Figure 1

The second derivative of the ground-state energy of the effective Hamiltonian $\overline{H}$ (solid), based on Eq. (6), and for the transverse Ising Hamiltonian, $H_{\mathit{\text{Ising}}}$ (dashed), as a function of $\varphi =\arg (\theta +i\chi )$. Note that both show singularities at $\theta =\pm \chi$. $\mid {\theta }^2+{\chi }^2\mid$ was chosen to be 1.9, and we set $N=100$. For smaller $\theta$ and $\chi$, the commutator between $H_{\theta }$ and $H_{\chi }$ becomes small, and the Hamiltonian $\overline{H}$ becomes asymptotically closer to the Ising Hamiltonian $H_{\mathit{\text{Ising}}}$.Reuse & Permissions

Figure 2

Determination of the eigenvalue spectrum of the unitary operator. The Fourier transform of ${\mathbf{\mid }⟨i\mid U^m\mid \Psi ⟩\mathbf{\mid }}^2$ is shown as a vertical density, and is shown as a function of the horizontal coordinate $\theta$, for a fixed $\chi =0.2$. Four qubits are used here so as to provide a complex, but not confusing, diagram. The white bands indicate a large Fourier component. The superimposed gray lines show all energy differences—note that some of these are disallowed. In this case, 2048 samples are used in the Fourier series, and simulations are taken in steps of 0.01 in $\theta$. From this diagram, we can see the energy gap between the ground state and first excited state approaching zero. We can also see a level crossing, where one of the gray lines is reflected through the origin at $\theta \simeq 0.23$.Reuse & Permissions