How is the Controlled Z Gate implemented?

How are 2 qubit Controlled Z gates (1 control / 1 target) typically implemented? Do we seek physical realizations of this gate, or have an alternate decomposition?

• This will generally be hardware dependent. Do you have a specific platform in mind? – Niel de Beaudrap Aug 22 at 23:37
• No - I was curious if there was a typical realization with CNOTs / Pauli gates. I'd also be interested on which platforms CZs can be directly implemented. – C. Kang Aug 23 at 17:21

1 Answer

On IBM QX architectures, the only entangling gate is the controlled not (CX), so controlled Z gates with control qubit $$n$$ and target qubit $$m$$ are decomposed into $$CZ^n_m = H_m CX^n_m H_m$$.

Although this is QX-specific, it demonstrates a principle that will likely hold on any architecture -- That the general method is to use a similarity transform to rotate Z onto the most conducive rotation axis for the hardware, perform that rotation, and rotate Z back into place.