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 4. Consider a quantum circuit in which the Hadamard gate is applied to a quantum state and then the outout is measured in the computational basis.  Show that when the state is input to this circuit that the output is with probability 1, and that when the state is input to this circuit the output is with probability 1.

Answer: First consider the case when the state is input to the circuit. We have

So the result of applying to the state is the state .

In the general case the state will produce the result 0 with probability and the value 1 with probability when measured in the computational basis. Since the result in this case is we have and so the result will be with probability , with the result having probability . So the output as measured will always be .

When the state is input to the circuit we have

So the result of applying to the state is the state , which when measured in the computational basis will always produce the value .