VQEs

image credit “Jeremy Retamar Arrieta”

What are Variational Quantum Egiensolvers?

Variational quantum eigensolver (VQE) is a quantum algorithm that can be used to find the ground state energy of a Hamiltonian, as well as excited states of the Hamiltonian. The VQE algorithm is based on the variational principle, which seeks to find an ansatz (an approximation) to the ground state that has lower energy than the ground state itself. It has been shown that the VQE allows for universal quantum computing, making it a versatile tool for tackling a wide range of tasks. The VQE algorithm is versatile and can be applied to a wide range of tasks, making it a promising candidate for use in many applications of quantum computers. One particularly useful application of the VQE algorithm is in finding the ground state energy of a Hamiltonian. In this application, the algorithm is used to find an approximation to the ground state energy that is lower than the ground state energy itself. This is accomplished by minimizing a cost function that is based on the expectation value of the Hamiltonian. The VQE algorithm can also be used to find excited states of a Hamiltonian. In this application, the algorithm is used to find an approximation to an excited state of the Hamiltonian that is lower than the excited state energy itself.

This is accomplished by minimizing a cost function that is based on the expectation value of the Hamiltonian plus an additional term that accounts for the overlap between the excited state and the ground state. The VQE algorithm has potential applications in many other areas, including machine learning and control theory.

It can also be used to find excited states of a system by employing an orthogonality constraint. Additionally, the VQE can be used to find low-energy excited states via the subspace expansion method. These are just a few examples of the many applications of the VQE; its versatility makes it a powerful tool for tackling a wide range of tasks in quantum computing.

The cost function is defined as 〈ψ|H|ψ〉, where ψ is a trial state and H is the Hamiltonian. VQE is aimed at finding the ground state energy of a Hamiltonian, where the cost function is defined as 〈ψ|H|ψ〉. According to the Rayleigh-Ritz variational principle, the cost is meaningful and faithful as C(θ) > EG, with equality holding if |ψ(θ)〉 is the ground state |ψG〉of H.

The quantum subroutine has two fundamental steps:

1. Prepare the quantum state |Ψ(vec(θ))⟩ often called the ansatz.
2. Measure the expectation value ⟨Ψ(vec(θ))|H|Ψ(vec(θ))⟩.

This bound allows us to use classical computation to run an optimization loop to find this eigenvalue:

1. Use a classical non-linear optimizer to minimize the expectation value by varying ansatz parameters vec(θ)
2. Iterate until convergence.

Another way to understand this is the classical algorithm controls the quantum children. The classical parent is then replaced by the quantum child that performs the best. Then the algorithm would be trained again to further improve accuracy and fidelity.