efficiently solving the time-dependent Schroedinger equation was first developed for distinguishable degrees of freedom [@MCTDH2000] and ML-X is an extension to indistinguishable particles such as bosons or fermions and mixtures thereof [@MLX2013; @MLX2013b; @MLX2017]. ML-X is an ab-initio method, whose power lies in expanding the wave-function in time-dependent basis functions. Let us demonstrate the underlying ansatz for the system at hand: $$\ket{\Psi(t)} =\sum_{i=1}^{S}\sqrt{\lambda_i(t)} 
    \ket{\Psi_i^M(t)} \otimes \ket{\Psi_i^I(t)},
\label{eq:wfn_ansatz_species_layer}$$ $$\ket{\Psi_i^{\sigma}(t)} = \sum_{\vec{n}^{\sigma}|N_\sigma}
    C_{i, \vec{n}^{\sigma}}(t)
    \ket{\vec{n}^{\sigma}(t)}.
\label{eq:wfn_ansatz_particle_layer}$$

The time-dependent many-body wave-function $\ket{\Psi(t)}$ has two layers: the so-called species layer and the particle layer . In the first step we separate majority and impurity species and assign them to $S \in \mathbb{N}$ corresponding species wave-functions $\ket{\Psi_i^{\sigma}(t)}$. The time-dependent coefficients $\lambda_i(t)$ are normalized $\sum_{i=1}^{S} \lambda_i(t) = 1$ and describe the degree of entanglement between the components. In case $\exists \ i\in\{1,\dots,S\} : \lambda_i(t) \approx 1$ the components are said to be disentangled. In the second step each species wave-function $\ket{\Psi_i^{\sigma}(t)}$, which depends on $N_{\sigma}$ indistinguishable coordinates, is expanded in terms of species-dependent symmetrized product states, also known as permanents or number states, $\ket{\vec{n}^{\sigma}}=\ket{n_1^{\sigma},\ldots, n_{s_{\sigma}}^{\sigma}}$ admitting $s_{\sigma}\in \mathbb{N}$ normalized single particle functions (SPF) $\ket{\varphi_{j}^{\sigma}(t)}$. The sum is over all possible configurations $\vec{n}^{\sigma}|N_\sigma$ fulfilling the constraint $\sum_{i=1}^{s_{\sigma}} 