# What is the main advantage of using the Variational Quantum Eigensolver over a classical algorithm?

What is the main advantage of using the Variational Quantum Eigensolver (quantum computing) over a classical algorithm? I know a key fact is the speed-up, but how is this speed-up quantised.

Both VQE and classical methods rely on the variational principle. The variational principle for quantum mechanics says that if you have a physical system $$H$$ and a wavefunction $$|\psi\rangle$$ describing a state in that system, it's expectation value $$\langle \psi | H | \psi \rangle$$ will be greater than the ground state energy, $$E_0$$. In other words, you can't ever find a state in your system with less energy than the ground state.
On a quantum computer, this is accomplished by positing a parameterized quantum circuit as a trial wavefunction, $$|\psi(\theta)\rangle$$. Then, the circuit is executed at a specific set of parameters and we compute the resultant expectation value. Based on the result, a classical optimization algorithm is employed to iteratively change parameters to reduce the expectation value. The goal is to find some parameter set $$\theta'$$, such that $$|\langle\psi(\theta')|H|\psi(\theta')\rangle - E_0 |< \epsilon$$. If the eigenvector for the ground state is in the parameter space of $$|\psi(\theta)\rangle$$ (a "good" ansatz), a properly functioning quantum computer and a suitable classical optimization algorithm should be able to provide a good estimate for the ground state of that system.
A similar thing happens on a classical computer. However, the trial wavefunction is implemented on a classical computer where the physics of the system are simulated. A trial wavefunction (ansatz) is suggested, and a classical computer then attempts to minimize the functional $$F[\psi] = \frac{\psi^* H\psi}{\psi^*\psi}$$.