Calculating excitation gaps of small molecules (ES-Gap)

Recent updates

Competition opens and initial records are added.

Competition summary

Participants are asked to calculate excitation energies of small molecules with various configurations. The result is evaluated as an average of relative errors between the submitted values of the gap calculated by quantum algorithms/methods and those by exact calculations.

Background information

Quantum Chemistry

See Background information of the similar problem. Again, we treat only electrons as quantum-mechanical objects and leave nuclei of a molecule as classical in this competition.

Excited state and excitations energy

Eigenstates of Hamiltonian of a molecular system other than the ground state (lowest-energy eigenstate) are called excited states. The excited states slightly above the ground state (i.e. eigenstates whose eigenenergies are close to the ground state energy) are important to know the chemical properties of the system such as

  • absorption/fluorescence spectrum of molecules,
  • photon-induced chemical reactions,
  • charge/energy transfer transition.

Those properties have several industrial interests as well as academic ones: for example, the absorption/fluorescence spectrum is critical to design organic electro-luminescence materials. However, calculation of excited states is typically by far difficult than that of the ground state.

Noisy Intermediate-Scale Quantum (NISQ) devices are again a nice candidate to overcome the difficulty. As an extension of the algorithm to compute the ground state, several algorithms for computing excited states are proposed. Interested readers can refer our brief introduction on several methods and the references therein.

Purpose of competition

This competition aims to search promising quantum algorithms/methods on NISQ devices for computing the excited states of molecules. As is the same with the competition for the ground states, this competition considers only small molecules that one can calculate exact results by classical computers. Benchmarking precision of quantum algorithms/methods will still be a useful milestone when we apply them to larger molecules that cannot be calculated by classical computers.


In this competition, participants calculate the first excitation gap of in the singlet sector (total spin $S^2=0$) of energy spectrum of the molecules specified in Competition_ES-Gap.csv, Evaluation is done by the following score $z$,

$$ z=\frac{1}{N}\sum_m\left(\frac{1}{N_m}\sum_{i=1}^{N_m}\frac{|\Delta_{QC}^m(R^m_i)-\Delta_{FCI}^m(R^m_i)|}{|\Delta_{FCI}^m(R^m_i)|}\right), $$ $$ \Delta_{QC,FCI}^m(R)=ES_{QC,FCI}^m(R)-GS_{QC,FCI}^m(R). $$

Here we use the notations

  • $N$ is the number of molecules in the dataset
  • $N_m$ is the number of data points (configurations) of each molecule $m$
  • $R_i^m$ is a coordinate of data point $i$ of molecule $m$
  • $GS_{QC(FCI)}^m(R)$ is the ground state energy of molecule $m$ at coordinate $R$ calculated by quantum algorithms/methods (exact diagonalization = full configuration interaction (FCI)) of the Hamiltonian
  • $ES_{QC(FCI)}^m(R)$ is the 1st excited state energy of the molecule $m$ at coordinate $R$ in the singlet (total spin $S^2=0$) sector calculated by quantum algorithms/methods (FCI).

In short, we evaluate the score by an average of the relative errors between the energy gaps calculated by quantum algorithms/methods and those by exact diagonalization.

Moreover, we consider four cases separately according to the way we simulate the result of the quantum algorithm. The definition can be found here.

  • Case 1: Expectation value
  • Case 2: Sampling without noise
  • Case 3: Sampling with noise
  • Case 4: Real NISQ devices


Dataset for this competition is based on Small_Molecules_1 dataset, which is available on this Github page The molecules is the same as those in the competition for the potential energy surface for small molecules,

  • H2 / H4 / H6 molecules with linear geometry and ring geometry
  • LiH
  • BeH2
  • H2O
Molecule Bond length Total # of points
$\rm H_2$ r=0.5, 0.6, ..., 2.0 16
$\rm H_4$ (line) r=0.5, 0.6, ..., 2.0 16
$\rm H_6$ (line) r=0.5, 0.6, ..., 2.0 16
$\rm H_4$ (ring) r=0.5, 0.6, ..., 2.0 16
$\rm H_6$ (ring) r=0.5, 0.6, ..., 2.0 16
$\rm LiH$ r=0.5, 0.6, ..., 2.0 16
$\rm BeH_2$ r=0.5, 0.6, ..., 2.0 16
$\rm H_2O$ r=0.5, 0.6, ..., 2.0, angle = 104.5 deg 16

Here, Hn (line) denotes the molecule where $n$ hydrogen atoms are aligned in line and have the same distance between each adjacent atom. For Hn (ring), $n$ hydrogen atoms are placed on the circle with the same adjacent bond length. Two bond lengths of Be-H in BeH2 are taken as the same and all atoms are aligned in line. The angle of two O-H bonds in H2O is taken as 104.5 degree and each bond length is taken as the same.

All data contained in Small_Molecules_1 dataset are openfermion.hamiltonian.MolecularData class in .hdf5 format [1]. We prepare Competition_ES-Gap.csv at that specifies the names of files to be computed in this competition. To ease calculations of the score, we also put a Python script at that calculates the score from the csv: participants have only to fill qc_energy_0 column and qc_energy_1 column in the csv by the obtained ground state energy and the excited state energy.

For further details of the dataset, See of the dataset directory,

Submission Guide

  • We accept submission in the form of Competition_ES-Gap.csv at, whose qc_energy_0 column and qc_energy_1 column are filled. We are happy if you also submit the script used for calculations.
  • We also accept submission in the form of programming codes, since benchmarking all points in the dataset needs relatively large computational power for personal computers. We will run the submitted code and add the records of it. We recommend the codes are written in Python with Qulacs, but use of any language/library is considered.

Competition host and contact


[1] Preparation of molecular data is performed by PySCF, OpenFermion, OpenFermion-PySCF.

Current result

Case 1: Expectation value

Rank Author Method Score Mean # of func. eval. Date
1 QunaSys VQD, SymmetryPreserving-20, BFGS, jordan-wigner 0.369094 94376.953 20191004
2 QunaSys MCVQE, SymmetryPreserving-20, BFGS, jordan-wigner 0.554986 34208.750 20191004
3 QunaSys QSE, HardwareEfficient-5, SMO, jordan-wigner 0.720857 nan 20190801
4 QunaSys QSE, HardwareEfficient-5, BFGS, jordan-wigner 0.723876 nan 20190801
5 QunaSys VQD, UCCSD, BFGS, jordan-wigner 1.107189 1426.891 20191004
6 QunaSys MCVQE, UCCSD, BFGS, jordan-wigner 1.427822 675.141 20191004
7 QunaSys VQD, HardwareEfficient-8, BFGS, jordan-wigner 1.553515 20218.672 20191005
8 QunaSys SSVQE, UCCSD, SMO, jordan-wigner, cis 1.577729 1177.328 20190730
9 QunaSys QSE, UCCSD, BFGS, jordan-wigner 1.598058 nan 20190801
10 QunaSys SSVQE, UCCSD, BFGS, jordan-wigner 1.617397 721.469 20191004
11 QunaSys MCVQE, HardwareEfficient-8, BFGS, jordan-wigner 3.615655 15593.281 20191005
12 QunaSys QSE, UCCSD, SMO, jordan-wigner 9.342742 nan 20190801

Case 2: Sampling without noise

To be benchmarked!

Case 3: Sampling with noise

To be benchmarked!

Case 4: Real NISQ devices

To be benchmarked!