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

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.

Figures (13)

• Figure 1: Standard plot of the magnetization signal. a) Case without attenuation, theoretical description of magnetization according to Eq. (\ref{['eq:M_eq']}) b) Case with an attenuation factor of $\eta = 0.005\pi$ in units of inverse of time, included, corresponding to the FID signal in Eq. (\ref{['eq:FID']}). Both subplots correspond to a simulation of a three-spin system, computed with $2^{14}$ temporal data points on a uniform grid with a time step of $\Delta t = 0.7$ in arbitrary time units. This resolution ensures accurate representation of the magnetization dynamics under the specified conditions.
• Figure 2: Standard plot of the NMR spectrum of a molecule with three spins. (a) Case of negligible coupling constants. Three peaks are observed centered at the values of the chemical shifts as described in the NMR Hamiltonian (\ref{['eq:NMR_ham']}) for all $J_{ij} = 0$. (b) Case of non-negligible coupling constants. Three groups of four different peaks are centered around the three different chemical shifts. The peak splittings due to the two-spin interaction are equal to the difference in the coupling constants of the spins involved. Since in this case a spin can interact with the other two, there are two splittings observed within the same group.
• Figure 3: The general form of the Hadamard test circuits used in this publication to generate the dataset, with a controlled-magnetization gate using a single ancilla qubit. $W$ is the identity gate or the $S^{\dagger}$ gate (with $S$ being the standard phase gate) if we want to measure the real or imaginary part of the magnetization operator, corresponding to the $M_x$ and $M_y$ components, respectively.
• Figure 4: The general form of the block encoding circuits used in this publication to encode the magnetization operator. Given the operator to be encoded as a linear combination of unitaries, the PREP operator encodes the coefficients and the SELECT operator encodes the unitaries.
• Figure 5: The general form of the SELECT operator circuits used in this publication to encode the magnetization operator. The different controlled states specify which term $M_i$ of the operator to be applied to $|{\psi}\rangle$