VQE (Variational Quantum Eigensolver) is a hybrid quantum-classical algorithm for estimating the ground state energy of a quantum system. It was proposed in 2014 by Peruzzo et al. and is one of the most important algorithms for near-term NISQ devices, with applications in quantum chemistry and materials science.

VQE is grounded in the variational principle: for any trial state $\lvert\psi(\boldsymbol\theta)\rangle$, the expectation value of the Hamiltonian $H$ is an upper bound on the true ground state energy $E_0$.

$$E_0 \leq \langle\psi(\boldsymbol\theta)\rvert H\lvert\psi(\boldsymbol\theta)\rangle$$

A parameterized quantum circuit (the ansatz) prepares a trial state $\lvert\psi(\boldsymbol\theta)\rangle$ depending on classical parameters $\boldsymbol\theta$. The quantum computer evaluates the expectation value $\langle H\rangle = \langle\psi(\boldsymbol\theta)\rvert H\lvert\psi(\boldsymbol\theta)\rangle$ by decomposing $H$ into Pauli strings and measuring each term. A classical optimizer then updates $\boldsymbol\theta$ to lower $\langle H\rangle$, and the loop repeats until convergence.

The quality of VQE depends critically on the choice of ansatz. A good ansatz should be expressive enough to represent the ground state, shallow enough to avoid decoherence, and structured to avoid barren plateaus (regions where gradients vanish exponentially). Common ansätze include the hardware-efficient ansatz (uses gates native to the hardware) and the UCCSD ansatz (unitary coupled cluster with singles and doubles, motivated by quantum chemistry).

VQE is primarily used in quantum chemistry to compute molecular ground state energies, enabling prediction of molecular properties, reaction energetics, and drug-molecule interactions. Classical methods like full configuration interaction scale exponentially in the number of electrons, making strongly correlated molecules intractable classically. VQE is also used in condensed matter physics to study strongly correlated systems such as the Hubbard model.