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We propose a quantum algorithm for inferring the molecular nuclear spinHamiltonian from time-resolved measurements of spin-spin correlators, which canbe obtained via nuclear magnetic resonance (NMR). We focus on learning theanisotropic dipolar term of the Hamiltonian, which generates dynamics that arechallenging-to-classically-simulate in some contexts. We demonstrate theability to directly estimate the Jacobian and Hessian of the correspondinglearning problem on a quantum computer, allowing us to learn the Hamiltonianparameters. We develop algorithms for performing this computation on both noisynear-term and future fault-tolerant quantum computers. We argue that the formeris promising as an early beyond-classical quantum application since it onlyrequires evolution of a local spin Hamiltonian. We investigate the example of aprotein (ubiquitin) confined in a membrane as a benchmark of our method. Weisolate small spin clusters, demonstrate the convergence of our learningalgorithm on one such example, and then investigate the learnability of theseclusters as we cross the ergodic to non-ergodic phase transition by suppressingthe dipolar interaction. We see a clear correspondence between a drop in themultifractal dimension measured across many-body eigenstates of these clusters,and a transition in the structure of the Hessian of the learning cost-function(from degenerate to learnable). Our hope is that such quantum computationsmight enable the interpretation and development of new NMR techniques foranalyzing molecular structure.

Quantum computation of molecular structure using data from challenging-to-classically-simulate nuclear magnetic resonance experiments