Digital quantum simulation of NMR experiments

Abstract

Simulations of nuclear magnetic resonance (NMR) experiments can be an important tool for extracting information about molecular structure and optimizing experimental protocols but are often intractable on classical computers for large molecules such as proteins and for protocols such as zero-field NMR. We demonstrate the first quantum simulation of an NMR spectrum, computing the zero-field spectrum of the methyl group of acetonitrile using four qubits of a trapped-ion quantum computer. We reduce the sampling cost of the quantum simulation by an order of magnitude using compressed sensing techniques. We show how the intrinsic decoherence of NMR systems may enable the zero-field simulation of classically hard molecules on relatively near-term quantum hardware and discuss how the experimentally demonstrated quantum algorithm can be used to efficiently simulate scientifically and technologically relevant solid-state NMR experiments on more mature devices. Our work opens a practical application for quantum computation.

Fig. 1.. Liquid-state NMR spectrum computed on quantum hardware.

Zero-field (ZF) spectrum of acetonitrile computed on an trapped-ion quantum computer (blue curve) compared with the NMR experiment (green curve) performed in (10). The inset shows the chemical structure of acetonitrile, highlighting the methyl group that was probed in the experiment. arb. units, arbitrary units.

Fig. 2.. Compressed sensing reconstruction and benchmarking.

(A) Comparison of the FID of a noisy quantum circuit emulation (blue line) and the nonuniform, sparsely sampled points experimentally measured on the trapped-ion quantum computer (green circles). The noise is modeled by two-qubit gates subject to both amplitude and phase damping with rates of 0.005 and 0.035 s, respectively. (B) NMR spectrum extracted from the digital quantum simulation, where the spectrum is the real part of the Fourier transform of the FID. Green dots show the spectrum after replacing unsampled points of the FID with zeros. Dashed blue line shows the best (under 𝓁₁-norm) Lorentzian fits to this zero-padded data. Solid yellow line shows the reconstructed spectrum after applying the IST-S algorithm. The y axis is rescaled (zoomed-in) compared to Fig. 1 to make the features more visible. (C) Fidelity of quantum simulation. The yellow crosses show the squared Bhattacharyya coefficient, and the green dots show a generalized cross entropy benchmark (gXEB) (19) as a function of the circuit depth measured in the number of two-qubit gates. arb. units, arbitrary units.

Fig. 3.. Scaling up to classically hard liquid-state zero-field NMR simulations.

(A) Chemical structures of (i) anti-3,4-difluoroheptane (38), (ii) a system with two coupled tert-butyl groups, and (iii) the B[ACR9]3 phosphorous system (21). Light green atoms do not contribute to the NMR signal, and dashed boxes indicate strongly interacting clusters where circuit synthesis can substantially speed up the quantum computation (see the Supplementary Materials). (B) Experimental design curves for (Me₃Si)₃P₇ [(A), iii], showing $1/\sqrt{D}$ scaling, where D is the circuit depth, of the frequency resolution up to a minimally achievable width set by the decoherence of the quantum computer. The circuit depth is measured by the number of (arbitrarily connected) two-qubit gates. (C) Optimal resolution for all three molecules. The circles indicate the resolution at optimal circuit depth, and the dashed black horizontal lines indicate the resolution accessible in NMR experiments.