Solucionador de estados excitados¶

Introdução¶

507566668eed4e56ac3d8d55b01e176a

Neste tutorial vamos discutir a interface de cálculo de estados excitados do Qiskit Chemistry. O objetivo é computar os estados excitados de um Hamiltoniano molecular. Este Hamiltoniano pode ser eletrônico ou vibrônico. Para saber mais sobre a preparação do Hamiltoniano, confira os tutoriais de Estrutura Eletrônica e Estrutura Vibrônica.

O primeiro passo é definir o sistema molecular. A seguir, pedimos a parte eletrônica de uma molécula de hidrogênio.

[1]:

from qiskit.chemistry.drivers import PySCFDriver, UnitsType, Molecule
from qiskit.chemistry.transformations import FermionicTransformation, FermionicQubitMappingType

molecule = Molecule(geometry=[['H', [0., 0., 0.]],
                              ['H', [0., 0., 0.735]]],
                     charge=0, multiplicity=1)
driver = PySCFDriver(molecule = molecule, unit=UnitsType.ANGSTROM, basis='sto3g')
transformation = FermionicTransformation(qubit_mapping=FermionicQubitMappingType.JORDAN_WIGNER)

Numpy 1.16 has memory leak bug  https://github.com/numpy/numpy/issues/13808
It is recommended to downgrade to numpy 1.15 or older

O Solucionador¶

Então precisamos definir um solucionador. O solucionador é o algoritmo através do qual o estado fundamental é computado.

Let’s first start with a purely classical example: the NumPy eigensolver. This algorithm exactly diagonalizes the Hamiltonian. Although it scales badly, it can be used on small systems to check the results of the quantum algorithms. Here, we are only interested to look at eigenstates with a given number of particle. To compute only those states a filter function can be passed to the NumPy eigensolver. A default filter function is already implemented in Qiskit and can be used in this way:

[3]:

from qiskit.chemistry.algorithms import NumPyEigensolverFactory

numpy_solver = NumPyEigensolverFactory(use_default_filter_criterion=True)

As energias de excitação também podem ser acessadas com o algoritmo qEOM [arXiv preprint arXiv:1910.12890 (2019)]. O método EOM encontra as energias de excitação (diferenças de energia entre o estado fundamental e todos os \(n\)th estados excitados) resolvendo o seguinte problema de pseudo-autovalor.

\[\begin{split}\begin{pmatrix} \text{M} & \text{Q}\\ \text{Q*} & \text{M*} \end{pmatrix} \begin{pmatrix} \text{X}_n\\ \text{Y}_n \end{pmatrix} = E_{0n} \begin{pmatrix} \text{V} & \text{W}\\ -\text{W*} & -\text{V*} \end{pmatrix} \begin{pmatrix} \text{X}_n\\ \text{Y}_n \end{pmatrix}\end{split}\]

com

\[M_{\mu_{\alpha}\nu_{\beta}} = \langle0| [(\hat{\text{E}}_{\mu_{\alpha}}^{(\alpha)})^{\dagger},\hat{\text{H}}, \hat{\text{E}}_{\nu_{\beta}}^{(\beta)}]|0\rangle\]

\[Q_{\mu_{\alpha}\nu_{\beta}} = -\langle0| [(\hat{\text{E}}_{\mu_{\alpha}}^{(\alpha)})^{\dagger}, \hat{\text{H}}, (\hat{\text{E}}_{\nu_{\beta}}^{(\beta)})^{\dagger}]|0\rangle\]

\[V_{\mu_{\alpha}\nu_{\beta}} = \langle0| [(\hat{\text{E}}_{\mu_{\alpha}}^{(\alpha)})^{\dagger}, \hat{\text{E}}_{\nu_{\beta}}^{(\beta)}]|0\rangle\]

\[W_{\mu_{\alpha}\nu_{\beta}} = -\langle0| [(\hat{\text{E}}_{\mu_\alpha}^{(\alpha)})^{\dagger}, (\hat{\text{E}}_{\nu_{\beta}}^{(\beta)})^{\dagger}]|0\rangle\]

Embora a equação anterior possa ser resolvida classicamente, cada elemento da matriz deve ser medido no computador quântico com o estado fundamental correspondente. Para usar o qEOM como um solucionador no Qiskit, temos que definir um cálculo do estado fundamental primeiro, explicando aos algoritmos como encontrar o estado fundamental. Com isso o solucionador qEOM pode ser inicializado:

[4]:

from qiskit import BasicAer
from qiskit.aqua import QuantumInstance
from qiskit.chemistry.algorithms.ground_state_solvers import (GroundStateEigensolver,
                                                              VQEUCCSDFactory)
from qiskit.chemistry.algorithms.excited_states_solvers import QEOM

# This first part sets the ground state solver
# see more about this part in the ground state calculation tutorial
quantum_instance = QuantumInstance(BasicAer.get_backend('statevector_simulator'))
solver = VQEUCCSDFactory(quantum_instance)
gsc = GroundStateEigensolver(transformation, solver)

# The qEOM algorithm is simply instantiated with the chosen ground state solver
qeom_excited_states_calculation = QEOM(gsc, 'sd')

O cálculo e os resultados¶

Os resultados são computados e impressos

[5]:

from qiskit.chemistry.algorithms.excited_states_solvers import ExcitedStatesEigensolver

numpy_excited_states_calculation = ExcitedStatesEigensolver(transformation, numpy_solver)
numpy_results = numpy_excited_states_calculation.solve(driver)

qeom_results = qeom_excited_states_calculation.solve(driver)

print(numpy_results)
print('\n\n')
print(qeom_results)

=== GROUND STATE ENERGY ===

* Electronic ground state energy (Hartree): -1.857275030202
  - computed part:      -1.857275030202
  - frozen energy part: 0.0
  - particle hole part: 0.0
~ Nuclear repulsion energy (Hartree): 0.719968994449
> Total ground state energy (Hartree): -1.137306035753

=== EXCITED STATE ENERGIES ===

  1:
* Electronic excited state energy (Hartree): -0.882722150245
> Total excited state energy (Hartree): -0.162753155796
  2:
* Electronic excited state energy (Hartree): -0.224911252831
> Total excited state energy (Hartree): 0.495057741618

=== MEASURED OBSERVABLES ===

  0:  # Particles: 2.000 S: 0.000 S^2: 0.000 M: 0.000
  1:  # Particles: 2.000 S: 0.000 S^2: 0.000 M: 0.000
  2:  # Particles: 2.000 S: 0.000 S^2: 0.000 M: 0.000

=== DIPOLE MOMENTS ===

~ Nuclear dipole moment (a.u.): [0.0  0.0  1.3889487]

  0:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

  1:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

  2:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.




=== GROUND STATE ENERGY ===

* Electronic ground state energy (Hartree): -1.857275030145
  - computed part:      -1.857275030145
  - frozen energy part: 0.0
  - particle hole part: 0.0
~ Nuclear repulsion energy (Hartree): 0.719968994449
> Total ground state energy (Hartree): -1.137306035696

=== EXCITED STATE ENERGIES ===

  1:
* Electronic excited state energy (Hartree): -1.244586753196
> Total excited state energy (Hartree): -0.524617758747
  2:
* Electronic excited state energy (Hartree): -0.882724353597
> Total excited state energy (Hartree): -0.162755359148
  3:
* Electronic excited state energy (Hartree): -0.224913456192
> Total excited state energy (Hartree): 0.495055538257

=== MEASURED OBSERVABLES ===

  0:  # Particles: 2.000 S: 0.000 S^2: 0.000 M: 0.000

=== DIPOLE MOMENTS ===

~ Nuclear dipole moment (a.u.): [0.0  0.0  1.3889487]

  0:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.38894909]
    - computed part:      [0.0  0.0  1.38894909]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  -0.00000039]  Total: 0.00000039
                 (debye): [0.0  0.0  -0.000001]  Total: 0.000001

Pode-se observar a partir desses resultados que um estado está faltando nos resultados do NumPy. A razão para isso é porque o spin também é usado como um filtro e apenas estados singleto são mostrados. A seguir, utilizamos uma função de filtro personalizada para verificar consistentemente nossos resultados e apenas filtrar estados com números incorretos de partículas (neste caso o número de partículas é 2).

[6]:

import numpy as np

def filter_criterion(eigenstate, eigenvalue, aux_values):
    return np.isclose(aux_values[0][0], 2.)

new_numpy_solver = NumPyEigensolverFactory(filter_criterion=filter_criterion)
new_numpy_excited_states_calculation = ExcitedStatesEigensolver(transformation, new_numpy_solver)
new_numpy_results = new_numpy_excited_states_calculation.solve(driver)

print(new_numpy_results)

=== GROUND STATE ENERGY ===

* Electronic ground state energy (Hartree): -1.857275030202
  - computed part:      -1.857275030202
  - frozen energy part: 0.0
  - particle hole part: 0.0
~ Nuclear repulsion energy (Hartree): 0.719968994449
> Total ground state energy (Hartree): -1.137306035753

=== EXCITED STATE ENERGIES ===

  1:
* Electronic excited state energy (Hartree): -1.244584549813
> Total excited state energy (Hartree): -0.524615555364
  2:
* Electronic excited state energy (Hartree): -1.244584549813
> Total excited state energy (Hartree): -0.524615555364
  3:
* Electronic excited state energy (Hartree): -1.244584549813
> Total excited state energy (Hartree): -0.524615555364
  4:
* Electronic excited state energy (Hartree): -0.882722150245
> Total excited state energy (Hartree): -0.162753155796
  5:
* Electronic excited state energy (Hartree): -0.224911252831
> Total excited state energy (Hartree): 0.495057741618

=== MEASURED OBSERVABLES ===

  0:  # Particles: 2.000 S: 0.000 S^2: 0.000 M: 0.000
  1:  # Particles: 2.000 S: 1.000 S^2: 2.000 M: 0.000
  2:  # Particles: 2.000 S: 1.000 S^2: 2.000 M: -1.000
  3:  # Particles: 2.000 S: 1.000 S^2: 2.000 M: 1.000
  4:  # Particles: 2.000 S: 0.000 S^2: 0.000 M: 0.000
  5:  # Particles: 2.000 S: 0.000 S^2: 0.000 M: 0.000

=== DIPOLE MOMENTS ===

~ Nuclear dipole moment (a.u.): [0.0  0.0  1.3889487]

  0:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

  1:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

  2:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

  3:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

  4:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

  5:
  * Electronic dipole moment (a.u.): [0.0  0.0  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
    - frozen energy part: [0.0  0.0  0.0]
    - particle hole part: [0.0  0.0  0.0]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.
                 (debye): [0.0  0.0  0.0]  Total: 0.

[1]:

import qiskit.tools.jupyter
%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit Software	Version
Qiskit	0.23.0
Terra	0.16.0
Aer	0.7.0
Ignis	0.5.0
Aqua	0.8.0
IBM Q Provider	0.11.0
System information
Python	3.6.9 \|Anaconda, Inc.\| (default, Jul 30 2019, 13:42:17) [GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)]
OS	Darwin
CPUs	2
Memory (Gb)	16.0
Tue Oct 20 18:06:26 2020 CEST

This code is a part of Qiskit

This code is licensed under the Apache License, Version 2.0. You may
obtain a copy of this license in the LICENSE.txt file in the root directory
of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.

Any modifications or derivative works of this code must retain this
copyright notice, and modified files need to carry a notice indicating
that they have been altered from the originals.

[ ]: