0 & 0 & 1 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & i \\
 0 & 0 & 0 & 0 & 0 & 0 & i & 0 \\
 0 & 0 & 0 & 0 & 0 & i & 0 & 0 \\
 0 & 0 & 0 & 0 & i & 0 & 0 & 0 \\
\end{array} \right) }, \end{aligned}$$ where $U_{\rm E} = e^{-\pi(\sigma_z \otimes \sigma_z \sigma_0 +
\sigma_z \otimes \sigma_0 \sigma_z)/4}$. $U_{\rm E}$ was implemented by waiting for the period of $\displaystyle \frac{n}{\Delta \omega_0}$, where $\Delta \omega_0$ is the Larmor frequency difference between CC and CSs. $n$ is an integer and selected so that $ \displaystyle \frac{n}{\Delta \omega_0} 
\approx \frac{\pi}{J^{\rm (CC, CSs)}}$, see Table \[H\_parameters\]. All $Z_k(\theta)$’s ($k = $ C or S) were virtually implemented by controlling $\phi$’s in $R_k(\phi, \theta)$’s ($k = $ C or S) [@Kondo2006]. We employed the jump-and-return pulses [@PhysRevA.74.052324] to realize $R_k(\phi, \theta)$ with concatenated composite pulses [@doi:10.7566/JPSJ.82.014004].

Magnetic field sensing is equivalent to measuring the phase difference between the initial and final state of CC, as shown in Fig. 