Updated example for ground state energy with QPE

documentation/source/reference/Examples/GroundStateEnergyQPE.rst

@@ -5,43 +5,58 @@ Ground State Energy with QPE
 
 .. currentmodule:: qrisp
 
+
+For a given Hamiltonian $H$ and an eigenstate $\ket{\lambda_j}$ (e.g., the ground state $\ket{\lambda_0}$), the goal is to find the eigenvalue $E_j$ such that
+
+$$e^{iH}\\ket{\\lambda_j} = e^{iE_j}\\ket{\\lambda_j}$$
+
+This is achieved by applying quantum phase estimation to the unitary $U=e^{iH}$ with the initial state $\ket{\lambda_j}$. 
+With the equation $e^{iE_j}=e^{2\pi i\theta_j}$, the energy $E_j$ can be calculated from the resulting phase $\theta_j\in [0,1)$.
+In practice, the exact eigenstate $\lambda_j$ is unknown, and one uses an approximation $\ket{\psi_0}$ which can be expressed in the eigenbasis of the Hamiltonian as
+
+$$\\ket{\\psi_0}=\\sum_j a_j\\ket{\\lambda_j}$$
+
+If the approximation $\ket{\psi_0}$ is sufficently close to the eigenstate $\ket{\lambda_j}$, the correct phase, and hence the eigenvalue $E_j$, is obtained with high probability.
+
+
 Example Hydrogen
 ================
 
-We caluclate the ground state energy of the Hydrogen molecule with :ref:`Quantum Phase Estimation <QPE>`. 
+We caluclate the ground state energy of the electronic Hamiltonian for the Hydrogen molecule with :ref:`Quantum Phase Estimation <QPE>`. 
 
 Utilizing symmetries, one can find a reduced two qubit Hamiltonian for the Hydrogen molecule. Frist, we define the Hamiltonian, and compute the 
 ground state energy classically.
 
 ::
 
     from qrisp import QuantumVariable, x, QPE
-    from qrisp.operators.pauli.pauli import X,Y,Z
+    from qrisp.operators import X,Y,Z
     import numpy as np
 
     # Hydrogen (reduced 2 qubit Hamiltonian)
     H = -1.05237325 + 0.39793742*Z(0) -0.39793742*Z(1) -0.0112801*Z(0)*Z(1) + 0.1809312*X(0)*X(1)
     E0 = H.ground_state_energy()
+    print(E0)
     # Yields: -1.85727502928823
 
-In the following, we utilize the ``trotterization`` method of the :ref:`QubitOperator` to obtain a function **U** that applies **Hamiltonian Simulation**
-via Trotterization. If we start in a state that is close to the ground state and apply :ref:`QPE`, we get an estimate of the ground state energy.
+The $\ket{10}$ state provides a good approximation of the ground state:
 
 ::
 
     # ansatz state
     qv = QuantumVariable(2)
     x(qv[0])
     E1 = H.get_measurement(qv)
-    E1
+    print(E1)
     # Yields: -1.83858104676077
 
-
-We calculate the ground state energy with quantum phase estimation. As the results of the phase estimation are modulo :math:`2\pi`, we subtract :math:`2\pi`.
+In the following, we utilize the :meth:`trotterization <qrisp.operators.qubit.QubitOperator.trotterization>` method of the :ref:`QubitOperator` to obtain a function **U** that applies **Hamiltonian Simulation**
+via Trotterization. If we start in a state that is close to the ground state and apply :ref:`QPE`, we get an estimate of the ground state energy.
+As the results of the phase estimation are modulo :math:`2\pi` and the searched for eigenvalue is between $-2\pi$ and 0, we subtract :math:`2\pi`.
 
 ::
 
-    U = H.trotterization()
+    U = H.trotterization(forward_evolution=False)
 
     qpe_res = QPE(qv,U,precision=10,kwargs={"steps":3},iter_spec=True)
 
@@ -50,5 +65,5 @@ We calculate the ground state energy with quantum phase estimation. As the resul
     
     phi = list(sorted_results.items())[0][0]
     E_qpe = 2*np.pi*(phi-1) # Results are modulo 2*pi, therefore subtract 2*pi 
-    E_qpe
+    print(E_qpe)
     # Yields: -1.8591847149174