I am currently trying to implement the VQSE algorithm. There the biggest eigenvalues and their corresponding eigenvectors of a density matrix $\rho$ are computed.
In contrast to VQE, the matrix $\rho$ is not used in the cost function (as observable), but as the input for the circuit.
Lets say $\rho$ is in $\mathbb{C}^{2^n \times 2^n}$ so the eigenvectors are in $\mathbb{C}^{2^n}$. I need $n$ qubits to represent the eigenvectors. I could just interpret the density matrix 'flattened' as a $\mathbb{C}^{{2^n}{2^n}} = \mathbb{C}^{2^{2n}}$ vector. Then I could do something like amplitude encoding on $2n$ qubits to represent the matrix. But I am very certain that this is not the idea of the paper. There $\rho$ has to be encoded in $n$ qubits. I looked into density matrix purification but I do not know if/how I can compute that for an arbitrary density matrix where I don't know how the density matrix got composed.
Edit:
Is purification of the density matrix the right idea? But then how can that be done for an arbitrary density matrix? (Let $\rho = \sum_i p_i \left|\psi_i\right\rangle\left\langle\psi_i\right|$, I know the values of $\rho$ but not of $\left|\psi_i\right\rangle$ or $p_i$). Or is there another way to encode a density matrix of $n$ qubit states in an $n$ qubit quantum register?
