Ising model example

Author: renezander90

documentation/source/_static/Ising_chain_N=6.png

@@ -0,0 +1,84 @@
+.. _IsingModel:
+
+Transverse Field Ising Model
+============================
+
+.. currentmodule:: qrisp
+
+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.pauli 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()
+
+.. figure:: /_static/Ising_chain_N=6.png
+   :alt: Ising chain
+   :align: center
+

documentation/source/reference/Examples/IsingModel.rst

@@ -39,7 +39,9 @@ In this section, we provide a glimpse into the diverse range of applications tha
      - Calculating molecular potential energy curves with :ref:`VQE`.
    * - :ref:`GroundStateEnergyQPE` 
      - Calculating ground state energies with quantum phase estimation.                    
-      
+   * - :ref:`IsingModel` 
+     - Hamiltonian simulation of a transverse field Ising model. 
+
 
 .. toctree::
    :hidden:
@@ -60,3 +62,4 @@ In this section, we provide a glimpse into the diverse range of applications tha
    Shor
    MolecularPotentialEnergyCurve
    GroundStateEnergyQPE
+   IsingModel