N-Qubit Hadamard vs Quantum Fourier Transform

Question

Both Simon's algorithm and the algorithm for period finding begin by placing qubits in the equal superposition state, but Simon's algorithm uses the n-qubit Hadamard $H^{otimes n}$ while the period finding algorithm uses the quantum Fourier transform. My understanding is both QFT and  the n-qubit Hadamard perform the same operation on the $|00...0rangle$ state, creating the $frac{1}{sqrt{2^n}} sum_{xin{0,1}^n}|xrangle$ state. I'm reading this from the Qiskit textbook.
When the result is the same, why do the two algorithms use different ways to achieve the equal superposition? More generally, when would one use the n-qubit Hadamard, and when would one use the QFT?

Martin Vesely · Answer

As you pointed out correctly, both $H^{otimes n}$ and QFT applied on input state $|0rangle^{otimes n}$ return state
$$
|psirangle = frac{1}{sqrt{2^n}}sum_{i=0}^{2^n}|irangle.
$$
There is no special reason why $n$ qubit Hadamard gate cannot replace QFT for $|0rangle^{otimes n}$ input. Maybe, an author of article about Simon's algorithm wanted to be more precise.
Overall, it does not matter which approach you use. Both return same result for $|0rangle^{otimes n}$ input.
EDIT (based on DaftWullie comment): However, in practice $n$ qubits Hadamard is prefered as it is very simple circuit in comparison with QFT.

N-Qubit Hadamard vs Quantum Fourier Transform

One Answer

Add your own answers!

Ask a Question