QDET demo (#1327)

**Title:**
QDET demo 

**Summary:**

**Relevant references:**

**Possible Drawbacks:**

**Related GitHub Issues:**

----
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- [ ] Tutorial
- [ ] Demo
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---------

Co-authored-by: Diksha Dhawan <ddhawan@umich.edu>
Co-authored-by: Diksha Dhawan <40900030+ddhawan11@users.noreply.github.com>
Co-authored-by: agran2018 <45397799+agran2018@users.noreply.github.com>
Co-authored-by: ixfoduap <40441298+ixfoduap@users.noreply.github.com>
Co-authored-by: Josh Izaac <josh146@gmail.com>
Co-authored-by: Daniela Angulo <42325731+daniela-angulo@users.noreply.github.com>

Author: 6 people

_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_how_to_build_qdet_hamiltonian.png

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+r"""Quantum Defect Embedding Theory (QDET)
+=========================================
+Many interesting problems in quantum chemistry and materials science feature a strongly correlated
+region embedded within a larger environment. Example of such systems include point defects in
+materials [#Galli]_, active site of catalysts [#SJRLee]_ and surface phenomenon such as adsorption
+[#Gagliardi]_. Such systems can be accurately simulated with **embedding theories**, which effectively
+capture the strong electronic correlations in the active region with high accuracy, while accounting
+for the environment in a more approximate manner.
+
+In this demo, we show how to implement quantum defect embedding theory (QDET) [#Galli]_. This method
+has been successfully applied to study systems such as defects in calcium oxide [#Galli]_ and to calculate
+excitations of the negatively charged nitrogen-vacancy defect in diamond [#Galli2]_. QDET can be used to calculate ground states,
+excited states and dynamic properties of materials. These make QDET a powerful method for affordable quantum simulation
+of materials. Another important advantage
+of QDET is the compatibility of the method with quantum algorithms as we explain in the following
+sections.
+
+.. figure:: ../_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_how_to_build_qdet_hamiltonian.png
+    :align: center
+    :width: 70%
+    :target: javascript:void(0)
+
+"""
+
+#############################################
+# The main component of a QDET simulation is to construct an effective Hamiltonian that describes
+# the impurity subsystem and its interaction with the environment. In second quantization, the
+# effective Hamiltonian can be represented in terms of electronic creation, :math:`a^{\dagger}`, and
+# annihilation , :math:`a`, operators as
+#
+# .. math::
+#
+#     H^{eff} = \sum_{ij} t_{ij}^{eff}a_i^{\dagger}a_j + \frac{1}{2}\sum_{ijkl} v_{ijkl}^{eff}a_i^{\dagger}a_{j}^{\dagger}a_ka_l,
+#
+# where :math:`t_{ij}^{eff}` and :math:`v_{ijkl}^{eff}` represent the effective one-body and
+# two-body integrals, respectively, and the indices :math:`ijkl` span over the orbitals inside the impurity.
+# This Hamiltonian describes a simplified representation of the quantum system that is more
+# computationally tractable, while properly capturing the essential physics of the problem.
+#
+# Implementation
+# --------------
+# A QDET simulation typically starts by obtaining a mean field approximation of the whole system
+# using efficient quantum chemistry methods such as density functional theory (DFT). These
+# calculations provide a set of orbitals that are partitioned into **impurity** and **bath** orbitals.
+# The effective Hamiltonian is constructed from the impurity orbitals and is subsequently solved
+# by using either a high accuracy classical method or a quantum algorithm. Let's implement these
+# steps for an example!
+#
+# Mean field calculations
+# ^^^^^^^^^^^^^^^^^^^^^^^
+# We implement QDET to compute the excitation energies of a negatively charged nitrogen-vacancy
+# defect in diamond [#Galli2]_. We use DFT to obtain a mean field description of the whole system.
+# The DFT calculations are performed with the `QUANTUM ESPRESSO <https://www.quantum-espresso.org/>`_
+# package. This requires downloading pseudopotentials [#Modji]_ for each atomic species
+# in the system from the QUANTUM ESPRESSO `database <https://www.quantum-espresso.org/pseudopotentials/>`_.
+# To prepare our system, the necessary carbon and nitrogen pseudopotentials can be downloaded by executing
+# the following commands through the terminal or command prompt:
+#
+# .. code-block:: bash
+#
+#    wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/C_ONCV_PBE-1.2.upf
+#    wget -N -q http://www.quantum-simulation.org/potentials/sg15_oncv/upf/N_ONCV_PBE-1.2.upf
+#
+# Next, we need to create the input file for running QUANTUM ESPRESSO. This contains
+# information about the system and the DFT calculations. More details on how to construct the input
+# file can be found in the QUANTUM ESPRESSO `documentation <https://www.quantum-espresso.org/Doc/INPUT_PW.html>`_
+# page. For the system taken here, the input file can be downloaded with
+#
+# .. code-block:: bash
+#
+#    wget -N -q https://west-code.org/doc/training/nv_diamond_63/pw.in
+#
+# DFT calculations can now be initiated using the ``pw.x`` executable in ``WEST``, taking ``pw.in`` as the input file
+# and directing the output to ``pw.out``. This process is parallelized across 2 cores using ``mpirun``.
+#
+# .. code-block:: bash
+#
+#    mpirun -n 2 pw.x -i pw.in > pw.out
+#
+# Identify the impurity
+# ^^^^^^^^^^^^^^^^^^^^^
+# Once we have obtained the mean field description, we can identify our impurity by finding
+# the states that are localized around the defect region in real space. To do that, we compute the
+# localization factor :math:`L_n` for each state ``n``, defined as:
+#
+# .. math::
+#
+#     L_n = \int_{V \in \Omega} d^3 r |\Psi_n(r)|^2,
+#
+# where :math:`V` is the identified volume including the impurity within the supercell volume
+# :math:`\Omega` and :math:`\Psi` is the wavefunction [#Galli2]_. We will use the
+# `WEST <https://pubs.acs.org/doi/10.1021/ct500958p>`_ program to compute the localization factor.
+# This requires the ``westpp.in`` input file, example for which is shown below. Here, we specify the
+# box parameters within which the localization factor is being computed; the vectors for this box are provided in
+# in atomic units as ``[x_start, x_end, y_start, y_end, z_start, z_end]``.
+#
+# .. rst-class:: sphx-glr-script-out
+#
+#   .. code-block:: none
+#
+#    westpp_control:
+#      westpp_calculation: L # triggers the calculation of the localization factor
+#      westpp_range:         # defines the range of states to compute the localization factor
+#      - 1                   # start from the first state
+#      - 176                 # use all 176 states
+#      westpp_box:           # specifies the parameter of the box in atomic units for integration
+#      - 6.19
+#      - 10.19
+#      - 6.28
+#      - 10.28
+#      - 6.28
+#      - 10.28
+#
+# The calculation can now be performed by running the ``westpp.x`` executable from WEST using mpirun to
+# parallelize it across two cores.
+#
+# .. code-block:: bash
+#
+#    mpirun -n 2 westpp.x -i westpp.in > westpp.out
+#
+# This creates the file ``westpp.json`` which contains the information we need here. Since
+# computational resources required to run the calculation are large, for the purpose of this tutorial we just
+# download a pre-computed file with:
+#
+# .. code-block:: bash
+#
+#    mkdir -p west.westpp.save
+#    wget -N -q https://west-code.org/doc/training/nv_diamond_63/box_westpp.json -O west.westpp.save/westpp.json
+#
+# We can plot the computed localization factor for each of the states:
+#
+# .. code-block:: bash
+#
+#    import json
+#    import numpy as np
+#    import matplotlib.pyplot as plt
+#
+#    with open('west.westpp.save/westpp.json','r') as f:
+#        data = json.load(f)
+#
+#    y = np.array(data['output']['L']['K000001']['local_factor'],dtype='f8')
+#    x = np.array([i+1 for i in range(y.shape[0])])
+#
+#    plt.plot(x,y,'o')
+#    plt.axhline(y=0.08,linestyle='--',color='red')
+#
+#    plt.xlabel('Kohn-Sham orbital index')
+#    plt.ylabel('Localization factor')
+#
+#    plt.show()
+#
+#
+# .. figure:: ../_static/demonstration_assets/qdet/localization.jpeg
+#    :align: center
+#    :width: 70%
+#    :target: javascript:void(0)
+#
+# From this plot, it is easy to see that the orbitals can be categorized as orbitals with low and
+# high localization factors. For the purpose of defining an impurity, we need highly localized
+# orbitals, so we set a cutoff of :math:`0.06`, illustrated by the red dashed line, and choose the orbitals that have a localization
+# factor larger than :math:`0.06`. We'll use these orbitals for the calculation of the parameters
+# needed to construct the effective Hamiltonian.
+#
+# Electronic Integrals
+# ^^^^^^^^^^^^^^^^^^^^
+# The next step in QDET is to define the effective one-body and two-body integrals for the impurity.
+# The effective two-body integrals :math:`v^{eff}` are computed first as matrix elements of the
+# partially screened static Coulomb potential :math:`W_0^{R}`.
+#
+# .. math::
+#
+#     v_{ijkl}^{eff} = [W_0^{R}]_{ijkl},
+#
+# where :math:`W_0^R` results from screening the bare Coulomb potential :math:`v` with the reduced
+# polarizability :math:`P_0^R = P - P_{imp}`, where :math:`P` is the system's polarizability and
+# :math:`P_{imp}` is the impurity's polarizability. Since solving the effective Hamiltonian
+# accounts for the exchange and correlation interactions between the active electrons, we remove
+# these interactions from the Kohn-Sham Hamiltonian :math:`H^{KS}` to avoid double counting them.
+#
+# The one-body term :math:`t^{eff}` is obtained by subtracting from the Kohn-Sham Hamiltonian the
+# double-counting term accounting for electrostatic and exchange-correlation interactions in the
+# active space.
+#
+# .. math::
+#
+#     t_{ij}^{eff} = H_{ij}^{KS} - t_{ij}^{dc}.
+#
+# We use the WEST program to compute these parameters. WEST will first compute the
+# quasiparticle energies, then the partially screened Coulomb potential, and finally
+# the parameters of the effective Hamiltonian. The software offers several execution modes through the
+# :code:`wfreq_calculation` keyword. We choose ``XWGQH`` to ensure the full workflow is executed, which computes
+# both the quasiparticle corrections and the final parameters for the QDET effective Hamiltonian. The input file
+# for such a calculation is shown below:
+#
+# .. rst-class:: sphx-glr-script-out
+#
+#   .. code-block:: none
+#
+#    wstat_control:
+#      wstat_calculation: S           # starts the calculation from scratch
+#      n_pdep_eigen: 512              # number of eigenpotentials; matches number of electrons
+#      trev_pdep: 0.00001             # convergence threshold for eigenvalues
+#
+#    wfreq_control:
+#      wfreq_calculation: XWGQH       # compute quasiparticle corrections and Hamiltonian params
+#      macropol_calculation: C        # include long-wavelength limit for condensed systems
+#      l_enable_off_diagonal: true    # calculate off-diagonal elements of G_0-W_0 self-energy
+#      n_pdep_eigen_to_use: 512       # number of PDEP eigenvectors to be used
+#      qp_bands: [87,122,123,126,127,128] # impurity orbitals
+#      n_refreq: 300                  # number of frequencies on the real axis
+#      ecut_refreq: 2.0               # cutoff for the real frequencies
+#
+# We can now execute the calculation with:
+#
+# .. code-block:: bash
+#
+#    mpirun -n 2 wfreq.x -i wfreq.in > wfreq.out
+#
+# This calculation takes some time and requires computational resources, therefore we download a
+# pre-computed output file with
+#
+# .. code-block:: bash
+#
+#    mkdir -p west.wfreq.save
+#    wget -N -q https://west-code.org/doc/training/nv_diamond_63/wfreq.json -O west.wfreq.save/wfreq.json
+#
+# This output file contains all the information we need to construct the effective Hamiltonian.
+#
+# Effective Hamiltonian
+# ^^^^^^^^^^^^^^^^^^^^^
+# We now construct the effective Hamiltonian by importing the electron integral results and using
+# WEST:
+#
+# .. code-block:: python
+#
+#    from westpy.qdet import QDETResult
+#
+#    effective_hamiltonian = QDETResult(filename='west.wfreq.save/wfreq.json')
+#
+# The effective Hamiltonian can be solved using a high level method such as the full configuration
+# interaction (FCI) algorithm from WEST as:
+#
+# .. code-block:: python
+#
+#    solution = effective_hamiltonian.solve()
+#
+# Using :code:`solve()` prints the excitation energies, spin multiplicity and relative occupation of
+# the active orbitals.
+#
+# .. rst-class:: sphx-glr-script-out
+#
+#   .. code-block:: none
+#
+#    ======================================================================
+#    Building effective Hamiltonian...
+#    nspin: 1
+#    occupations: [[2. 2. 2. 2. 1. 1.]]
+#    =====================================================================
+#                   diag[1RDM - 1RDM(GS)]
+#       E [eV] char                    87    122    123    126   127   128
+#    0  0.000   3-                 0.000  0.000  0.000  0.000 0.000 0.000
+#    1  0.436   1-                -0.001 -0.009 -0.018 -0.067 0.004 0.091
+#    2  0.436   1-                -0.001 -0.009 -0.018 -0.067 0.092 0.002
+#    3  1.251   1-                -0.002 -0.019 -0.023 -0.067 0.054 0.057
+#    4  1.939   3-                -0.003 -0.010 -0.127 -0.860 1.000 0.000
+#    5  1.940   3-                -0.003 -0.010 -0.127 -0.860 0.000 1.000
+#    6  2.935   1-                -0.000 -0.032 -0.043 -0.855 0.929 0.002
+#    7  2.936   1-                -0.000 -0.032 -0.043 -0.855 0.002 0.929
+#    8  4.661   1-                -0.006 -0.054 -0.188 -1.672 0.960 0.960
+#    9  5.080   3-                -0.014 -0.698 -0.213 -0.075 1.000 0.000
+#    ----------------------------------------------------------------------
+#
+# The solution object is a dictionary that contains information about the FCI eigenstates of the
+# system, which includes various excitation energies, spin multiplicities, eigenvectors etc.
+# More importantly, while FCI handles small embedded effective Hamiltonians with ease, it quickly
+# hits a wall with larger impurities. This is precisely where quantum computing steps in, offering
+# the scalability needed to tackle such complex systems. The first step to solving these effective
+# Hamiltonians via quantum algorithms in PennyLane, is to convert them to qubit Hamiltonians.
+#
+# Quantum Simulation
+# ^^^^^^^^^^^^^^^^^^
+# We now map the effective Hamiltonian to the qubit basis. Note that the two-electron integrals obtained
+# before are represented in Chemists' notation and need to be converted to a notation
+# that is compatible with PennyLane. Here's how to construct the `qubit Hamiltonian <https://pennylane.ai/qml/demos/tutorial_fermionic_operators>`__:
+#
+# .. code-block:: python
+#
+#    from pennylane.qchem import one_particle, two_particle, observable
+#    import numpy as np
+#
+#    effective_hamiltonian = QDETResult(filename="west.wfreq.save/wfreq.json")
+#
+#    one_e, two_e = effective_hamiltonian.h1e, effective_hamiltonian.eri
+#
+#    t = one_particle(one_e[0])
+#    v = two_particle(np.swapaxes(two_e[0][0], 1, 3))
+#    qubit_op = observable([t, v], mapping="jordan_wigner")
+#    print("Qubit Hamiltonian: ", qubit_op)
+#
+# .. rst-class:: sphx-glr-script-out
+#
+#   .. code-block:: none
+#
+#       Qubit Hamiltonian:  (2.40331309905556+0j) * I(0) + (-0.12208093833046951+0j) * Z(0) + (-0.12208093833046951+0j) * Z(1) + (-0.003330743747901097+0j) * (Y(0) @ Z(1) @ Y(2)) +  ...
+#
+# We can use this Hamiltonian in a quantum algorithm such as `quantum phase estimation (QPE) <https://pennylane.ai/qml/demos/tutorial_qpe>`__.
+# You can compare the results and verify that the computed energies from quantum algorithm
+# match those that we obtained before.
+#
+# Conclusion
+# ----------
+# Quantum defect embedding theory is a novel framework for simulating strongly correlated
+# quantum systems and has been successfully used for studying defects in solids. Applicability of
+# QDET however is not limited to defects: it can be used for other systems where a strongly
+# correlated subsystem is embedded in a weakly correlated environment. Additionally, QDET is able to
+# correct the interaction double counting issue within the active space faced by a variety of
+# other embedding theories. The Green's function based formulation of QDET ensures
+# exact removal of double counting corrections at the `GW <https://en.wikipedia.org/wiki/GW_approximation>`__
+# level of theory thus removing the
+# approximation present in the initial DFT based formulation. This formulation also helps to capture
+# the response properties and provides access to excited state properties. Another major advantage
+# of QDET is the ease with which it can be used with quantum computers in a hybrid framework [#Baker]_.
+# Given its ability to access excited states and its hybrid quantum compatibility, potential future extensions
+# could involve applying QDET to the wider field of spectroscopy, such as investigating
+# core-level excited states relevant to `X-ray Absorption Spectroscopy (XAS) <https://pennylane.ai/qml/demos/tutorial_xas>`__.
+# In conclusion, QDET is a powerful embedding approach for simulating complex quantum systems.
+#
+# References
+# ----------
+#
+# .. [#Galli]
+#    J. Davidsson, M. Onizhuk, C. Vorwerk, G. Galli,
+#    "Discovery of atomic clock-like spin defects in simple oxides from first principles",
+#    `arXiv:2302.07523 <https://arxiv.org/abs/2302.07523>`__.
+#
+# .. [#SJRLee]
+#    S. J. R. Lee, F. Ding, F. R. Manby, T. F. Miller III,
+#    "Analytical Gradients for Projection-Based Wavefunction-in-DFT Embedding"
+#    `arXiv:1903.05830 <https://arxiv.org/abs/1903.05830>`__.
+#
+# .. [#Gagliardi]
+#    A. Mitra, Matthew Hermes, M. Cho, V. Agarawal, L. Gagliardi,
+#    "Periodic Density Matrix Embedding for CO Adsorption on the MgO(001)Surface"
+#    `J. Phys. Chem. Lett. 2022, 13, 7483 <https://pubs.acs.org/doi/10.1021/acs.jpclett.2c01915>`__.
+#
+# .. [#Galli2]
+#    N. Sheng, C. Vorwerk, M. Govoni, G. Galli,
+#    "Green's function formulation of quantum defect embedding theory",
+#    `arXiv:2203.05493 <https://arxiv.org/abs/2203.05493>`__.
+#
+# .. [#Modji]
+#    M. S. Zini, A. Delgado, *et al.*,
+#    "Quantum simulation of battery materials using ionic pseudopotentials"
+#    `arXiv:2302.07981 <https://arxiv.org/abs/2302.07981>`__.
+#
+# .. [#Baker]
+#    Jack S. Baker, Pablo A. M. Casares, *et al.*,
+#    "Simulating optically-active spin defects with a quantum computer"
+#    `arXiv:2405.13115 <https://arxiv.org/abs/2405.13115>`__.
+#