Two Qubit Gates

Two nuclear spins in a molecule interact with each other, as one would expect of two magnets. But the details of the spins' interaction are more complicated because they are mediated by the electrons. In liquid state, the interaction is also modulated by the rapid motions of the molecule. The resulting effective interaction is called the $J$-coupling. When the difference of the precession frequencies between the coupled nuclear spins is large compared to the strength of the coupling, it is a good approximation to write the coupling Hamiltonian as a product of the $z$-Pauli operators for each spin: $H_J=C{\sigma_{z}}^{({\mathsf {1}})}{\sigma_{z}}^{({\mathsf {2}})}$. This is the ``weak coupling'' regime. With this Hamiltonian, an initial state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_0}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ of two nuclear-spin qubits evolves as $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_t}\mbox{$\rangle\hspa... ...t}\vert$}{\psi_0}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, where a different rotating frame is used for each nuclear spin to eliminate the spin's internal evolution. (The use of rotating frames is compatible with the coupling Hamiltonian because the Hamlitonian is invariant under frame rotations.) Because the Hamiltonian is diagonal in the logical basis, the effect of the coupling can be understood as an increase of the (signed) precession frequency of the second spin if the first one is up and a decrease if the first one is down (Fig. 5). The changes in precession frequency for adjacent nuclear spins in organic molecules are typically in the range of $20$- $200{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$. They are normally much smaller for non-adjacent nuclear spins. The strength of the coupling is called the ``coupling constant'' and is given as the change in the precession frequency. In terms of the constant $C$ used above, the coupling constant is given by $J=2C/\pi$ in ${\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$. For example, the coupling constants in TCE are close to $100{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$ between the two carbons, $200{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$ between the proton and the adjacent carbon, and $9{\mathchoice{\mbox{Hz}}{\mbox{Hz}}{\mbox{\small Hz}}{\mbox{\tiny Hz}}}\,$ between the proton and the far carbon.

FIG. 5: Effect of the $J$-coupling. In the weak-coupling regime with a positive coupling constant, the coupling between two spins can be interpreted as an increase in precession frequency of the spin $\mathsf {2}$ when the spin $\mathsf {1}$ is ``up'' and a decrease when spin $\mathsf {1}$ is ``down''. The two diagrams depict the situation in which spin $\mathsf {2}$ is in the plane. The diagram on the left has spin $\mathsf {1}$ pointing up along the $z$ axis. In the rotating frame of spin $\mathsf {2}$, it precesses from the $x$-axis to the $y$-axis. The diagram on the right has spin $\mathsf {1}$ pointing down, causing a precession in the opposite direction of spin $\mathsf {2}$. Note that neither the coupling nor the external field change the orientation of a spin pointing up or down along the $z$-axis.

The $J$-coupling and the one-qubit pulses suffice for realizing the controlled-not operation usually taken as one of the fundamental gates of QIP. A pulse sequence for implementing the controlled-not in terms of the $J$-coupling constitutes the first quantum algorithm of Sect. 3. A problem with the $J$-coupling in liquid-state NMR is that it cannot be turned off when it is not needed for implementing a gate.