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

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

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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: $$ RX(2\pi) = \begin{bmatrix} \cos{\left(\frac{2\pi}{2}\right)} & -i\sin{\left(\frac{2\pi}{2}\right)} \\ -i\sin{\left(\frac{2\pi}{2}\right)} & \cos{\left(\frac{2\pi}{2}\right)} \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} $$

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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 $\left| 1 \right>$.

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