# How is the energy/eigenvalue gap plot drawn for adiabatic quantum computation?

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 $$\tilde{H}(s) = \begin{pmatrix} s & \epsilon (1-s) \\ \epsilon (1-s) & 1-s \end{pmatrix}$$ 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?

If the parameter $s$ is varied adiabatically there should be no need to solve the time-dependent Schrodinger equation. The adiabatic theorem implies the instantaneous eigenvalues are always those of the parameterized Hamiltonian, provided its spectrum is separated.
Thus your final statement and hadsed's conclusion are correct: The figure just shows the eigenvalues of the matrix as a function of $s$.
So you just plug in the right $s$ values and find the eigenvalues of that $H(s)$, and plotting that gives you the eigenspectrum in Fig. 1.