A hybrid quantum algorithm to detect conical intersections

Emiel Koridon1,2, Joana Fraxanet3, Alexandre Dauphin3,4, Lucas Visscher2, Thomas E. O'Brien5,1, and Stefano Polla5,1

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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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 (${H_2C=NH}$).

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.

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[1] Yuchen Wang and David A. Mazziotti, "Quantum simulation of conical intersections", Physical Chemistry Chemical Physics (2024).

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