Algorithms

Variational Quantum Eigensolver (VQE)

A hybrid quantum-classical algorithm optimized for finding the ground state energy of molecules on NISQ devices.

QWiki Team 1 min read
#algorithms#chemistry#nisq#vqe
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Introduction

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm designed to find the eigenvalues of a large matrix, specifically the ground state energy of a molecular Hamiltonian. It is highly optimized for Noisy Intermediate-Scale Quantum (NISQ) computers.

Variational Principle

VQE is based on the variational method in quantum mechanics. For any ansatz state ∣ψ(θ⃗)⟩|\psi(\vec{\theta})\rangle parameterized by classical parameters θ⃗\vec{\theta}, the expectation value of the Hamiltonian HH is always greater than or equal to the ground state energy E0E_0: ⟨H⟩θ⃗=⟨ψ(θ⃗)∣H∣ψ(θ⃗)⟩⟨ψ(θ⃗)∣ψ(θ⃗)⟩≥E0\langle H \rangle_{\vec{\theta}} = \frac{\langle\psi(\vec{\theta})|H|\psi(\vec{\theta})\rangle}{\langle\psi(\vec{\theta})|\psi(\vec{\theta})\rangle} \ge E_0

Hybrid Architecture

VQE delegates work between quantum and classical processors:

  1. Quantum Processor: Prepares the parameterized state ∣ψ(θ⃗)⟩|\psi(\vec{\theta})\rangle and measures the expectation values of the Hamiltonian terms.
  2. Classical Processor: Evaluates the energy and uses classical optimization algorithms (e.g., COBYLA, Adam) to update the parameters θ⃗\vec{\theta} to minimize the energy.
  3. This loop repeats until convergence is reached.