qcr:2606.12874.1

Ground State Solvers for Molecules

This Qiskit Nature tutorial shows how to actually solve for the ground-state energy of a molecule once its electronic-structure problem has been defined, the step where quantum algorithms do the work. It introduces the GroundStateEigensolver, which ties together the three pieces of a quantum chemistry calculation: the problem (the molecular Hamiltonian), a qubit mapper that converts the fermionic operators into qubit operators, and a solver (such as VQE with a chemistry-aware ansatz, or an exact classical eigensolver for reference). The tutorial walks through configuring a GroundStateEigensolver with VQE and the UCCSD (unitary coupled-cluster) ansatz, running it to compute a molecule's ground-state energy, and comparing the result against the exact classical solution and the Hartree-Fock baseline. It explains how the solver returns not just the energy but the full result object with associated properties, and how the choice of ansatz and mapper affects accuracy and qubit count. By assembling the end-to-end ground-state workflow, the tutorial delivers the core capability most users come to Qiskit Nature for: estimating molecular energies on a quantum computer.

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# --- Setup cell added by QCR (not part of the original tutorial) ---
# Source: qiskit-community/qiskit-nature @ 0.8, Apache License 2.0.
# Installs the example's dependencies. If a later cell still reports a missing
# package, restart the runtime/kernel and run again from the top.
%pip install -q qiskit-nature==0.8.0 pyscf qiskit-algorithms

Ground state solvers

Introduction

In this tutorial we are going to discuss the ground state calculation interface of Qiskit Nature. The goal is to compute the ground state of a molecular Hamiltonian. This Hamiltonian can for example be electronic or vibrational. To know more about the preparation of the Hamiltonian, check out the Electronic structure and Vibrational structure tutorials.

It should be said, that in the electronic case, we are actually computing purely the electronic part. When using the Qiskit Nature stack as presented in this tutorial, the nuclear repulsion energy will be added automatically, to obtain the total ground state energy.

The first step is to define the molecular system. In the following we ask for the electronic part of a hydrogen molecule.

In [1]:

from qiskit_nature.units import DistanceUnit
from qiskit_nature.second_q.drivers import PySCFDriver

driver = PySCFDriver(
    atom="H 0 0 0; H 0 0 0.735",
    basis="sto3g",
    charge=0,
    spin=0,
    unit=DistanceUnit.ANGSTROM,
)

es_problem = driver.run()

We will also be sticking to the Jordan-Wigner mapping. To learn more about the various mappers available in Qiskit Nature, check out the Qubit Mappers tutorial.

In [2]:

from qiskit_nature.second_q.mappers import JordanWignerMapper

mapper = JordanWignerMapper()

The Solver

After these steps, we need to define a solver. The solver is the algorithm through which the ground state is computed.

Let's first start with a purely classical example: the NumPyMinimumEigensolver. 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.

In [3]:

from qiskit_algorithms import NumPyMinimumEigensolver

numpy_solver = NumPyMinimumEigensolver()

To find the ground state we could also use the Variational Quantum Eigensolver (VQE) algorithm. The VQE algorithm works by exchanging information between a classical and a quantum computer as depicted in the following figure.

Let's initialize a VQE solver.

In [4]:

from qiskit_algorithms import VQE
from qiskit_algorithms.optimizers import SLSQP
from qiskit.primitives import StatevectorEstimator
from qiskit_nature.second_q.circuit.library import HartreeFock, UCCSD

ansatz = UCCSD(
    es_problem.num_spatial_orbitals,
    es_problem.num_particles,
    mapper,
    initial_state=HartreeFock(
        es_problem.num_spatial_orbitals,
        es_problem.num_particles,
        mapper,
    ),
)

vqe_solver = VQE(StatevectorEstimator(), ansatz, SLSQP())
vqe_solver.initial_point = [0.0] * ansatz.num_parameters

To define the VQE solver one needs three essential elements:

An Estimator primitive: these were released as part of Qiskit Terra 0.22. To learn more about primitives, check out this resource.
A variational form: here we use the Unitary Coupled Cluster (UCC) ansatz (see for instance [Physical Review A 98.2 (2018): 022322]). Since it is a chemistry standard, a factory is already available allowing a fast initialization of a VQE with UCC. The default is to use all single and double excitations. However, the excitation type (S, D, SD) as well as other parameters can be selected. We also prepend the UCCSD variational form with a HartreeFock initial state, which initializes the occupation of our qubits according to the problem which we are trying solve.
An optimizer: this is the classical piece of code in charge of optimizing the parameters in our variational form. See the corresponding documentation for more information.

One could also use any available ansatz / initial state or even define one's own. For instance,

In [5]:

from qiskit_algorithms import VQE
from qiskit.circuit.library import TwoLocal

tl_circuit = TwoLocal(
    rotation_blocks=["h", "rx"],
    entanglement_blocks="cz",
    entanglement="full",
    reps=2,
    parameter_prefix="y",
)

another_solver = VQE(StatevectorEstimator(), tl_circuit, SLSQP())

The calculation and results

We are now ready to put everything together to compute the ground-state of our problem. Doing so requires us to wrap our mapper and quantum algorithm into a single GroundStateEigensolver like so:

In [6]:

from qiskit_nature.second_q.algorithms import GroundStateEigensolver

calc = GroundStateEigensolver(mapper, vqe_solver)

This will now take of the entire workflow:

generating the second-quantized operators stored in our problem (here referred to as es_problem)
mapping (and potentially reducing) the operators in the qubit space
running the quantum algorithm on the Hamiltonian qubit operator
once converged, evaluating the additional observables at the determined ground state

In [7]:

res = calc.solve(es_problem)
print(res)

=== GROUND STATE ENERGY ===
 
* Electronic ground state energy (Hartree): -1.857275030145
  - computed part:      -1.857275030145
~ Nuclear repulsion energy (Hartree): 0.719968994449
> Total ground state energy (Hartree): -1.137306035696
 
=== 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  1.38894893]
    - computed part:      [0.0  0.0  1.38894893]
  > Dipole moment (a.u.): [0.0  0.0  -0.00000023]  Total: 0.00000023
                 (debye): [0.0  0.0  -0.00000058]  Total: 0.00000058

We can compare the VQE results to the NumPy exact solver and see that they match.

In [8]:

calc = GroundStateEigensolver(mapper, numpy_solver)
res = calc.solve(es_problem)
print(res)

=== GROUND STATE ENERGY ===
 
* Electronic ground state energy (Hartree): -1.857275030202
  - computed part:      -1.857275030202
~ Nuclear repulsion energy (Hartree): 0.719968994449
> Total ground state energy (Hartree): -1.137306035753
 
=== 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  1.3889487]
    - computed part:      [0.0  0.0  1.3889487]
  > Dipole moment (a.u.): [0.0  0.0  0.0]  Total: 0.0
                 (debye): [0.0  0.0  0.0]  Total: 0.0

Using a filter function

Sometimes the true ground state of the Hamiltonian is not of interest because it lies in a different symmetry sector of the Hilbert space. In this case the NumPyEigensolver can take a filter function to return only the eigenstates with for example the correct number of particles. This is of particular importance in the case of vibrational structure calculations where the true ground state of the Hamiltonian is the vacuum state. A default filter function to check the number of particles is implemented in the different problems and can be used as follows:

In [9]:

from qiskit_algorithms import NumPyMinimumEigensolver
from qiskit_nature.second_q.drivers import GaussianForcesDriver
from qiskit_nature.second_q.mappers import DirectMapper
from qiskit_nature.second_q.problems import HarmonicBasis

driver = GaussianForcesDriver(logfile="aux_files/CO2_freq_B3LYP_631g.log")
basis = HarmonicBasis([2, 2, 2, 2])
vib_problem = driver.run(basis=basis)
vib_problem.hamiltonian.truncation_order = 2

mapper = DirectMapper()

solver_without_filter = NumPyMinimumEigensolver()
solver_with_filter = NumPyMinimumEigensolver(
    filter_criterion=vib_problem.get_default_filter_criterion()
)

gsc_wo = GroundStateEigensolver(mapper, solver_without_filter)
result_wo = gsc_wo.solve(vib_problem)

gsc_w = GroundStateEigensolver(mapper, solver_with_filter)
result_w = gsc_w.solve(vib_problem)

print(result_wo)
print("\n\n")
print(result_w)

=== GROUND STATE ===
 
* Vibrational ground state energy (cm^-1): -8e-12
The number of occupied modals for each mode is: 
- Mode 0: 0.0
- Mode 1: 0.0
- Mode 2: 0.0
- Mode 3: 0.0



=== GROUND STATE ===
 
* Vibrational ground state energy (cm^-1): 2432.106954036546
The number of occupied modals for each mode is: 
- Mode 0: 1.0
- Mode 1: 1.0
- Mode 2: 1.0
- Mode 3: 1.0

In [10]:

import tutorial_magics

%qiskit_version_table
%qiskit_copyright

Version Information

Qiskit Software	Version
`qiskit-terra`	0.24.0.dev0+2b3686f
`qiskit-aer`	0.11.2
`qiskit-ibmq-provider`	0.19.2
`qiskit-nature`	0.6.0
System information
Python version	3.9.16
Python compiler	GCC 12.2.1 20221121 (Red Hat 12.2.1-4)
Python build	main, Dec 7 2022 00:00:00
OS	Linux
CPUs	8
Memory (Gb)	62.50002670288086
Thu Apr 06 08:55:38 2023 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.

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Jun 16, 2026

qcr:2606.12874.1

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Keywords

ground-state

vqe

quantum-chemistry

qiskit

uccsd