Ising coupling of two qubits ($H_I=J{\sigma }_z\otimes {\sigma }_z$)

An Ising coupling generates $e^{-i\varphi Z\otimes Z}$ with $\varphi =\int J(t)dt$ in units $\hslash =1$. It is entangling precisely when $\varphi$ is not an integer multiple of $\pi /2$. A $\pi$ rotation on one qubit reverses its sign, permitting an ideal echo to cancel an always-on coupling.

A $\pi$ rotation about the x-axis on one qubit conjugates $Z\otimes Z$ to its negative. With ideal instantaneous pulses, two equal free-evolution intervals separated by this pulse and its inverse cancel an Ising coupling exactly. Finite-duration pulses require accounting for the coupling during the pulses; the error is small only when $|J|$ times the pulse duration is small, or when the control is explicitly designed to compensate it.