Competition
Calculating potential energy surface of ground states of small molecules (GS-PES)

Competition summary

In this competition, participants are asked to estimate the ground state energies of several small molecules with different geometric configurations. Participants are given a dataset containing Hamiltonians describing those molecules, and will compute the lowest energy eigenvalue of the Hamiltonians with algorithms/methods for quantum computers. Results are evaluated by a weighted sum of their deviations from the exact values of energy. This competition will measure how well the quantum algorithms/methods can capture and describe the electronic structure of molecular systems.

Background information

Quantum chemistry

Quantum chemistry treats a molecular system as a quantum-mechanical object and tries to calculate various properties of it based on quantum mechanics. Especially, the quantum nature of electrons of molecules are vital to understand the system, and we often treat only electrons quantum-mechanically in this competition while nuclei of the molecules are left as classical object.

In quantum chemistry, the electronic structure of the system is described by a matrix called Hamiltonian. Eigenvalues/eigenvectors of the Hamiltonian are called eigenenergies/eigenstates. The smallest eigenenergy is dubbed as the ground state energy and highly important because, for example, one can infer the most stable state of the system and the stability from it. However, the dimension of the Hamiltonian becomes exponentially large in the size of the system (or the number of atoms), so it is practically impossible to directly diagonalize the Hamiltonian and obtain the ground state energy even for modest sized molecules.

Quantum chemistry on NISQ

Noisy Intermediate-Scale Quantum (NISQ) device is a promising candidate to make computations in quantum chemistry faster drastically. NISQ devices can handle a molecular Hamiltonian more efficiently than classical computers do, because, roughly speaking, they are also quantum-mechanical objects and can emulate the quantum-mechanical molecular system with small overhead. They are believed to be able to calculate the (approximate) ground state of the Hamiltonian that cannot practically be obtained with classical computers. In the past few years, there is an increasing number of proposals to utilize NISQ devices in quantum chemistry problems. Most of the proposals are, however, heuristic and it makes their accuracy/performance not so obvious.

Potential energy surface

In this competition, we benchmark the accuracy of algorithms/methods on NISQ devices by evaluating how well the ground-state Potential Energy Surface (PES) of a small molecule is reproduced.

PES is one of the most important concepts in chemistry to investigate and predict chemical stability, reactions, etc. of a given molecular system. It is defined as a function of eigenenergy of electronic states of a particular system, $$ E(R_1,R_2,\cdots), $$ where $R_1, R_2, ...$ are coordinates of the molecule which specify the geometric configuration of atoms (such as a bond length and angle). As an example, the PES of hydrogen molecules is shown in Fig. 1. From this figure, one can know

• the bond length of ~0.74 Angstrom is the most stable structure of the hydrogen molecule with the lowest ground state energy.
• the energy of ~0.33 Hartree will be released when two isolated hydrogen atoms form the hydrogen molecule since the energy difference between the bond length of ~0.74 Angstrom and infinity is around 0.33 Hartree.

Reference

For more details of "background information", you can refer to nice review papers:

• "Quantum computational chemistry", S. McArdle et al., https://arxiv.org/abs/1808.10402
• "Quantum Chemistry in the Age of Quantum Computing", Y. Cao et al., https://arxiv.org/abs/1812.09976

Purpose of competition

This competition aims to find promising quantum algorithms/methods that will be run on NISQ devices to obtain the electric ground state of molecules. Although this competition handles only small molecules that one can easily calculate exact results with classical computers, benchmarking precision on quantum algorithms/methods will be of quite an importance when we apply them to larger molecules that cannot be calculated by classical computers.

Evaluation

In this competition, participants use a dataset described in the next section, which includes Hamiltonians of several small molecules with various geometries. Participants calculate the ground state energy of each Hamiltonian based on the algorithms/methods that will be run on NISQ devices in the near future.

Evaluation is given by the following score $z$, $$ z = \frac{1}{N} \sum_m \left( \frac{1}{N_m} \sum_{i=1}^{N_m} \frac{|\left(E_{QC}^m(R^m_i)-\delta_m\right)-E_{FCI}^m(R^m_i)|}{|E_{FCI}^m(R^m_i)-E_{FCI,\rm{min}}^m|} \right), $$ $$ \delta_m=E_{QC}^m(R_{\rm{min}}^m)-E_{FCI,\rm{min}}^m. $$ Here we use the notations

• $N$ is the number of molecules in the dataset
• $N_m$ is the number of data points (configurations) for each molecule $m$
• $R_i^m$ is the coordinates of data points $i=1,...,N_m$ of the molecule $m$
• $E_{QC(FCI)}^m(R)$ is the ground state energy of the molecule $m$ whose coordinates are $R$, calculated by quantum algorithms/methods (exact diagonalization = full-configuration interaction (FCI) ) of the Hamiltonian
• $R_{\rm{min}}^m$ and $E_{FCI,\rm{min}}^m=E_{FCI}^m(R_{\rm{min}}^m)$ are the coordinate of the most stable structure of the molecule $m$ and its ground state energy
• $\delta_m$ is the energy shift applied to $E_{QC}^m$ to set the energies at $R_{\rm{min}}^m$ to be the same between QC and FCI.

The denominator in the summation indicates that we put more weights on the energy around the most stable configuration of each molecule. The numerator in the summation means the difference of the energies between QC and FCI after the energy of QC is sifted by $\delta_m$ so that two energies are identical at the most stable configuration $R_{\rm{min}}^m$. The reason for including the shift in the definition of the score is to avoid divergence of the score around the most stable configuration $R_{\rm{min}}^m$. This definition may seem a bit technical, but we believe it will evaluate the precision of the PES in a balanced manner. The meaning of the score and the variables in the equations is depicted in Fig. 2.

Moreover, we consider four cases separately according to the way we simulate the result of the quantum algorithm.

Case 1: Expectation value

Measurement outcome of a given observable $O$ for a quantum state $|\psi\rangle$ is given directly by its expectation value $\langle \psi |O|\psi \rangle$.

Case 2: Sampling without noise

Since one can only do a projective measurement in Z-basis on most NISQ devices, we consider a case where measurement outcomes of a given observable $O$ are obtained by summing the of measurement outcomes of a single Pauli matrix. Specifically, the outcome of measuring the observable $O$ will be calculated in the following way:

1. $O$ is decomposed as a linear combination of Pauli matrices, $O=\sum_i c_i P_i$, where $c_i$ is a coefficient and $P_i$ is a Pauli matrix.
2. each term $c_i P_i$ yields a measurement outcome of $\pm c_i$ according to the probability distribution determined by $\langle \psi |P_i|\psi \rangle$. We sample the outcomes for $n_{\rm{sample}}$ times and take the average of them to estimate $\langle c_i P_i \rangle$.
3. repeat step 2 for all $i$ and sum up the results.

When $n_{\rm{sample}}$ is infinity, the results of case 1 and case 2 become identical.

Case 3: Sampling with noise

This is a case where the effect of noise is introduced to Case 2. The basis scheme for simulating the measurement outcomes is the same, but the quantum state $|\psi\rangle$ is prone to error under some noise model. Then the state becomes a mixed state $\rho$, so the outcome of the step 2 in Case 2 are computed from ${\rm Tr}(\rho P_i)$.

Case 4: Real NISQ devices

We run the algorithms on real NISQ devices such as Rigetti Qunatum Cloud Servises and IBM Q.

Dataset

Dataset for the competition is based on our Small_Molecules_1 dataset, which is available on this Github page, https://github.com/qulacs/Quantaggle_dataset/tree/master/datasets/Small_Molecules_1 .

Among the dataset, we have chosen following small molecules,

• H₂ / H₄ / H₆ molecules with linear geometry and ring geometry
• LiH
• BeH₂
• H₂O

to evaluate the accuracy of PES. The data points (configurations) are shown in the following table.

Molecule	Bond length	Total # of points	Optimal bond length
H2	r=0.5, 0.6, ..., 2.0	16	0.7349
H4 (line)	r=0.5, 0.6, ..., 2.0	16	0.8882
H6 (line)	r=0.5, 0.6, ..., 2.0	16	0.9255
H4 (ring)	r=0.5, 0.6, ..., 2.0	16	1.2774
H6 (ring)	r=0.5, 0.6, ..., 2.0	16	0.9882
LiH	r=0.5, 0.6, ..., 2.0	16	1.5475
BeH2	r=0.5, 0.6, ..., 2.0	16	1.3164
H2O	r=0.5, 0.6, ..., 2.0, angle = 104.5 deg	16	1.0236

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

All of the data contained in Small_Molecules_1 dataset is openfermion.hamiltonian.MolecularData class in .hdf5 format [1]. We prepare Competition_GS-PES.csv at https://github.com/qulacs/Quantaggle_dataset/tree/master/competitions/GS-PES which specifies the names of files to be computed in this competition. Moreover, to ease the calculations of the score, we put a Python script at https://github.com/qulacs/Quantaggle_dataset/tree/master/competitions/GS-PES which calculates the score from the csv: participants only have to fill qc_energy column in the csv file with the obtained ground state energies. We note that participants have to calculate the energy of the most stable configurations of the molecules (the last column in the table above) in addition to the bond lengths .

For further details of the dataset, See README.md of the dataset directory, https://github.com/qulacs/Quantaggle_dataset/tree/master/datasets/Small_Molecules_1 .

Submission Guide

• We accept the submission in the form of Competition_GS-PES.csv at https://github.com/qulacs/Quantaggle_dataset/tree/master/competitions/GS-PES, whose qc_energy column is filled. We are happy if you also submit the script used for calculations.
• We also accept the submission in the form of programming codes, since benchmarking all of the points in the dataset needs a 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

• Host: QunaSys Inc.
• Contact: contact form

Reference

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