Transverse Field Ising Model#

In this example, we study Hamiltonian dynamics of the transverse field Ising model defined by the Hamiltonian

$$H = -J\sum_{(i,j)\in E}Z_iZ_j + B\sum_{i\in V}X_i$$

for a lattice graph $G=(V,E)$ and real parameters $J, B$. We investigate the total magnetization of the system as it evolves under the Hamiltonian.

Here, we consider an Ising chain.

import matplotlib.pyplot as plt
import networkx as nx
import numpy as np

def generate_chain_graph(N):
    coupling_list = [[k,k+1] for k in range(N-1)]
    G = nx.Graph()
    G.add_edges_from(coupling_list)
    return G

G = generate_chain_graph(6)

We implement methods for creating the Ising Hamiltonian and the total magnetization observable for a given graph.

from qrisp import QuantumVariable
from qrisp.operators import X, Y, Z

def create_ising_hamiltonian(G, J, B):
    H = sum(-J*Z(i)*Z(j) for (i,j) in G.edges()) + sum(B*X(i) for i in G.nodes())
    return H

def create_magnetization(G):
    H = (1/G.number_of_nodes())*sum(Z(i) for i in G.nodes())
    return H

With all the necessary ingredients, we conduct the experiment: For varying evolution times $T$:

Prepare the $\ket{0}^{\otimes N}$ state.
Perform Hamiltonian simulation via Trotterization.
Measure the total magnetization.

T_values = np.arange(0, 2.0, 0.05)
M_values = []

M = create_magnetization(G)

for T in T_values:
    H = create_ising_hamiltonian(G,1.0,1.0)
    U = H.trotterization()

    qv = QuantumVariable(G.number_of_nodes())
    U(qv,t=-T,steps=5)
    M_values.append(M.get_measurement(qv,precision=0.005))

Finally, we visualize the results. As expected, the total magnetization decreases in the presence of a transverse field with increasing evolution time $T$.

import matplotlib.pyplot as plt
plt.scatter(T_values, M_values, color='#6929C4', marker="o", linestyle='solid', s=10, label='Magnetization')
plt.xlabel("Time", fontsize=15, color="#444444")
plt.ylabel("Magnetization", fontsize=15, color="#444444")
plt.legend(fontsize=12, labelcolor="#444444")
plt.tick_params(axis='both', labelsize=12)
plt.grid()
plt.show()