in ${\mathcal{A}}$ and $\kappa_k =
\bra{\psi_k} \sigma_3 \ket{\psi_k}$. The sample space for $\omega$ can be chosen as $\Omega = \Omega_1 \cup \Omega_2$, where $$\Omega_1 = \{ k \mid k = 1,\cdots,N \}, \quad
\Omega_2 = \{ (j,k) \mid j,k = 1,\cdots,N, \, j < k \},$$ and the operator $A(t, \omega)$ is set to be $$A(t, \omega) = \left\{ \begin{array}{ll}
  |\Omega| (m_1^{(k)} \sigma_1^{(k)} + m_2^{(k)} \sigma_2^{(k)}),
    & \text{if } \omega = k \in \Omega_1, \\
  |\Omega| \gamma_{jk}(t) (\sigma_1^{(j)} \sigma_1^{(k)} + \sigma_2^{(j)} \sigma_2^{(k)}),
    & \text{if } \omega = (j,k) \in \Omega_2,
\end{array} \right.$$ where $|\Omega| = N(N+1)/2$ is the cardinality of $\Omega$. For all $\omega_0 \in \Omega$, the probabilities are assigned the same: $${\mathbb{P}}(\omega = \omega_0) = \frac{1}{\abs{\Omega}},$$ which gives the defintion of $\mu_{\omega}$. The above definitions immediately lead to $${\mathbb{E}}_{\omega} A(t,\omega) = H_{\mathrm{hard}} := H - H_{\mathrm{easy}}.$$ In our implementation, the intensity function $\lambda(t,\omega)$ is chosen as $$\lambda(t, \omega) = \left\{ \begin{array}{ll}
  |\Omega| \bigl\lvert m_1^{(k)} + {\mathrm{i}}m_2^{(k)} \bigr\rvert
    & \text{if } \omega = k \in \Omega_1, \\[5pt]
  2 |\Omega| |\gamma_{jk}(t)| & \text{if } \omega = (j,k) \in \Omega_2,
\end{array} \right.$$ so that the operator ${\widetilde{A}}(t,\omega)$ defined in does not change the magnitude of the state.

The $Z$-basis can be considered 