Project: Molecular Ground State

Computing molecular ground-state energies is the application most often cited as a near-term payoff for quantum computers. This capstone walks the smallest real example — the hydrogen molecule $\mathrm{H}_2$ — from its electronic Hamiltonian down to a two-qubit state you can prepare and check.

From molecule to qubits

The electronic structure problem asks for the lowest eigenvalue $E_0$ of the molecular Hamiltonian in a chosen basis. For $\mathrm{H}_2$ in a minimal (STO-3G) basis, the second-quantized Hamiltonian maps — via the Jordan–Wigner or Bravyi–Kitaev transform — to a sum of Pauli strings on four qubits. Exploiting particle-number and parity symmetries reduces this to an effective two-qubit Hamiltonian of the form

H = g_0 I + g_1 Z_0 + g_2 Z_1 + g_3 Z_0 Z_1 + g_4 X_0 X_1 + g_5 Y_0 Y_1,

with real coefficients $g_i$ that depend on the bond length $R$ . This is the model used by O'Malley et al. (2016) in the first VQE measurement of an $\mathrm{H}_2$ curve.

The relevant subspace

Number conservation confines the ground state to the single-excitation sector spanned by $|01\rangle$ and $|10\rangle$ . The ansatz

|\psi(\theta)\rangle = \cos\tfrac{\theta}{2}\,|01\rangle - \sin\tfrac{\theta}{2}\,|10\rangle

is therefore expressive enough to reach the true ground state with a single parameter $\theta$ . The classical optimizer varies $\theta$ to minimize $\langle\psi(\theta)|H|\psi(\theta)\rangle$ .

The converged state

At the equilibrium bond length the $X_0X_1$ and $Y_0Y_1$ coupling drives the optimizer to the symmetric mixing angle $\theta = \pi/2$ , producing the singlet-like state

|\psi\rangle = \frac{1}{\sqrt{2}}\bigl(|01\rangle - |10\rangle\bigr).

This is the lowest-energy combination of the two configurations: the antisymmetric superposition maximizes the favorable exchange term, exactly as the variational principle predicts.

Try it

Prepare the converged ground state $(|01\rangle - |10\rangle)/\sqrt{2}$ with the supplied gate recipe. The grader checks the two-qubit statevector.

Run your code to see the quantum state.