Different approach

Condition (1) requires that the coefficients $\{ \alpha_i^{(n)} \}$ depend upon the Hamiltonian eigenvalues $\{ \varepsilon_i \}$. Resulting condition number $\kappa$ of the Hessian of $Q[\rho]$ is

$$\kappa \sim \frac{\varepsilon_{\rm max} - \varepsilon_{\rm min}} {\varepsilon_{\rm gap}}$$

Instead start by imposing condition (2), making the $\{ \alpha_i^{(n)} \}$ independent of state $i$ (so that $\kappa \approx 1$) e.g.

$$Q[\rho] = E[\rho] + \alpha {\rm Tr}\left[ \rho^2 \left( 1 - \rho \right)^2 \right]$$

For $\alpha \gg \mathop{\rm max}\limits _i{\left\vert \tilde{\varepsilon}_i - \lambda \right\vert}$, the minimum occurs for

i.e. for approximately idempotent density-matrices.