The Hadamard gate cannot be replaced by a resource state in universal quantum computation

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.