IBM Quantum Platform - VQE

Hybrid Classical-Quantum Algorithm

Operator $H$ (Hamiltonian): we need "ground state" eigenvector - eigenvector of this operator that corresponds to the minimum eigenvalue.
- can be expressed as linear combination

H = \sum_{α = 1}^{T} c_{α} h_{α}

Ansatz (approach): Quantum circuit that prepares a quantum state approximating ground state eigenvector
- Starting guess which does not contain variational parameters is called reference state

| ψ_{ref} ⟩ = U_{ref} (\vec{θ}) {| 0 ⟩}^{\otimes N}

U_{ansatz} (\vec{θ}) = U_{var} (\vec{θ}) U_{ref} (\vec{θ})

Estimator: means of estimating the expectation value of operator H

{⟨ ψ (\vec{θ}) | H | ψ (\vec{θ}) ⟩}_{N} \approx \frac{1}{N} \sum_{j = 1}^{N} E_{j}

Classical optimiser: minimizes the cost function. Especially important at the presence of loss.

We’ve shifted from a process whose complexity depends only on matrix dimension to one that depends on required precision and the number of non-commuting Pauli operators that make up the matrix. The last bit has no analog in classical computing.