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Simulating NMR Spectra with a Quantum Computer

Joaquín Ossorio-Castillo, Alexandre Rodríguez-Coello

Abstract

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.

Simulating NMR Spectra with a Quantum Computer

Abstract

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.

Paper Structure

This paper contains 18 sections, 74 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Constructing the Hamiltonian Matrix of a Spin System
  3. Brief Quantum Formalism
  4. Hamiltonian of a Spin System
  5. Quantum Algorithm for Calculating the Eigenvalues of a Hamiltonian Matrix
  6. Approach 1: Quantum Phase Estimation
  7. $\mathbb{SETUP}$
  8. $\mathbb{STEP}$ 1
  9. $\mathbb{STEP}$ 2
  10. $\mathbb{STEP}$ 3
  11. $\mathbb{STEP}$ 4
  12. Approach 2: Variational Quantum Eigensolver
  13. Example: Sulfanol
  14. Calculation of the Eigenvalues Using the QPE Algorithm
  15. Calculation of the Eigenvalues and Eigenstates Using the VQE
  16. ...and 3 more sections

Figures (5)

  • Figure 1: Sulfanol molecule
  • Figure 2: Classically calculated $^1$H-NMR spectrum of the sulfanol molecule
  • Figure 3: Comparison of the computed lowest eigenvalue with and without error mitigation. The extrapolation curve corresponds to the values of the last iteration of the optimization process.
  • Figure 4: $^1$H-NMR spectrum of the sulfanol molecule using the $D_H$ and $V_H$ matrices calculated with our quantum algorithm
  • Figure 5: $^1$H-NMR spectrum of the sulfanol molecule simulated by MestReNova