# Which IBM composer gate realizes the 2x2 single-qubit $-I$ matrix?

Which IBM Quantum composer gate is used to realize the 2x2 matrix $$-I$$ acting on a single qubit? And how is it called?

• If I understand your question correctly the matrix you give is only a (-1) global phase factor away from the identity matrix Apr 4, 2023 at 18:19
• Yes, thats true, so is there a gate for that in IBM composer ? Apr 5, 2023 at 4:51

The 2x2 matrix $$-I$$ represents a gate that performs an identity transformation and applies a global phase $$e^{i\pi} = -1$$ on a single-qubit state. In the IBM Quantum composer you can use any single-qubit rotation gate with an angle $$\theta = 2\pi$$ to implement such operation. For example, let's consider the $$RX(\theta)$$ gate: $$RX(2\pi) = \begin{bmatrix} \cos{\left(\frac{2\pi}{2}\right)} & -i\sin{\left(\frac{2\pi}{2}\right)} \\ -i\sin{\left(\frac{2\pi}{2}\right)} & \cos{\left(\frac{2\pi}{2}\right)} \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$$
Upto my knowledge there is no such special gate in IBM composer to represent the gate $$\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$$ which can also be represented as ($$e^{iπ}$$)$$I$$
where $$I$$ is the identity gate and $$e^{iπ}$$ is a global phase.
There is no special gate to perform this operation but you can use $$Z$$ gate which is a phase gate to perform this operation however it works when qubit is in state $$\left| 1 \right>$$.