  We consider models of quantum computation that involve operations performed
on some fixed resourceful quantum state. Examples that fit this paradigm
include magic state injection and measurement-based approaches. We introduce a
framework that incorporates both of these cases and focus on the role of
coherence (or superposition) in this context, as exemplified through the
Hadamard gate. We prove that given access to incoherent unitaries (those that
are unable to generate superposition from computational basis states, e.g.
CNOT, diagonal gates), classical control, computational basis measurements, and
any resourceful ancillary state (of arbitrary dimension), it is not possible to
implement any coherent unitary (e.g. Hadamard) exactly with non-zero
probability. We also consider the approximate case by providing lower bounds
for the induced trace distance between the above operations and $n$ Hadamard
gates. To demonstrate the stability of this result, this is then extended to a
similar no-go result for the case of using $k$ Hadamard gates to exactly
implement $n>k$ Hadamard gates.
