Anti-crossings and spectral gap during quantum adiabatic evolution

Abstract

We aim to give more insights into adiabatic evolution concerning the occurrence of anti-crossings and their link to the spectral minimum gap $\Delta _{\min }$. We study in detail adiabatic quantum computation applied to a specific combinatorial problem called weighted max k-clique. A clear intuition of the parametrization introduced by V. Choi is given which explains why the characterization is not general enough. We show that the instantaneous vectors involved in the anti-crossing vary brutally through it making the instantaneous ground-state hard to follow during the evolution. This result leads to a relaxation of the parametrization to be more general.

A Appendices

To illustrate the intuition on the $a_k$’s, this graph (Fig. 9) is the same structure of the graph (Fig. 2) where nodes 1 and 3 are swapped as well as nodes 5 and 6. We keep the same weights vector $w=[1,1,1,1.5,1.5,1.5]$.

Fig. 9

Toy example 2

For $\alpha =0.2$, this produces the following final states according to their energy (Fig. 10a) and the eigenvalues evolution (Fig. 10b):

Fig. 10

States a with their energy and $E_i(s)$ b during evolution for toy example 9 with $\alpha =0.2$

With this instance, we observe that the final slope of $E_0(s)$ comes from the slope of $E_2(s)$; therefore, we expect that $g_2(s)$ becomes dominant before transmitting to $g_1(s)$ after the first anti-crossing between $E_2$ and $E_1$. Eventually, the anti-crossing of $E_1$ and $E_0$ will produce a crossing of $g_1$ and $g_0$. The latter will become dominant till being equal to 1 at the end. Now, focusing on the slope of $E_0(s)$ before the anti-crossing, following the different successive anti-crossings, the jumps end up in the 4th energy level. In terms of $a_k$’s and $b_k$’s, this means that $a_3(s)$ becomes important just before the anti-crossing and crosses $a_0(s)$ at the anti-crossing. The same goes for $b_3(s)$ and $b_0(s)$. The plots below support these previous analyses.

Fig. 11

$a_k$ (a), $b_k$ (b) and $g_k$ (c) during evolution for toy Example 9 with $\alpha =0.2$

Remarks:

• 1. In Fig. 11a, we only see one specific situation, namely the crossing of $a_3$ with $a_0$ at the anti-crossing point $s^*$ between $E_0$ and $E_1$. This means that the curve $E_0(s)$ was going toward the 4th energy level in terms of the slope before $s^*$ and immediately change its slope toward its final direction of the 1st energy level. Hypothetically, we could observe $a_3$ crossing $a_1$, which will indicate that $E_0(s)$ change its direction toward the 2nd energy level, then $a_1$ crossing $a_0$. In this hypothetical case, $E_0(s)$ undergoes two anti-crossings.

• 2. In Fig. 11b, we focus on the behavior of $|E_1(s)\rangle$. Here, we understand that $E_1(s)$ first went toward the 3rd energy level before changing direction toward the lowest energy level. This is $b_2$ crossing $b_0$ when the first anti-crossing between $E_1$ and $E_2$ occurs. Then, it takes the direction of the 4th energy level when $b_0$ crosses $b_3$. Indeed, it fetches the direction of $E_0$ before anti-crossing (remember it was $a_3$ which was dominant at this point). Then, again it takes back its initial direction with $b_2$ becoming dominant at the second anti-crossing between $E_1$ and $E_2$. Eventually, it smoothly changes its direction toward its final goal.

• 3. In Fig. 11c, the point of view is quite different as we look from the final lowest energy position $E_0(1)$ and see from where it comes. We see in Fig. 10b that the final blue slope undergoes two anti-crossing before becoming blue. Indeed, it starts green and then jumps to red and finally blue. These successive anti-crossings appear on the plot of $g_k$; first, $g_2$ is dominant, and then at the first anti-crossing between $E_2$ and $E_1$, $g_2$ crosses with $g_1$. Now, the final slope of $E_0$ is transported by $E_1$. Eventually, $E_1$ anti-crosses $E_0$ so $g_1$ crosses $g_0$ and the evolution (at least for the ground-state) can finish peacefully.