# How to decompose the Quantum Fourier Inverse matrix into elementary quantum gates?

I am not sure how to find the following matrix (the inverse of Quantum Fourier Transform) in terms of elementary quantum gates? I am using Qiskit to implement it.

Read the circuit for the QFT on 2 qubits and reverse and dagger everything

$$(1 \otimes H)(Controlled(2,S_1^\dagger))(H \otimes 1)$$

Edit:

For how: Implementation of inverse QFT?

This decomposition is $$m=2$$ on https://en.wikipedia.org/wiki/Quantum_Fourier_transform Note that $$R_2 = S$$

$$1 \otimes H$$ to denote Hadamard on the second qubit. $$H \otimes 1$$ for Hadamard on the first. $$Controlled(2,S_1^\dagger)$$ for use qubit 2 as the control and assuming that apply $$S^\dagger$$ on the first qubit.

• I am not sure I understand the notation. By (1⊗H), do you mean applying a Hadamard gate to the first qubit? And I am not sure what (Controlled(2,S1)) means? Also, could you explain how you came up with this? Thank you in advance. – Rahaf Jul 12 at 20:03
• This might help you youtu.be/uuBgK44JrnA – Aman Jul 15 at 7:08
• Thank you! I have one last question, how can I represent the controlled-S† gate? I can only find a single-qubit S† gate. – Rahaf Jul 15 at 16:18
• if you're implementing using QISKit the video shows the code also. – Aman Jul 16 at 8:05