# Exact meaning of "parity control" in quirk

I am trying to understand this distillation circuit (related to this question), but I am confused by the meaning of the "parity control Z" operations represented with this symbol:

The explanation in the simulator is "Include a qubit's Z observable in the column parity control. Gates in the same column only apply if an odd number of parity controls are satisfied".

Does it mean that if the the first $$k$$ qubits are affected with this operator, and some unitary $$U$$ is applied on the last $$n-k$$ qubit, the resulting operation is a controlled unitary operation only applied if the first $$k$$ qubits have an odd number of $$1$$ once the state is expanded in the computational basis?

It seems to be the case with simple examples but I don't know if I just found out particular cases.

• Oh ok: so in different terms it corresponds to the unitary is $(1+P)/2 \otimes I + (1-P)/2 \otimes U$, where $P$ is the Pauli operator corresponding to the $X_{\text{par}}$, $Y_{\text{par}}$, $Z_{\text{par}}$ that have been put on the "control" side. Aug 29, 2023 at 16:27