Shorthand notation for the sign flip gate

Question

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.

Answer 1

A unitary $U$ and $e^{i\phi}U$, which differs from it by a phase, act exact identically on any quantum state. Thus, they should really be considered the "same" unitary in terms of their action.

You can therefore use $X$ instead of your unitary, which is $-X$. This will have exactly the identical action in any circuit.

(Why is this? There are different ways to see this: Either since $|\psi\rangle$ and $e^{i\phi}|\psi\rangle$ describe the same quantum state, or by working with density operators on which $U$ acts as $\rho\mapsto U\rho U^\dagger$, such that phases cancel. Also, note that this is not true for controlled-unitaries -- but this is an entirely different question.)