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arXiv:2509.24664 · Physical Chemistry Chemical Physics

Study of nuclear magnetic resonance spectra with the multi-modal multi-level quantum complex exponential least squares algorithm

A. M. Romero, Josh J. M. Kirsopp, Giuseppe Buonaiuto, Michał Krompiec

Published Jan 1, 2026• 1 citation
arXivHTMLPDF

Abstract

We present a novel application of the multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) algorithm, a state-of-the-art, early fault-tolerant quantum phase estimation (QPE) technique, to the simulation and analysis of nuclear magnetic resonance (NMR) of spin systems. By leveraging the robustness and precision of MM-QCELS, we demonstrate enhanced phase resolution in quantum simulations of spin dynamics, also in systems with complex coupling topologies. Our approach enables accurate extraction of spectral features with up to an order of magnitude fewer evaluations of the time series signal in comparison with the conventional Fourier transform, making a significant step toward scalable quantum simulations of NMR Hamiltonians. This work bridges an advanced quantum algorithm design with a practical spectroscopic application, offering a promising new approach for a quantum-based chemical analysis.

DOI
10.1039/d5cp04124d

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We present a novel application of the multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) algorithm, a state-of-the-art, early fault-tolerant quantum phase estimation (QPE) technique, to the simulation and analysis of nuclear magnetic resonance (NMR) of spin systems. By leveraging the robustness and precision of MM-QCELS, we demonstrate enhanced phase resolution in quantum simulations of spin dynamics, also in systems with complex coupling topologies. Our approach enables accurate extraction of spectral features with up to an order of magnitude fewer evaluations of the time series signal in comparison with the conventional Fourier transform, making a significant step toward scalable quantum simulations of NMR Hamiltonians. This work bridges an advanced quantum algorithm design with a practical spectroscopic application, offering a promising new approach for a quantum-based chemical analysis.
We introduce a multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) method to simultaneously estimate multiple eigenvalues of a quantum Hamiltonian on early fault-tolerant quantum computers. Our theoretical analysis demonstrates that the algorithm exhibits Heisenberg-limited scaling in terms of circuit depth and total cost. Notably, the proposed quantum circuit utilizes just one ancilla qubit, and with appropriate initial state conditions, it achieves significantly shorter circuit depths compared to circuits based on quantum phase estimation (QPE). Numerical results suggest that compared to QPE, the circuit depth can be reduced by around two orders of magnitude under several settings for estimating ground-state and excited-state energies of certain quantum systems.
Quantum computing is used to improve NMR spectroscopy sensitivity and resolution. Many scientific fields employ NMR to analyze molecular structures and interactions. Typical NMR data processing algorithms have limitations, especially in recognizing low-abundance compounds and resolving overlapping signals. To solve NMR data processing problems, the proposed research uses quantum algorithms, simulations, and a hybrid quantum-classical approach. Quantum Fourier transform (QFT) enhances sensitivity and quantum phase estimation (QPE) improves resolution. The QFT accelerates data analysis using quantum parallelism to detect low-concentration chemicals. QPEs accurately estimate phases, resolve overlapping peaks, and improve peak assignments. Quantum-assisted NMR data processing improvements are shown numerically. Quantum algorithms improve sensitivity and resolution, allowing delicate signals and correct structural assessments. This study also addresses quantum hardware restrictions, noise, and efficient quantum algorithm design. Quantum-assisted NMR data processing has the potential to transform NMR spectroscopy. Researchers can acquire new accuracy and sensitivity into molecule structures, interactions, and dynamics by linking quantum computers and NMR data analysis. This research advances quantum computing and NMR spectroscopy and lays the groundwork for future studies on quantum-assisted approaches in real-world NMR applications. Quantum-assisted data processing will enable novel molecular characterisation methods and groundbreaking scientific discoveries as quantum technologies advance.
We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to non-ergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal 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 computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure.
Simulating nuclear magnetic resonance (NMR) spectra is a promising application of quantum simulation. Using Quantinuum's System Model H2, a trapped-ion quantum computer, we demonstrate an end-to-end, large-scale digital NMR simulation of a classically challenging benchmark molecule, 1,2-di-tert-butyl-diphosphane. We implement a hardware-efficient reduction of the nuclear-spin Hamiltonian, enabling Trotterized real-time evolution of an effective 21-spin model with tailored error suppression to reduce the effects of device noise. The reconstructed liquid-state proton NMR spectrum agrees with classical reference calculations and reproduces key spectroscopic features that previous quantum hardware demonstrations did not capture. Given the widespread use of NMR in chemical analysis and industrial research, these results advance digital quantum simulation of NMR spectra towards practical quantum utility.
Advanced atomic magnetometers have made it possible to acquire nuclear magnetic resonance spectra in zero to ultralow magnetic fields. This regime carries the benefit of compact, low-cost instrumentation with reduced spin relaxation effects and the ability to probe phenomena that are inaccessible in conventional high-field experiments. A trade-off is that the resulting spectra must be interpreted using simulations that are taxing for classical computation. Working by example for small-molecule and protein spectroscopy, we demonstrate that these simulations are a promising target for fault-tolerant quantum computation. Our holistic analysis spans from input selection to the construction of explicit circuits for qubitized quantum dynamics. By maintaining parity with experimental requirements, we demonstrate how certain cases might be especially promising for early fault-tolerant architectures. Notably, valuable natural products and small proteins could be simulated with a few hundred logical qubits and less than 10 12 T gates, corresponding to days of runtime with foreseeable hardware based on spin or superconducting qubits.
The procedure for simulating the nuclear magnetic resonance spectrum linked to the spin system of a molecule for a certain nucleus entails diagonalizing the associated Hamiltonian matrix. As the dimensions of said matrix grow exponentially with respect to the spin system's atom count, the calculation of the eigenvalues and eigenvectors marks the performance of the overall process. The aim of this paper is to provide a formalization of the complete procedure of the simulation of a spin system's NMR spectrum while also explaining how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance. Two well-known quantum algorithms for calculating the eigenvalues of a matrix are analyzed and put to the test in this context: quantum phase estimation and the variational quantum eigensolver. Additionally, we present simulated results for the later approach while also addressing the hypothetical noise found in a physical quantum computer.