Adiabatic computation: A toy model

We discuss a toy model for adiabatic quantum computation which displays some phenomenological properties expected in more realistic implementations. This model has two free parameters: the adiabatic evolution parameter $s$ and the $\alpha$ parameter, which emulates many-variable constraints in the classical computational problem. The proposed model presents, in the $s\text{−}\alpha$ plane, a line of first-order quantum phase transition that ends at a second-order point. The relation between computation complexity and the occurrence of quantum phase transitions is discussed. We analyze the behavior of the ground and first excited states near the quantum phase transition, the gap, and the entanglement content of the ground state.

Figure 1

(Color online) Density of states as a function of the energy for $s=1$ for different values of $\alpha$. The solid lines correspond $n=100$ and dashed lines to $n=300$, plotted for $\alpha ∊\{0,1,2,3\}$. The maxima of the curves shift to higher values of $\omega$ for increasing $\alpha$.

Figure 2

(Color online) Mean value of $s_x=S_x∕n$ plotted in the $\alpha \text{−}s$ plane.

Figure 3

(Color online) Energy levels as a function of $s$ computed for different values of $n$ and $\alpha$. The series I presents the energy levels for $n=10$ and $\alpha ∊\{0,3,10\}$, respectively, for {i, ii, iii}. The series II presents the energy levels for $n=20$; the values of $\alpha$ are labeled as before.

Figure 4

(Color online) Different regions where the ground state (lower curve, blue) and the first excited state (red) undergo level anticrossings, plotted for $n=30$ and $\alpha =5$.

Figure 5

(Color online) Square of the modulus of the inner product of the two lowest levels with the states ∣⇒⟩ and ∣⇑⟩ which are the ground states of $H\left(s\right)$ for $s=0$ and 1, respectively, plotted here for $n=50$ and $\alpha =5$. (a) and (b) present, respectively, the evolution of the ground- and first-excited-state projections along the Hamiltonian path. (c) combines (a) and (b), and clearly displays the exchange between these two states at $s=s_c$.

Figure 6

(Color online) Square modulus of the projections of the ground state immediately after the anticrossing and of the first excited state immediately before the anticrossing along the Dicke basis, plotted for $n=100$ and $\alpha =5$. Note that a logarithmic scale is used.

Figure 7

(Color online) Reduced concurrence in the thermodynamic limit obtained by tracing over $n-2$ arbitrary qubits, as a function of $\alpha$ and $s$. For $\alpha >{\alpha }_c$ the concurrence presents a discontinuity which increases with $\alpha$.

Figure 8

(Color online) Scaling of the energy gap at the QPT point with $n$ for different values of the $\alpha$ parameter. The curves range from $\alpha =0$ to 3. One observes a clear crossover at the critical value ${\alpha }_c=3\sqrt{3}∕2$ (yellow) with a $\nu$ value close to $4∕3$.