[WIP] How to build a vibrational Hamiltonian

demonstrations/tutorial_how_to_build_vibrational_hamiltonians.metadata.json

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+{
+    "title": "How to build vibrational Hamiltonians",
+    "authors": [
+        {
+            "username": "ddhawan"
+        },
+        {
+            "username": "austingmhuang"
+        },
+        {
+            "username": "soran"
+        }
+    ],
+    "dateOfPublication": "2025-07-10T00:00:00+00:00",
+    "dateOfLastModification": "2025-07-10T00:00:00+00:00",
+    "categories": [
+        "Getting Started",
+        "How-to"
+    ],
+    "tags": [],
+    "previewImages": [
+        {
+            "type": "thumbnail",
+            "uri": "/_static/demo_thumbnails/regular_demo_thumbnails/thumbnail_how_to_build_spin_hamiltonians.png"
+        },
+        {
+            "type": "large_thumbnail",
+            "uri": "/_static/demo_thumbnails/large_demo_thumbnails/thumbnail_large_how_to_build_spin_hamiltonians.png"
+        }
+    ],
+    "seoDescription": "Learn how to build vibrational Hamiltonians with PennyLane.",
+    "doi": "",
+    "references": [
+        {
+            "id": "loaiza",
+            "type": "article",
+            "title": "Simulating near-infrared spectroscopy on a quantum computer for enhanced chemical detection",
+            "authors": "I. Loaiza *et al.*",
+            "journal": "arXiv:2504.10602",
+            "year": "2025",
+            "url": "https://arxiv.org/abs/2504.10602"
+        },
+        {
+            "id": "motlagh",
+            "type": "article",
+            "title": "Quantum Algorithm for Vibronic Dynamics: Case Study on Singlet Fission Solar Cell Design",
+            "authors": "D. Motlagh *et al.*",
+            "journal": "arXiv:2411.13669",
+            "year": "2024",
+            "url": "https://arxiv.org/abs/2411.13669"
+        }
+    ],
+    "basedOnPapers": [],
+    "referencedByPapers": [],
+    "relatedContent": [
+        {
+            "type": "demonstration",
+            "id": "tutorial_quantum_chemistry",
+            "weight": 1.0
+        },
+        {
+            "type": "demonstration",
+            "id": "tutorial_how_to_build_spin_hamiltonians",
+            "weight": 1.0
+        }
+    ]
+}

demonstrations/tutorial_how_to_build_vibrational_hamiltonians.py

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+r"""How to build vibrational Hamiltonians
+=========================================
+Vibrational motions are crucial for describing the quantum properties of molecules and
+materials. Molecular vibrations can affect the outcome of chemical reactions and there are several
+vibrational spectroscopy techniques that provide valuable insight into understanding chemical
+systems and processes [#loaiza]_ [#motlagh]_. Classical quantum
+computations have been routinely implemented to describe vibrational motions of molecules. However,
+for challenging vibrational systems, classical methods typically have fundamental theoretical
+limitations that prevent their practical implementation. This makes quantum algorithms an ideal
+choice where classical methods are not efficient or accurate.
+
+Quantum algorithms require a precise description of the system Hamiltonian to compute vibrational
+properties. In this demo, we learn how to use PennyLane features to construct and manipulate
+different representations of vibrational Hamiltonians. We also briefly discuss the implementation of
+the Hamiltonian in an interesting quantum algorithm for computing the vibrational dynamics of a
+molecule.
+
+.. figure:: ../_static/demo_thumbnails/opengraph_demo_thumbnails/OGthumbnail_vibrational_hamiltonian.png
+    :align: center
+    :width: 70%
+    :target: javascript:void(0)
+"""
+
+######################################################################
+# Vibrational Hamiltonian
+# -----------------------
+# A molecular vibrational Hamiltonian can be defined in terms of the kinetic energy operator of the
+# nuclei, :math:`T`, and the potential energy operator, :math:`V`, which describes the interactions
+# between the nuclei as:
+#
+# .. math::
+#
+#     H = T + V.
+#
+# The kinetic and potential energy operators can be written in terms of momentum and position
+# operators, respectively. There are several ways to construct the potential energy operator which
+# lead to different representations of the vibrational Hamiltonian. Here we explain some of these
+# representations and provide PennyLane codes for constructing them.
+#
+# Christiansen representation
+# ^^^^^^^^^^^^^^^^^^^^^^^^^^^
+# The Christiansen representation of the vibrational Hamiltonian relies on the n-mode expansion of
+# the potential energy surface :math:`V` over vibrational normal coordinates :math:`Q`.
+#
+# .. math::
+#
+#     V({Q}) = \sum_i V_1(Q_i) + \sum_{ij} V_2(Q_i,Q_j) + ...,
+#
+# This provides a general representation of the potential energy surface where the terms :math:`V_n`
+# depend on :math:`n` vibrational modes at most.
+#
+# The Christiansen Hamiltonian is then constructed in second-quantization based on this potential
+# energy surface in terms of bosonic creation :math:`b^{\dagger}` and annihilation :math:`b`
+# operations:
+#
+# .. math::
+#
+#     H = \sum_{i}^M \sum_{k_i, l_i}^{N_i} C_{k_i, l_i}^{(i)} b_{k_i}^{\dagger} b_{l_i} +
+#         \sum_{i<j}^{M} \sum_{k_i,l_i}^{N_i} \sum_{k_j,l_j}^{N_j} C_{k_i k_j, l_i l_j}^{(i,j)}
+#         b_{k_i}^{\dagger} b_{k_j}^{\dagger} b_{l_i} b_{l_j},
+#
+# where :math:`M` represents the number of normal modes and :math:`N` is the number of modals.
+# Recall that a modal is a one-mode vibrational wave function defined as a function of a normal
+# coordinate. The coefficients :math:`C` represent n-mode integrals which depend on the
+# :math:`n`-mode contribution of the potential energy, :math:`V_n`, defined above.
+#
+# PennyLane provides a set of functions to construct the Christiansen Hamiltonian, either directly
+# in one step or from its building blocks step by step. An important step in both methods is to
+# construct the potential energy operator which is done based on
+# single-point energy calculations along the normal modes of the molecule. The
+# :func:`~.pennylane.qchem.vibrational_pes` function in PennyLane provides a convenient way to
+# perform the potential energy calculations for a given molecule. The calculations are typically
+# done by performing single-point energy calculations at small distances along the
+# normal mode coordinates. Computing the energies for each mode separately provides the term
+# :math:`V_1` while displacing atoms along two different modes simultaneously gives the term
+# :math:`V_2` and so on.
+#
+# Let's look at an example for the HF molecule.
+
+import numpy as np
+import pennylane as qml
+
+symbols  = ['H', 'F']
+geometry = np.array([[0.0, 0.0, 0.0], [0.0, 0.0, 1.79]])
+mol = qml.qchem.Molecule(symbols, geometry)
+pes = qml.qchem.vibrational_pes(mol, optimize=False)
+
+######################################################################
+# The :func:`~.pennylane.qchem.vibrational_pes` function creates a
+# :class:`~.pennylane.qchem.VibrationalPES` object that stores the potential energy surface
+# and vibrational information. This object is the input for several functions that we learn about
+# here. For instance, the :func:`~.pennylane.qchem.christiansen_integrals` function accepts this
+# object to compute the integrals needed to construct the Christiansen
+# Hamiltonian defined above.
+
+integrals = qml.qchem.vibrational.christiansen_integrals(pes, n_states=4)
+h_bosonic = qml.qchem.christiansen_bosonic(integrals[0])
+print(h_bosonic)
+
+######################################################################
+# The bosonic Hamiltonian constructed with :func:`~.pennylane.qchem.christiansen_bosonic` can be
+# mapped to its qubit form by using the :func:`~.pennylane.qchem.christiansen_mapping` function.
+
+h_qubit = qml.bose.christiansen_mapping(h_bosonic)
+h_qubit
+
+######################################################################
+# This provides a vibrational Hamiltonian in the qubit basis that can be used in any desired quantum
+# algorithm in PennyLane.
+#
+# Note that PennyLane also provides the :func:`~.pennylane.qchem.christiansen_hamiltonian` function
+# that uses the :class:`~.pennylane.qchem.VibrationalPES` object directly and builds the 
+# Christiansen Hamiltonian in qubit representation.
+
+h_christiansen = qml.qchem.vibrational.christiansen_hamiltonian(pes,n_states=4)
+
+######################################################################
+# You can verify that the two Hamiltonians are identical.
+#
+# Taylor representation
+# ^^^^^^^^^^^^^^^^^^^^^
+# The Taylor representation of the vibrational Hamiltonian relies on a Taylor expansion of
+# the potential energy surface :math:`V` in terms of the vibrational mass-weighted normal
+# coordinate operators :math:`q`.
+#
+# .. math::
+#
+#     V = V_0 + \sum_i F_i q_i + \sum_{ij} F_{ij} q_i,q_j + ....
+#
+# Note that the force constants :math:`F` are derivatives of the potential energy surface.
+#
+# The Taylor Hamiltonian can then be constructed by defining the kinetic and potential energy
+# components in terms of the momentum and position operators.
+#
+# .. math::
+#
+#     H = \sum_{i \ge j} K_{ij} p_i p_j + \sum_{i\ge j} \Phi_{ij}^{(2)} q_i q_j +
+#          \sum_{i \ge j \ge k} \Phi_{ijk}^{(3)} q_i q_j q_k + ...,
+#
+# where the coefficients :math:`\Phi` are obtained from the force constants :math:`F` after
+# rearranging the terms in the potential energy operator according to the number of different modes.
+#
+# The :func:`~.pennylane.qchem.taylor_coeffs` function computes the coefficients :math:`\Phi` from
+# a :class:`~.pennylane.qchem.VibrationalPES` object.
+
+one, two = qml.qchem.taylor_coeffs(pes, min_deg=2, max_deg=4)
+
+######################################################################
+# We can then use these coefficients to construct a bosonic form of the Taylor Hamiltonian.
+
+h_bosonic = qml.qchem.taylor_bosonic(coeffs=[one, two], freqs=pes.freqs, uloc=pes.uloc)
+print(h_bosonic)
+
+######################################################################
+# This bosonic Hamiltonian can be mapped to the qubit basis using mapping schemes of
+# :func:`~.pennylane.bose.binary_mapping` or :func:`~.pennylane.bose.unary_mapping` functions.
+
+h_qubit = qml.binary_mapping(h_bosonic, n_states=4)
+h_qubit
+
+######################################################################
+# This Hamiltonian can be used in any desired quantum algorithm in PennyLane.
+#
+# Note that PennyLane also provides the :func:`~.pennylane.qchem.taylor_hamiltonian` function
+# that uses the :class:`~.pennylane.qchem.VibrationalPES` object directly and builds the qubit
+# Taylor Hamiltonian.
+
+h_taylor = qml.qchem.vibrational.taylor_hamiltonian(pes, n_states=4)
+
+######################################################################
+# You can verify that the two Hamiltonians are identical.
+#
+# Conclusion
+# ----------
+# The `qchem module <https://docs.pennylane.ai/en/latest/code/qml_qchem.html>`__ in PennyLane
+# provides a set of tools that can be used to construct several representations of vibrational
+# Hamiltonians. Here we learned how to use these tools to build vibrational Hamiltonians
+# step-by-step and also use the PennyLane built-in functions to easily construct the Hamiltonians in
+# one step. The qchem module provides
+# `features <https://docs.pennylane.ai/en/latest/code/api/pennylane.qchem.vibrational_pes.html>`__
+# for constructing a vibrational potential energy surface efficiently using parallel executor
+# options. The modular design of the vibrational features also facilitate further extension of the
+# tools to support building other Hamiltonian types and representations and develop novel mapping
+# methods intuitively.
+#
+# References
+# ----------
+#
+# .. [#loaiza]
+#
+#     I. Loaiza *et al.*,
+#     "Simulating near-infrared spectroscopy on a quantum computer for enhanced chemical detection",
+#     arXiv:2504.10602, 2025.
+#
+# .. [#motlagh]
+#
+#     D. Motlagh *et al.*,
+#     "Quantum Algorithm for Vibronic Dynamics: Case Study on Singlet Fission Solar Cell Design",
+#     	arXiv:2411.13669, 2024.
+#
+# About the authors
+# -----------------
+#