How to construct a multi-control quantum gate whose control bit relation is XOR?

Question

So far, the control bits of most control gates have an AND relationship. How to construct a quantum gate whose control bits are XOR with the fewest auxiliary qubits? And then generalize it, is it possible to realize multi-control quantum gates of arbitrary control logic with different relationships?

Answer 1

I think what you are asking for is a block unitary matrix. To be concrete I'll discuss a three-qubit case, the generalization is straightforward. Consider $$U=|00\rangle\langle00|\otimes U_{00}+|01\rangle\langle01|\otimes U_{01}+|10\rangle\langle10|\otimes U_{10}+|11\rangle\langle11|\otimes U_{11}$$ or, in matrix representation $$ U=\begin{pmatrix}U_{00}&0&0&0\\0&U_{01}&0&0\\0&0&U_{10}&0\\0&0&0&U_{11}\end{pmatrix} $$ where $U_{00}-U_{11}$ are $2\times2$ matrices.

So, the matrix $U_{00}$ is applied to the last qubit if the first two are in the state $|00\rangle$, $U_{01}$ is applied when the control qubits are in the state $|01\rangle$ etc. Using these block unitaries, you can realize your arbitrary control logic.

The case when $U_{00}=U_{10}=U_{01}=\mathbb{1}$ and $U_{11}=V$ is the standard controlled-$V$ on the last qubit.

$$ U=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&V\end{pmatrix} $$

To insert $V$ at another place, say at $|01\rangle\langle01|$ you just need to swap $|01\rangle$ and $|11\rangle$. This is easy, since $|01\rangle=X_0 |11\rangle$. Explicitly, start with a standard $$C^2V=|11\rangle\langle11|\otimes V+(1-|11\rangle\langle11|)\otimes 1$$ and conjugate it with $X_0$ to get $$X_0C^2VX_0=|01\rangle\langle01|\otimes V+(1-|01\rangle\langle01|)\otimes 1=\begin{pmatrix}1&0&0&0\\0&V&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$

General block unitary can be constructed as a product of these simpler ones with only a single control state. This method uses no extra ancilla qubits. Note however that it may be very inefficient in terms of depth/gate counts.