励起状態ソルバー¶

はじめに¶

610a8ab6125544a595279f0859541d0f

このチュートリアルでは、 Qiskit Chemistry の励起状態計算インターフェースについて説明します。目標は、分子ハミルトニアンの励起状態を計算することです。このハミルトニアンは、電子的あるいは振動的なものである可能性があります。ハミルトニアンの準備についての詳細は、電子構造と振動構造のチュートリアルをチェックしてください。

まず分子系の定義から始めます。以下では、水素分子の電子部分を求めています。

[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

ソルバー¶

次に、ソルバーを定義する必要があります。ソルバーとは、基底状態を求めるアルゴリズムのことです。

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)

励起エネルギーは、 qEOM アルゴリズム [arXiv preprint arXiv:1910.12890 (2019)] によっても求めることができます。 EOM 法は、以下の擬似固有値問題を解くことにより、励起エネルギー (基底状態と全ての第 \(n\) 励起状態との間のエネルギー差) を求めます。

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

ただし

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

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

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

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

上の方程式は古典的に解くことができますが、各行列要素は対応する基底状態を量子コンピューター上で測定しなければなりません。 qEOM を Qiskit のソルバーとして使用するには、まず基底状態の計算を定義し、基底状態を得るためのアルゴリズムを設定します。これにより、 qEOM ソルバーを初期化することができます。

[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')

計算と結果¶

結果が計算されて、表示されます。

[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

これらの結果から、NumPy の結果と比べて状態が一つ足りないことがわかります。これは、スピンもフィルターとして使用され、一重項状態のみが表示されているためです。以下では、一貫性をもって結果を確認し、粒子数が正しくない状態 (今回は粒子数が 2) だけをフィルタリングするように、カスタムのフィルター関数を用いています。

[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.

[ ]: