# Standard name for CZ gate conjugated by Hadamards?

Is there a standard name for a $$CZ$$ gate conjugated by $$H \otimes H$$ gates? See below for circuit

In stim this is called the XCX gate (short for X-controlled X gate). Stim has A-controlled-B for all paulis A, B.

In Quirk you can make this gate by combining an X-axis control and an X gate. X-axis controls condition on the control being $$|-\rangle$$, instead of on the control being $$|1\rangle$$. Looks like this:

The concept of "controlling" can be generalized to work on any pair of commuting operations:

$$\text{Control}(A, B) = \text{Control}(B, A) = \exp(-i \ln(A) \ln(B) / \pi)$$

For example, you can check that:

\begin{aligned} \text{CNOT} &= \text{Control}(Z_0, X_1) \\&= \text{Control}(Z \otimes I, I \otimes X) \\ \\ \text{CZ} &= \text{Control}(Z_0, Z_1) \\ \text{HadamardConjugatedCZ} &= \text{Control}(X_0, X_1) \end{aligned}

There’s no standard name, but if you wanted to draw it, you’d draw two $$\oplus$$ signs connected by a vertical line since the H will convert Z to X.