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

HH CZ HH

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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:

enter image description here

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:

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

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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.

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