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# Analogue quantum chemistry simulation

## Abstract

Computing the electronic structure of molecules with high precision is a central challenge in the field of quantum chemistry. Despite the success of approximate methods, tackling this problem exactly with conventional computers remains a formidable task. Several theoretical1,2 and experimental3,4,5 attempts have been made to use quantum computers to solve chemistry problems, with early proof-of-principle realizations done digitally. An appealing alternative to the digital approach is analogue quantum simulation, which does not require a scalable quantum computer and has already been successfully applied to solve condensed matter physics problems6,7,8. However, not all available or planned setups can be used for quantum chemistry problems, because it is not known how to engineer the required Coulomb interactions between them. Here we present an analogue approach to the simulation of quantum chemistry problems that relies on the careful combination of two technologies: ultracold atoms in optical lattices and cavity quantum electrodynamics. In the proposed simulator, fermionic atoms hopping in an optical potential play the role of electrons, additional optical potentials provide the nuclear attraction, and a single-spin excitation in a Mott insulator mediates the electronic Coulomb repulsion with the help of a cavity mode. We determine the operational conditions of the simulator and test it using a simple molecule. Our work opens up the possibility of efficiently computing the electronic structures of molecules with analogue quantum simulation.

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## Code and data availability

The computer code developed for this study is available from the corresponding authors upon reasonable request. All data supporting the findings of this study can be generated using the numerical methods described within Methods and Supplementary Information and are available upon reasonable request.

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## Acknowledgements

We thank S. Blatt and N. Meyer for useful discussions regarding the experimental implementation of this proposal. We acknowledge support from the ERC Advanced Grant QENOCOBA under the EU Horizon 2020 programme (grant agreement 742102). J.A.-L. acknowledges support from “la Caixa” Foundation (ID 100010434) under fellowship LCF/BQ/ES18/11670016. Work at ICFO is supported by the Spanish Ministry of Economy and Competitiveness through the “Severo Ochoa” programme (SEV-2015-0522), Fundació Privada Cellex, and by Generalitat de Catalunya through the CERCA programme. A. G.-T. acknowledges support from the Spanish project PGC2018-094792-B-100 (MCIU/AEI/FEDER, EU) and from the CSIC Research Platform on Quantum Technologies PTI-001. T.S. is grateful to the Thousand-Youth-Talent Program of China. P.Z. was supported by the ERC Synergy Grant UQUAM and by SFB FoQus of the Austrian Science Foundation.

## Author information

Authors

### Contributions

J.A.-L., A.G.-T. and T.S. designed the model, developed the methods and analysed the data. J.A.-L. and A.G.-T. performed the calculations. J.A.-L., A.G.-T. and J.I.C. wrote the manuscript with input from all authors. J.I.C. and P.Z. conceived the study and were in charge of the overall direction and planning.

### Corresponding authors

Correspondence to Alejandro González-Tudela or J. Ignacio Cirac.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Results of the G-S algorithm.

a, b, We apply the G-S algorithm to identify the phase mask associated with a holographic 3D Coulomb potential on a lattice of N3 sites. Fixing the origin at the central site, we choose the nucleus position as n = [(2ndiv)−1, 0, 0] (the first coordinate is shifted so that the lattice induces a natural cutoff). a, An axial central cut of the potential (yellow markers) in direction z (see aligned set of sites in b; red), created by a phase mask composed of (ndivN) × (ndivN) cells for ndiv = 3 (see inset), compared to the objective Coulomb potential (blue solid line). In step (ii) of the algorithm, the Ewald sphere is discretized using a parallel projection, as in ref. 31. The field is initiated with random phases. Parameters: N = 30 and 7,000 iterations of the G-S algorithm. b, Location of the axial cut shown in a.

### Extended Data Fig. 2 Numerical simulation of the adiabatic preparation of the ground state of H2 with the simulator (particularized for a 2D lattice).

Red dashed lines follow the adiabatic evolution of the initial state $$\left|{\psi }_{0}\right\rangle$$ and arrows point to the direction of evolution. a, Preparation of the bosonic state through steps I(a)–I(c) (see Methods and Extended Data Table 1). Continuous lines indicate the exact energy of the two lowest energy states. For the adiabatic evolution we use Trotterized time as ΔtU = 0.5 and evolution with |ΔU|/(U2Δt) = 3 × 10−4. b, Steps II–III of the preparation of the fermionic state. In the simulation, we use the Trotterized time evolution in intervals of ΔtV0 = 0.05. In step II, the kinetic term is adiabatically introduced in steps of $$\Delta {t}_{{\rm{F}}}/({{\rm{V}}}_{0}^{2}\Delta t)=0.005$$. In step III, the electronic repulsion is tuned up as $$\Delta V/({{\rm{V}}}_{0}^{2}\Delta t)=0.02$$. Here yellow (blue) continuous lines follow the exact energy levels of Hqc, as calculated by imaginary-time evolution with (without) the effect of electronic repulsion. The top insets show the Mott excitation (a) fermionic population (b) in the lattice at the times indicated in the figure. The final point of the evolution shown in b corresponds to d/a0 = 1. Parameters: N = 60, U/Jc = 1.5, d/a = 10.

## Supplementary information

### Supplementary Information

This file contains Supplementary Text, Supplementary Figures 1 and 2, and additional references. This presents the numerical methods employed in this Letter, together with a derivation of the simulated Hamiltonian and conditions introduced in the main text

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Argüello-Luengo, J., González-Tudela, A., Shi, T. et al. Analogue quantum chemistry simulation. Nature 574, 215–218 (2019). https://doi.org/10.1038/s41586-019-1614-4

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• DOI: https://doi.org/10.1038/s41586-019-1614-4

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