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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.

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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 $F[\psi] = \frac{\psi^* H\psi}{\psi^*\psi}$.

For most practical applications, scientists will use a classical method to determine the ground state because quantum computers are still in their very early stages. However, for problems of greater complexity we hope algorithms such as VQE, and its variants, will be able to provide an advantage over purely classical approaches.

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