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I need to use the following matrix gate in a quantum circuit:

$$\text{Sign Flip}=\left[\begin{matrix}0 & -1 \\ -1 & 0\end{matrix}\right]$$

$\text{Sign Flip}$ can be decomposed as (in terms of Pauli-$X$,$Y$,$Z$):

$$\begin{bmatrix} 0 & -1\\ -1 & 0\end{bmatrix} = \begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix} \begin{bmatrix} 0 & -i\\ i & 0\end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix} \begin{bmatrix} 0 & -i\\ i & 0\end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix}$$

Is there any standard shorthand notation for the $\text{Sign Flip}$ gate? I don't really want to replace one simple custom gate by $5$ quantum gates in my circuit.

1Is it physically different from Pauli-X gate? – kludg – 2018-07-28T09:34:13.433

Looks to me like -X but I must be seeing it totally wrong, what am I missing? – user1271772 – 2018-07-28T18:32:37.923

2@user1271772 Norbert and Blue discussed this in comments that have since been deleted. The conclusion was that this gate is physically equivalent to Pauli X. – James Wootton – 2018-07-30T12:37:11.610