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# Quantum approximate Bayesian computation for NMR model inference

## Abstract

Recent technological advances may lead to the development of small-scale quantum computers that are capable of solving problems that cannot be tackled with classical computers. A limited number of algorithms have been proposed and their relevance to real-world problems is a subject of active investigation. Analysis of many-body quantum systems is particularly challenging for classical computers due to the exponential scaling of the Hilbert space dimension with the number of particles. Hence, solving the problems relevant to chemistry and condensed-matter physics is expected to be the first successful application of quantum computers. In this Article, we propose another class of problems from the quantum realm that can be solved efficiently on quantum computers: model inference for nuclear magnetic resonance (NMR) spectroscopy, which is important for biological and medical research. Our results are based on three interconnected studies. First, we use methods from classical machine learning to analyse a dataset of NMR spectra of small molecules. We perform stochastic neighbourhood embedding and identify clusters of spectra, and demonstrate that these clusters are correlated with the covalent structure of the molecules. Second, we propose a simple and efficient method, aided by a quantum simulator, to extract the NMR spectrum of any hypothetical molecule described by a parametric Heisenberg model. Third, we propose a simple variational Bayesian inference procedure for estimating the Hamiltonian parameters of experimentally relevant NMR spectra.

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## Data availability

The data and code to numerically generate the NMR data sets used in this manuscript can be found at https://github.com/dsels/QuantumNMR.

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## Acknowledgements

D.S. acknowledges support from the FWO as post-doctoral fellow of the Research Foundation—Flanders and from a 2019 grant from the Harvard Quantum Initiative Seed Funding programme. S.M. is supported by a research grant from the National Heart, Lung, and Blood Institute (K24 HL136852). O.D. and H.D. are supported by a research award from the National Heart, Lung, and Blood Institute, (5K01HL135342) and (T32 HL007575) respectively. E.D. acknowledges support from the Harvard–MIT CUA, ARO grant number W911NF-20-1-0163, the National Science Foundation through grant number OAC-1934714, AFOSR Quantum Simulation MURI.The authors acknowledge useful discussions with P. Mehta and M. Lukin.

## Author information

Authors

### Contributions

D.S. and E.D. conceived the presented idea in consultation with H.D., S.M. and O.D. D.S. developed the theoretical formalism, performed the analytic calculations and performed the numerical simulations. H.D. compiled the NMR data used in the manuscript. All authors provided critical feedback and helped shape the manuscript.

### Corresponding author

Correspondence to Dries Sels.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary Discussion and Figs. 1–5.

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Sels, D., Dashti, H., Mora, S. et al. Quantum approximate Bayesian computation for NMR model inference. Nat Mach Intell 2, 396–402 (2020). https://doi.org/10.1038/s42256-020-0198-x

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• DOI: https://doi.org/10.1038/s42256-020-0198-x