I was going through arXiv:quant-ph/0001106v1, the first paper by Farhi on adiabatic quantum computation.
 Equation 2.24 says, $$\tilde{H}(s) = (1-s)H_B + sH_P$$ which means the adiabatic evolution starts from the ground state of $H_B$ and slowly evolves until it arrives at the ground state of $H_P$. In section 3.1, the one qubit example has the adiabatic Hamiltonian as $$\tilde{H}(s) = \begin{pmatrix}
s &amp; \epsilon (1-s) \\ 
\epsilon (1-s) &amp; 1-s   
\end{pmatrix}$$
I don't see how the plot of Figure 1 is drawn. In figure 1, Farhi plotted eigenvalues of the Hamiltonian for s while the range for s was 0 to 1. The Hamiltonian is supposed to evolve according to the Schrodinger equation (eq 2.1), $$i \frac{d}{dt} |\Psi (t) \rangle = H(t) |\Psi (t) \rangle$$
Was this evolution solved to draw the plot? Or did Farhi derive the formula for eigenvalues in terms of s using just matrix math and plotted accordingly?


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