# Simulation of time evolution in an integrable system, XXZ model.

In this article, we will simulate the time evolution of an integrable system, the XXZ system, using adiabatic quantum simulation. The probability of starting the time evolution from a certain state and coming back to it (initial state is observed) is called the Loschmidt echo. This probability can be derived by hand calculation, but it is faster to use a quantum computer to solve it, since it can also calculate the existence probabilities of other states. The Hamiltonian is

﻿$H=\sum_{j}(S_j^xS_{j+1}^x+S_j^yS_{j+1}^y+\Delta S_{j}^zS_{j+1}^z)$﻿ ,

and the periodic boundary condition is applied. In this case, we will set N=4. The time range to be sampled was calculated as ﻿$4\hbar*2$﻿ in increments of 600.

As a result, as shown in Fig. 1, the initial state ﻿$\mid 1010 \rangle$﻿ , the initial state ﻿$\mid 0101 \rangle$﻿ , and the state ﻿$\mid 1100 \rangle$﻿ changed at different periods as shown in Fig. 1 for Δ=2.

Among them, the state ﻿$\mid 1010\rangle$﻿ and ﻿$\mid 0101\rangle$﻿ became a composite wave with smaller waves and changed periodically like a Rabi oscillation. However, as shown in Figure 2, when Δ=0, the probability of existence in state ﻿$\mid 0000\rangle$﻿ increases monotonically, and eventually only this state is found, although there is a periodic change.

The difference between an integrable system and a non-integrable system can be seen in the existence probability. Therefore, we may eventually find a new integrable system on a quantum computer.

Fig. 1 Existence probability of each state at Δ=2 for time t/T.

Fig. 2 Time evolution at Δ=1 for time t/T.