Quantum Hackathon Challenge: Molecular Geometry Optimization

1. Introduction and Motivation

The ability to precisely predict the structure of molecules is central to fields like chemistry, materials science, and drug discovery. While classical computation struggles with the exponential complexity of large quantum systems, quantum computers offer a path forward.

Motivation: Google's Willow QC Research

The recent paper, "Quantum computation of molecular geometry via many-body nuclear spin echoes," demonstrated a breakthrough by using the Willow superconducting quantum computer in a hybrid approach to determine molecular geometry.

i. The Research Question: The researchers' goal was to find the precise atomic distances (geometry) of molecules like toluene and 3',5'-dimethylbiphenyl (DMBP). ii. The Method: Instead of directly solving for the geometry, they simulated the complex Out-of-Time-Ordered Correlator (OTOC) dynamics of the molecule's nuclear spins. iii. The Hybrid Approach: They used a classical-quantum feedback loop where the quantum computer ran a Trotterized simulation of the spin Hamiltonian (encoded with a candidate geometry), and a classical optimizer adjusted the geometry until the simulated results matched experimental data.

The Hackathon Challenge: Your task is to implement a downscaled version of this core problem—molecular geometry optimization—using the Variational Quantum Eigensolver (VQE) algorithm and the Qiskit Nature toolkit. You will find the stable bond length of simple molecules by minimizing their ground state energy.

2. The Core Challenge: $H_2$ Bond Length Optimization

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm used to find the ground state energy of a molecular Hamiltonian. By performing VQE at various bond lengths, you can map the molecule’s Potential Energy Surface (PES) and find the equilibrium bond length where the energy is minimized.

Task 1: Hydrogen Molecule ($H_2$) - The Standard Model

The simplest molecule, $H_2$, allows for rapid iteration and testing on simulators or real quantum hardware (using just 2-4 qubits).

Goal: Determine the equilibrium bond length ($r_{eq}$) of the $H_2$ molecule.

Steps:

1. Define Molecular Geometry Range: Use a classical chemistry driver (e.g., PySCF) within Qiskit Nature to define the $H_2$ molecule for a range of inter-atomic distances (e.g., $r = 0.5 \AA$ to $2.5 \AA$).
2. Generate the Hamiltonian: For each distance $r_i$, generate the corresponding Electronic Structure Hamiltonian in the second-quantized form, map it to a Qubit Hamiltonian (using mappers like Jordan-Wigner or Parity), and reduce the number of qubits using symmetries.
3. Implement VQE: i. Ansatz: Design a variational circuit ($\vert\Psi(\vec{\theta})\rangle$). The Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz is standard, but a simpler EfficientSU2 can be used for initial testing. ii. Optimizer: Use a classical optimizer (e.g., COBYLA, SLSQP) to find the parameters ($\vec{\theta}^*$) that minimize the expectation value $\langle\Psi(\vec{\theta})\vert H_i \vert\Psi(\vec{\theta})\rangle$.
4. Analyze Results: Plot the minimum energy found ($E_i$) against the bond length ($r_i$) to generate the Potential Energy Surface (PES) curve. The minimum point of this curve is the $r_{eq}$.
5. Deliverable: Plot the PES curve and state the calculated $r_{eq}$ value.

3. Stretch Goal: Scaling the Problem

For teams that successfully and efficiently complete the $H_2$ challenge, the next level is to tackle a slightly more complex system that requires a larger number of qubits and features a more complex geometry.

Task 2: Trihydrogen Cation ($H_3^+$)

The trihydrogen cation ($H_3^+$) is a 3-atom molecule and the most abundant molecular ion in the universe. Its ground state geometry is an equilateral triangle.

Goal: Determine the equilibrium bond length ($r_{eq}$) of the $H_3^+$ molecule in its equilateral configuration.

Steps:

1. Setup: Define the $H_3^+$ molecule in the geometry of an equilateral triangle, where the only variable is the side length $r$.
2. Qubit Challenge: $H_3^+$ is more demanding on qubits (typically 6-8 qubits before reduction). Teams must carefully select their mapper and utilize symmetry reduction to run the problem on a 5-7 qubit simulator/device.
3. Execution and Optimization: Repeat the VQE workflow (Steps 2-4 from Task 1) to generate the PES curve and find the minimum energy bond length. Deliverable: Plot the PES curve for $H_3^+$ and state the calculated $r_{eq}$.

Bonus Error Mitigation: Implement error mitigation techniques like Zero Noise Extrapolation on simulated noisy backends (using Aer noise models)

4. Tools and Resources

i. Quantum SDK: Qiskit (specifically qiskit-nature) or any other preferred Python-based SDK. ii. Backend: Qiskit Aer simulators or any other preferred quantum simulator. iii. Classical Driver: PySCF (integrated via Qiskit Nature) or any other preferred.