and robust Boolean functions. We also prove that an exponential large set of functions in the first group are exponentially difficult to compute by multiple-layer perceptrons (hence incomputable by single-layer perceptrons) and a comparatively large set of functions in the second one, but not all, are computable by single-layer perceptrons. This suggests that the success of NNs, or lack of it, are in part determined by properties of the learned data sets. Finally we use polynomial threshold perceptrons to compute all Boolean functions with quantum annealing and present in detail a QUBO computation on the D-Wave Advantage. [ABSTRACT FROM AUTHOR]