This is one in a series of posts working through the exercises in the Quantum Country online introduction to quantum computing and related topics. The exercises in the original document are not numbered; I have added my own numbers for convenience in referring to them.
Exercise 2. Suppose that instead of the Hadamard matrix we’d defined a matrix . Explain why would not make a suitable quantum gate by applying it to the quantum state .
Answer: We have
so that applying to gives us
So applied to gives us the zero state. That is not a result we’d want from applying a quantum gate.