Variational Quantum Eigensolver
The Variational Quantum Eigensolver (VQE) is a quantum algorithm that aims to find the ground state energy of a given quantum system. It was first introduced by Peruzzo et al. in 2014 and has since become an important tool in quantum chemistry and material science.
VQE is designed to leverage the capabilities of both classical and quantum computation. It combines a classical optimization loop with a parameterized quantum circuit to find an approximation of the ground state energy of a target Hamiltonian. Here's an overview of how VQE works:
1. Hamiltonian Preparation: The VQE algorithm starts by encoding the problem's Hamiltonian into a quantum circuit. The Hamiltonian represents the energy of the quantum system under investigation and is typically represented as a sum of terms corresponding to the system's observables.
2. Ansatz Initialization: A parameterized quantum circuit, known as the ansatz, is prepared. The ansatz is a sequence of quantum gates and rotations with adjustable parameters. It represents a trial wavefunction that is expected to approximate the ground state of the system. The choice of ansatz depends on the specific problem and the available resources.
3. Quantum Circuit Execution: The parameterized ansatz is implemented on a quantum computer, and measurements are performed to estimate the expectation values of the terms in the Hamiltonian. This involves running the circuit multiple times and collecting statistical data from the measurements.
4. Energy Estimation: The expectation values obtained from the measurements are used to estimate the energy of the trial wavefunction. The energy estimation is performed classically using the measurement outcomes and the known form of the Hamiltonian.
5. Classical Optimization: A classical optimization algorithm is used to update the parameters of the ansatz based on the estimated energy. The objective is to minimize the energy of the trial wavefunction, driving it closer to the ground state energy.
6. Iteration: Steps 3 to 5 are repeated iteratively, updating the ansatz parameters and estimating the energy until convergence criteria are met. The convergence criteria could be a specified number of iterations, achieving a desired energy threshold, or satisfying other termination conditions.
The final energy obtained by VQE provides an approximation to the ground state energy of the system under consideration. The corresponding trial wavefunction can also be used to extract additional information about the system, such as molecular properties or excited states.
VQE is particularly useful for problems in quantum chemistry, where determining the ground state energy of molecules is a challenging task. By utilizing quantum resources and classical optimization, VQE offers a promising approach to tackle these problems on quantum computers.
