# Tensorial notation for this quantum XOR circuit

Suppose I have this quantum XOR circuit; this is a quantum circuit for the classical XOR operation $$\begin{eqnarray*} x_1'= x_1\oplus x_2\oplus x_3\oplus x_4,\\ x_2'=x_2\oplus x_3\oplus x_4\oplus x_5,\\ x_3'=x_3\oplus x_4\oplus x_5\oplus x_6. \end{eqnarray*}$$ Now suppose I had $$l$$ quibts instead of $$3$$. What would be tensorial notation for this generalized circuit, what I think is $$U=\prod_{i=0}^{2}\prod_{k=0}^{q-1}I^{\otimes k}\otimes X\otimes I^{\otimes i}\otimes P_1\otimes I^{\otimes l-2-i-k}$$ is this the correct notation. The first $$I$$ tells how many qubits will be unaltered, the second $$I$$ adjusts the correct control, $$P_1$$ is the projection operator, and last $$I$$ does nothing on the remaining qubits. Ignore the part of this operator when the state is $$0$$ i.e. projection operator $$P_0$$ is omitted.

• You cannot just ignore $P_0$, if you do your operation is neither unitary nor entangling! – bRost03 Jun 10 '19 at 4:27