I don't know of any explicit use of the Hadamard gate in circuits other than to introduce or take away superposition, but here are some interesting applications:
- The Hadamard gate is the single-qubit version of the Fourier Transform*, and the Fourier Transform is very useful.
- Instead of using the Hadamard at the end of a computation to get back the original basis, you can measure the system in the Hadamard basis instead of the standard basis.
- The set of $\{\text{CNOT}, \text{Hadamard}, \text{rotation by 45 degrees}, \frac{\pi}{8}\text{-gate} \}$ is a universal quantum gate set. That is to say, any quantum gate can be approximated arbitrarily well by repeated application of gates from this set.
- If you prepend and append two Hadamard gates to a CNOT gate, you reverse the role of control bit and target bit.
*Demonstration: The quantum Fourier transform is defined as follows. Take a system of $n$ particles in the standard basis, and put it in state $|k\rangle, 0 \leq k \leq 2^n-1$. Then applying the Fourier Transform $F$ yields a big superposition:
$$ F|k\rangle = \sum_{u=0}^{2^n-1}e^{i2\pi\frac{u\cdot k}{2^n-1}}|u\rangle $$
Take $n=1$ and $k\in\{0,1\}$, and you see that $F|0\rangle = |+\rangle$ and $F|1\rangle=|-\rangle$. As you can see, it is performs exactly the same transformation as a Hadamard gate. Moreover, for any number of qubits $n$, the Fourier still behaves as the Hadamard only for the state $0$, on which it acts as follows:
$$ F|0\rangle_n = \sum_{u=0}^{2^n-1}e^{i2\pi\frac{u\cdot 0}{2^n-1}}|u\rangle_n = \sum_{u=0}^{2^n-1}e^{0}|u\rangle_n = \sum_{u=0}^{2^n-1}|u\rangle_n $$
I have not normalized my states in these examples, but you should definitely do so on an exam.
