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=j(SjxSj+1x+SjySj+1y+ΔSjzSj+1z)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 424\hbar*2 in increments of 600.

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

Among them, the state 1010\mid 1010\rangle and 0101\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 0000\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.




Hikaru Wakaura
個人研究者の若浦 光です。量子アルゴリズムの実装結果や論文の紹介などを載せていきます。 mail: hikaruwakaura@gmail.com
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Hikaru Wakaura
個人研究者の若浦 光です。量子アルゴリズムの実装結果や論文の紹介などを載せていきます。 mail: hikaruwakaura@gmail.com
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