A Hybrid Quantum Algorithm To Detect Conical Intersections

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Emiel Koridon^1,2, Joana Fraxanet³, Alexandre Dauphin^3,4, Lucas Visscher², Thomas E. O’Brien^5,1, and Stefano Polla^5,1

¹Instituut-Lorentz, Universiteit Leiden, 2300RA Leiden, The Netherlands
²Theoretical Chemistry, Vrije Universiteit, 1081HV Amsterdam, The Netherlands
³ICFO – Institut de Ciències Fotòniques, 08860 Castelldefels (Barcelona), Spain
⁴PASQAL SAS, 2 av. Augustin Fresnel Palaiseau, 91120, France
⁵Google Research, Munich, 80636 Bavaria, Germany

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Abstract

Conical intersections are topologically protected crossings between the potential energy surfaces of a molecular Hamiltonian, known to play an important role in chemical processes such as photoisomerization and non-radiative relaxation. They are characterized by a non-zero Berry phase, which is a topological invariant defined on a closed path in atomic coordinate space, taking the value $pi$ when the path encircles the intersection manifold. In this work, we show that for real molecular Hamiltonians, the Berry phase can be obtained by tracing a local optimum of a variational ansatz along the chosen path and estimating the overlap between the initial and final state with a control-free Hadamard test. Moreover, by discretizing the path into $N$ points, we can use $N$ single Newton-Raphson steps to update our state non-variationally. Finally, since the Berry phase can only take two discrete values (0 or $pi$), our procedure succeeds even for a cumulative error bounded by a constant; this allows us to bound the total sampling cost and to readily verify the success of the procedure. We demonstrate numerically the application of our algorithm on small toy models of the formaldimine molecule (${H_2C=NH}$).

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Featured image: The image illustrates our algorithm, used to detect a conical intersection in the energy landscape of the Formaldimine molecule, shown in panel (d).

Panel (a) shows the energy gap as a function of nuclear coordinates, highlighting the conical intersection and three loops on which we test our algorithm. The algorithm will return a Berry phase of $pi$ only for the loop that contains the conical intersection.

Panel (b) shows the algorithm approximately tracking the energy of the ground state along each of the loops; note that the energy of the excited state does not need to be resolved.
Panel (c) shows follows one parameter of the quantum ansatz, continously tracking the state along each of the loops.

Finally, panel (e) shows the measured phase for the loop containing the conical intersection, as a function of the number of points we use to discretize the loop. The value $-1$ represents a Berry phase of $pi$ and a value of $+1$ represents a Berry phase of $0$.
This panel shows that $N=9$ points, and a corresponding number of Newton-Raphson parameter updates, are sufficient for resolving the conical intersection.

Popular summary

In the last decade, variational quantum algorithms (VQAs) have been in the spotlight as a potential paradigm for tackling quantum simulation problems on noisy small-scale quantum computers. The typical requirement for high-precision results strongly hinders the application of these algorithms to computational chemistry. Achieving this high precision is extremely expensive due to the cost of sampling, made worse by the need for error mitigation and complex optimization. We identify a problem in quantum chemistry that can bypass the high precision requirement, we design an algorithm to solve it and benchmark it on a small molecular model.

In our work, we develop a VQA that detects the presence of a conical intersection by tracking the ground state around a loop in nuclear coordinate space. Conical intersections play a key role in photochemical reactions, for example in the process of vision. Identifying the presence of a conical intersection in a molecular model can be an important step in understanding or predicting the photochemical properties of a system.

The question we pose has a discrete answer (yes/no); this lifts the requirement of high precision. Furthermore, we simplify the optimization problem by using fixed-cost updates to track the ground state approximately, to the required level of precision. This allows to prove bounds on the cost of the algorithm, which is rare in the context of VQAs.

We perform numerical benchmarks of the algorithm, demonstrating its resilience to different levels of sampling noise. We release publicly the code we developed for this task, which includes a framework for orbital-optimized quantum circuit ansätze that supports automatic differentiation.

► BibTeX data

@article{Koridon2024hybridquantum, doi = {10.22331/q-2024-02-20-1259}, url = {https://doi.org/10.22331/q-2024-02-20-1259}, title = {A hybrid quantum algorithm to detect conical intersections}, author = {Koridon, Emiel and Fraxanet, Joana and Dauphin, Alexandre and Visscher, Lucas and O’Brien, Thomas E. and Polla, Stefano}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1259}, month = feb, year = {2024}<br /> }

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Cited by

[1] Kumar J. B. Ghosh and Sumit Ghosh, “Exploring exotic configurations with anomalous features with deep learning: Application of classical and quantum-classical hybrid anomaly detection”, Physical Review B 108 16, 165408 (2023).

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