Implementing a Variational Quantum Eigensolver
to Model DNA
Anjali Pillai
Studying molecular structure and interaction can be complicated especially when working with larger molecules because complexity increases quickly as molecules become larger and larger. Despite this, molecular interactions are extremely important particularly in the fields of medicine and materials sciences. My inspiration for this project was an articles titled: “Using Quantum Computing to provide insight into COVID evolution” (Mathur, 2021). The article discussed using ab initio quantum modeling to model COVID spike proteins. These models provided insight into new strains of COVID and what characteristics new vaccine boosters should have. This article encouraged me to pursue quantum modeling for biological molecules.
A variational quantum eigensolver is a hybrid classical and quantum technique. To start, a Hamiltonian, or matrix of energy values, is created using Pauli matrices (3 matrices that represent a particles spin in the x, y, and z directions). Next, an ansatz or educated guess about the matrices eigenvalue or possible energy states is created using molecular properties such as bond length or molecular geometry. A classical optimization or minimization technique is then used to slightly change the ansatz until the minimum eigenvalue is achieved.
A quantum approach was used for this project because quantum is often the best approach when large numbers are used. An n particle system would have 2N possible combinations of spin up and spin down. Additionally, 2< becomes the dimension of the Hilbert space that the molecule lies in and 2Nx2N becomes the dimension of the Hamiltonian matrix created. Since these numbers get large very quickly, a variational quantum eigensolver was used. The classical counterpart to VQE is the Hartree Fock algorithm. Hartree Fock uses classical techniques to find the wave function and energy of a system by solving the time dependent schrodinger equation. The Hamiltonian is created using the Born-Oppenheimer Approximation and the Fock Operator behaves similarly to the ansatz, iterating through each energy until the minimum eigenvalue is found.
The research question is: How can we use a variational quantum eigensolver to model DNA?
The first step was to create the variational quantum computing code. The Qiskit textbook was referenced and the code provided for the bond length of Lithium Hydride was modified to account for the new version of IBM Quantum.
In the code above, the coordinates represent the atomic coordinates of each element in the diatomic interaction. Additionally, the code iterates from 0.5 to 4.0 angstroms with 0.2 angstrom jumps. For each of these interatomic distances an energy is calculated and the minimum energy corresponds with the most optimal bond length since systems tend towards lower energy.
In order to modify the VQE code provided by qiskit to properly model DNA, the atomic coordinates of the atoms in DNA were required. In this project, I focussed on 2 of the nucleic acids in DNA, namely guanine and cytosine. The software Mathematica was used in order to model these two molecules and find the coordinates of the atoms with units angstroms.
Once the coordinates of the atoms were found, the toms were then plotted using a 3D graphing software and each atom was labeled with the corresponding coordinates from Mathematica. Once all the coordinates were labeled, the atoms that would form hydrogen bonds were circled with the same color. For this project, the middle hydrogen bond was focussed on. Namely, the hydrogen bond between the nitrogen on the cytosine molecule and the hydrogen on the guanine molecule. The color coordinated bonds allowed for easy visualization of where the atomic interaction would occur.
The labeled coordinates then had to be rotated in order to ensure that they were on the same plane. The nitrogen on the cytosine was extremely close to the y-axis, so in order to account for that, the corresponding atom of the guanine was rotated onto the y-axis as well. The rotation code utilized the formula: tangent inverse of y over x in order to find the angle of rotation, theta. The angle was then put into the standard rotation matrix - cos(theta), -sin(theta), sin(theta), cos(theta). The rotation matrix was applied to the set of coordinates that represented guanine and then the new rotated coordinates of the hydrogen atom in guanine were selected.
The new rotated coordinates were then plugged into the variational quantum eigensolver and the code was run.
The results for Lithium Hydride provided through the Qiskit Textbook were first verified. The lowest energy corresponds to the most likely bond length as systems tend towards lower energy. As seen by the graph, the lowest energy is around 1.5 angstroms and as seen by the table, the lowest energy calculated through VQE was -7.88210 eV which means an interatomic distance of 1.5 angstroms The actual bond length between lithium and hydrogen is given as 1.596 angstroms. Additionally, as seen by the graph, the results from the variational quantum eigensolver and the results from the Hartree Fock approximation are very similar.
The guanine and cytosine interaction was modeled next. As seen by the graph, the lowest energy is around 3.5 angstroms and as seen by the table, the lowest energy calculated through VQE was -54.01574 eV which means an interatomic distance of 3.5 angstroms The actual bond length between guanine and cytosine is given as 3.4 angstroms. Additionally, in this graph the VQE and Hartree Fock results seem to differ more, however, the scale on the graph shows that the difference is only a mere 0.2 eV.
IBM Quantum was used in order to run the variational quantum eigensolver. Additionally, the Qiskit textbook provided by IBM Quantum was referenced. Mathematica was used to find atomic coordinates.
The variational quantum eigensolver approach proves to be very similar when compared to the traditional, Hartree Fock classical approach. Additionally, both approaches prove to be accurate in calculating the interatomic distance of diatomic molecules or interactions.
The next step in this project would be to model the interactions between the other set of nucleic acids - adenine and thymine. When trying to do this, atoms of both molecules needed to be rotated to an axis due to the shape of the molecule. The molecule was not able to be created on the VQE so this is an area of improvement.
For future studies, smaller steps can be taken when optimizing the bond length in order to get more accurate results. However, smaller steps would come at the expense of more computation time. In addition to this, future steps should be taken to create a molecular dependency between all three hydrogen bonds in this interaction. As of right now, the best approach is to compute the values separately and average them, however, the interaction is much more complex and each bond will depend on the other. The current limitations of quantum computing don’t allow more than diatomic interactions to be modeled but future studies should focus on higher order molecules. Finally, future studies should also focus on modeling more complex molecules such as viruses and different types of materials.
Introduction
Materials
Methodology
Results
Conclusion
I would like to thank my lab director Mr. Mark Hannum for his continued support and guidance throughout this project. This truly would not have been possible without his help. Additionally I would like to thank Mr. Frank Harkins from IBM Quantum for providing working VQE code despite the updates in IBM Quantum. I would also like to thank Suraj Vaddi for helping me with the rotation code and finding coordinates, Yuktha Vemula for helping me understand the biological interactions of viruses, Rohan Kompella for his excellent lecture on VQE, and Neha Asuri for helping me label coordinates. Further acknowledgements go to Chinmay Bhandaru, Neha Chandran, Sherry Yu, Mira Singh, Pi Rogers, and Anisha Talreja.
Acknowledgements
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