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Variational quantum eigensolver for chemical molecules

Luca Ion, Adam Smith

TL;DR

This work evaluates variational quantum eigensolvers for computing ground-state energies of the He-H+ and H2O molecules on quantum simulators and an IBM device, constructing Hamiltonians via exact methods and PennyLane. It employs a layered circuit ansatz with RX, RZ, and CZ entangling gates and benchmarks several gradient-descent schemes (FOGD, SOGD, SPSA, PS) against exact diagonalization. Results show that parameter-shift (PS) and second-order finite difference (SOGD) converge well in simulation, while SPSA offers fewer circuit calls but can hinder convergence, with hardware runs limited by noise. The findings inform practical VQE deployment, including initialization, learning-rate tuning, and scaling strategies to larger, more realistic active-space models.

Abstract

Solving interacting multi-particle systems is a central challenge in quantum chemistry and condensed matter physics. In this work, we investigate the computation of ground states and ground-state energies for the He-H+ and H2O molecules using quantum computing techniques. We employ the variational quantum eigensolver (VQE), implemented both on a quantum computer simulator and on an IBM quantum device. The resulting energies are benchmarked against exact ground-state energies obtained via classical methods. Simulations of the H2O molecule were performed on Nottingham's High Performance Computing (HPC) facilities.

Variational quantum eigensolver for chemical molecules

TL;DR

This work evaluates variational quantum eigensolvers for computing ground-state energies of the He-H+ and H2O molecules on quantum simulators and an IBM device, constructing Hamiltonians via exact methods and PennyLane. It employs a layered circuit ansatz with RX, RZ, and CZ entangling gates and benchmarks several gradient-descent schemes (FOGD, SOGD, SPSA, PS) against exact diagonalization. Results show that parameter-shift (PS) and second-order finite difference (SOGD) converge well in simulation, while SPSA offers fewer circuit calls but can hinder convergence, with hardware runs limited by noise. The findings inform practical VQE deployment, including initialization, learning-rate tuning, and scaling strategies to larger, more realistic active-space models.

Abstract

Solving interacting multi-particle systems is a central challenge in quantum chemistry and condensed matter physics. In this work, we investigate the computation of ground states and ground-state energies for the He-H+ and H2O molecules using quantum computing techniques. We employ the variational quantum eigensolver (VQE), implemented both on a quantum computer simulator and on an IBM quantum device. The resulting energies are benchmarked against exact ground-state energies obtained via classical methods. Simulations of the H2O molecule were performed on Nottingham's High Performance Computing (HPC) facilities.
Paper Structure (8 sections, 6 equations, 7 figures)

This paper contains 8 sections, 6 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. EXACT DIAGONALIZATION
  3. FINDING THE HAMILTONIAN
  4. QUANTUM CIRCUIT ANSATZ
  5. METHOD
  6. HE-H$^{+}$ RESULTS
  7. H$_2$O RESULTS
  8. DISCUSSION

Figures (7)

  • Figure 1: Circuit ansatz for $N=3$ and $M=2$. $R_X$ and $R_Z$ are rotation gates and the connection between the wires is done through Control-Z gates.
  • Figure 2: Simulated He-H$^{+}$ optimization for $R=0.9$Å, $M=1$, $N=2$, $\eta=0.8$ and for all the different gradient descent variants.
  • Figure 3: IBM He-H$^{+}$ optimization for $R=0.9$Å, $M=1$, $N=2$, $\eta=0.8$ and SOGD.
  • Figure 4: He-H$^{+}$: Energy($E$) against $R$ in picometres plot for the exact, VQE-simulation (SOGD), and IBM results using $M=1$ ansatz and $\eta=0.8$.
  • Figure 5: The geometry of the H$_2$O molecule. $R$ is the O-H bond length and $\phi$ is the HOH angle.
  • ...and 2 more figures