Here, ${\mathcal{L}}^{†}$ is the adjoint Lindblad map defined by $\mathrm{Tr}({\rho }_1^{†}\mathcal{L}[{\rho }_2])=\mathrm{Tr}({\mathcal{L}}^{†}[{\rho }_1^{†}]{\rho }_2)$, for arbitrary density matrices ${\rho }_1$, ${\rho }_2$.

The full diagonalization of the Liouvillian is not required; it is sufficient to identify the (slow or fast) eigenmodes to be suppressed. Our discussion can be generalized to nondiagonalizable Liouvillians, mappable into Jordan canonical form; however, the recipe we present will not apply to Liouvillian modes in nondiagonalizable blocks.

More formally, ${\ell }_a^{†}\in \mathcal{A}$, $\forall {\ell }_a\in \mathcal{A}$. This forbids the inclusion in $\mathcal{A}$ of a single element from a pair of Hermitian-conjugated left eigenmatrices, $\{{\ell }_i,{\ell }_{i+1}\},{\ell }_i={\ell }_{i+1}^{†}$.

In the case of a pure initial state, ${\rho }_0=|{\psi }_0⟩⟨{\psi }_0|$, it is sufficient to compute the eigenvector of ${\overline{\rho }}_{\perp }$ associated with the largest eigenvalue; the full diagonalization of ${\overline{\rho }}_{\perp }$ is not necessary.

In Fig. 3, we neglect $T_U$ as it depends on the quantum hardware (types of gates available, duration of their execution, etc.).

The choice of the ansatz depends on the specific problem. Even though more sophisticated gates can also be considered, the ansatz should be kept as simple as possible to minimize the preparation time, $T_U$, and simplify the optimization procedure.