The Variational Quantum Eigensolver

The whole thing rests on one inequality you already know: for any normalized trial state, $⟨\psi |H|\psi ⟩\ge E_0$, the true ground-state energy. So parametrize the trial state by some knobs $\vec{\theta }$, evaluate the energy, and minimize:

VQE changes exactly one thing about this familiar recipe: the state $|\psi (\vec{\theta })⟩$ is prepared by a quantum circuit, and the energy is read off by measuring qubits. The minimization is still done by an ordinary classical optimizer. Watch it in the demo: the landscape $E(\theta )$ never dips below the green floor $E_0$, and the minimum sits exactly on it.

A quantum computer doesn't speak in electrons and orbitals; it speaks in qubits and Pauli operators. So the electronic Hamiltonian is rewritten — through a fermion-to-qubit map like Jordan–Wigner — as a weighted sum of Pauli strings:

For H₂ in the minimal basis this collapses, after using the molecule's symmetries, to a tiny two-qubit operator $H=c_{0}I+c_1{Z}_0+c_2{Z}_1+c_3{Z}_0{Z}_1+c_4{X}_0{X}_1$, whose coefficients $c_j$ are built from the same one- and two-electron integrals as Hartree–Fock. The energy is then $⟨H⟩={\sum }_j{c}_j⟨P_j⟩$ — each $⟨P_j⟩$ a number you get by measuring the qubits.

The trial state is whatever a parametrized circuit produces, $|\psi (\vec{\theta })⟩=U(\vec{\theta })|0\cdots 0⟩$. The art is choosing $U$ rich enough to contain the true ground state but shallow enough to actually run. For H₂, the singlet ground state lives in a two-dimensional space spanned by the Hartree–Fock configuration $|g^2⟩$ and its double excitation $|u^2⟩$, so one rotation angle suffices: $|\psi (\theta )⟩=\cos \theta |g^2⟩-\sin \theta |u^2⟩$.

That is the entire ansatz, and the demo's left panel is its energy landscape. The loop is hybrid: the quantum device prepares $|\psi (\theta )⟩$ and reports $E(\theta )$; a classical optimizer on your laptop proposes the next $\theta$. Quantum where it helps, classical where it doesn't.

Why bother with a quantum computer at all? Because the trial state is the bottleneck. Classically, a general $N$-orbital wavefunction has $\sim 2^N$ amplitudes — the same exponential wall that full configuration interaction hits and that DMRG tiptoes around. A quantum computer stores that state in $N$ qubits natively, and a few measurements estimate $⟨H⟩$ without ever writing the amplitudes down.

VQE is built for this trade. It keeps the circuit shallow and offloads the optimization to a classical computer, which is what makes it the leading candidate for near-term (noisy, few-qubit) hardware — you don't need a fault-tolerant machine to vary one angle and measure an energy.

Here is the physics payoff, in the demo's right panel. Slide the bond length out toward dissociation and watch restricted Hartree–Fock (orange) climb to the wrong energy — it cannot describe two atoms pulling apart, the textbook static-correlation failure that the RHF/UHF story is about.

VQE (the variational minimum, indigo) follows the correct curve, because its two-configuration ansatz can mix in the $|u^2⟩$ state that lets the bond break. Same molecule, same integrals as your from-scratch SCF — but by going one step past Hartree–Fock on a quantum circuit, you recover the correlation energy that makes the chemistry come out right.