Checkpoint: Ground-State Energy

This checkpoint ties the module together: starting from a molecular qubit Hamiltonian, you compute the electronic ground-state energy of molecular hydrogen — the quantity a VQE run on real hardware is designed to estimate.

The full pipeline, recalled

Everything in this module feeds one number:

Electronic structure problem — the Born–Oppenheimer electronic Hamiltonian.
Second quantization — rewrite it in fermionic creation/annihilation operators with molecular integrals.
Jordan–Wigner — map fermions to qubits; the Hamiltonian becomes a weighted sum of Pauli strings.
Tapering — symmetries reduce $\text{H}_2$ to two qubits: $\hat H = g_1 Z_0 + g_2 Z_1 + g_3 Z_0 Z_1 + g_4 Y_0 Y_1 + g_5 X_0 X_1$ .
Hartree–Fock — the reference state $|01\rangle$ and its (uncorrelated) energy.
VQE / UCC — a particle-preserving ansatz that mixes $|01\rangle$ and $|10\rangle$ to find the true ground state.

Reducing to a 2×2 eigenproblem

Because the Hamiltonian conserves particle number and spin, its ground state is confined to the single-excitation subspace $\{|01\rangle, |10\rangle\}$ . There $\hat H$ is the $2\times2$ matrix

\hat H_{\text{sub}} = \begin{pmatrix} a & c \\ c & b \end{pmatrix}, \qquad \begin{aligned} a &= \langle 01|\hat H|01\rangle = g_1 - g_2 - g_3,\\ b &= \langle 10|\hat H|10\rangle = -g_1 + g_2 - g_3,\\ c &= g_4 + g_5, \end{aligned}

where $a$ and $b$ come from the diagonal $Z$ terms (using $Z|0\rangle = +|0\rangle$ , $Z|1\rangle = -|1\rangle$ ) and $c$ from the off-diagonal $X_0X_1 + Y_0Y_1$ terms, both of which map $|10\rangle \to |01\rangle$ with coefficient $+1$ . The electronic ground-state energy is the lower eigenvalue,

E_0 = \frac{a+b}{2} - \sqrt{\left(\frac{a-b}{2}\right)^2 + c^2}.

This is exactly the energy the one-parameter VQE ansatz from the earlier lesson converges to — diagonalizing the $2\times2$ block analytically here gives the same answer VQE finds variationally.

Try it

Using the equilibrium $\text{H}_2$ coefficients $g_1 = 0.3435$ , $g_2 = -0.4347$ , $g_3 = 0.5716$ , $g_4 = g_5 = 0.0910$ (Hartree), build the $2\times2$ subspace block and return the lower eigenvalue — the electronic ground-state energy. The grader also checks that your answer lies below the Hartree–Fock energy $a$ , confirming you captured the correlation energy.

Run your code to see the quantum state.

You have now carried a molecule from its electronic Hamiltonian all the way to a ground-state energy — the complete quantum-chemistry workflow in miniature.