  The vast majority of quantum states and unitaries have circuit complexity
exponential in the number of qubits. In a similar vein, most of them also have
exponential minimum description length, which makes it difficult to pinpoint
examples of exponential complexity. In this work, we construct examples of
constant description length but exponential circuit complexity. We provide
infinite families such that each element requires an exponential number of
two-qubit gates to be generated exactly from a product and where the same is
true for the approximate generation of the vast majority of elements in the
family. The results are based on sets of large transcendence degree and
discussed for tensor networks, diagonal unitaries, and maximally coherent
states.
