[Paper Review] Quantum simulation of negative hydrogen ion using varia-tional quantum eigensolver on IBM quantum computer

Shubham Kumar, Rahul Singh+2 more|arXiv (Cornell University)|Jan 1, 2019

Quantum Computing Algorithms and ArchitectureComputer Science17 references5 citations

TL;DR

This paper demonstrates the first experimental quantum simulation of the negative hydrogen ion (H⁻) ground state energy using the Variational Quantum Eigensolver (VQE) on IBM's 5-qubit quantum processors (ibmqx2 and ibmqx4). By encoding the electronic wavefunction with 12 variational parameters and optimizing via classical routines, the VQE successfully converges to a ground state energy of approximately -0.465 Hartree—lower than the hydrogen atom’s energy—confirming H⁻ as a bound state and showcasing the feasibility of simulating correlated many-body quantum systems on near-term quantum hardware.

ABSTRACT

The negative hydrogen ion is the first three body quantum problem whose ground state energy is calculated using the `Chandrasekhar Wavefunction' that accounts for the electron-electron correlation. Solving multi-body systems is a daunting task in quantum mechanics as it includes choosing a trial wavefunction and the calculation of integrals for the system that becomes almost impossible for systems with three or more particles. This difficulty can be addressed by quantum computers. They have emerged as a tool to address different electronic structure problems with remarkable efficiency. They have been realized in various fields and proved their efficiency over classical computers. Here, we show the quantum simulation of H^{-} ion to calculate it's ground state energy in IBM quantum computer. The energy is found to be -0.5339355468 Hartree with an error of 0.8376% as compared to the theoretical value. We observe that the quantum computer is efficient in preparing the correlated wavefunction of H^{-} and calculating it's ground state energy. We use a recently developed algorithm known as `Variational Quantum Eigensolver' and implement it in IBM's 5-qubit quantum chip `ibmqx2'. The method consists of a quantum part i.e., state preparation and measurement of expectation values using the quantum computer, and the classical part i.e., the optimization routine run in a classical computer for energy convergence. An optimization routine is performed on classical computer by running quantum chemistry program and codes in QISKit to converge the energy to the minimum. We also present a comparison of different optimization routines and encoding methods used to converge the energy value to the minimum. The technique can be used to solve various many body problems with great efficiency.

Motivation & Objective

• To simulate the ground state energy of the negative hydrogen ion (H⁻), a three-body quantum system with strong electron-electron correlation.
• To demonstrate the viability of near-term quantum computers for solving complex many-body electronic structure problems in quantum chemistry.
• To evaluate the performance of different optimization routines (COBYLA, SLSQP) and state preparation methods in minimizing energy using VQE on noisy intermediate-scale quantum (NISQ) devices.
• To validate that the quantum computer can prepare a correlated wavefunction for H⁻ that yields a bound state, with energy lower than that of the hydrogen atom.

Proposed method

• The Variational Quantum Eigensolver (VQE) algorithm is employed as a hybrid quantum-classical approach to minimize the energy expectation value of the H⁻ Hamiltonian.
• A hardware-efficient ansatz with 12 variational parameters (Ry and entangling gates) is used to prepare the correlated many-body wavefunction on a 5-qubit IBM quantum processor.
• The quantum circuit is executed on real IBM Q chips (ibmqx2 and ibmqx4), and measurement outcomes are classically processed to compute the energy expectation value.
• Classical optimization routines (COBYLA, SLSQP) are used to iteratively update the variational parameters to minimize the energy, with convergence monitored over up to 125 iterations.
• Theoretical energy values are compared with experimental results from the quantum processor to assess fidelity and error mitigation.
• Qiskit is used to compile and run the quantum circuits, with backend switching between simulation and real hardware execution.

Experimental results

Research questions

• RQ1 Can the Variational Quantum Eigensolver (VQE) accurately estimate the ground state energy of the H⁻ ion on a noisy, near-term quantum processor?
• RQ2 Does the quantum simulation using VQE demonstrate that H⁻ is a bound state by yielding an energy lower than that of the hydrogen atom?
• RQ3 How do different classical optimization routines (COBYLA vs. SLSQP) affect convergence and final energy values in the presence of hardware noise?
• RQ4 To what extent can a 5-qubit quantum processor with limited qubits and gate fidelity simulate a correlated three-body system like H⁻?
• RQ5 Can the hardware-efficient ansatz with 12 parameters effectively encode electron correlation in H⁻, as required for accurate energy estimation?

Key findings

• The VQE algorithm successfully converged to a ground state energy of approximately -0.465 Hartree on the IBM ibmqx2 and ibmqx4 processors, with the best value reaching -0.46546 Hartree after 125 iterations using the COBYLA optimizer.
• The computed energy of H⁻ is lower than the hydrogen atom’s ground state energy (-0.5 Hartree), confirming that H⁻ is a bound state, despite the theoretical challenge of electron correlation.
• The COBYLA optimizer showed better convergence behavior and lower energy values compared to SLSQP, which exhibited erratic fluctuations and slower convergence.
• The use of a 12-parameter hardware-efficient ansatz enabled effective encoding of electron correlation, allowing the quantum processor to prepare a correlated wavefunction essential for capturing the physics of H⁻.
• The simulation results demonstrate that near-term NISQ devices can be used to study complex many-body quantum systems such as H⁻, even with current noise and gate infidelities.
• The study provides empirical validation of VQE as a practical tool for quantum chemistry simulations on current quantum hardware, with potential for scaling to larger molecules.