VQE

Variational Quantum Eigensolver: a hybrid quantum-classical algorithm for finding ground state energies, especially useful in quantum chemistry.

VQE (Variational Quantum Eigensolver) is the flagship algorithm for near-term quantum computers. It’s a hybrid approach that uses a quantum computer to evaluate energies and a classical computer to optimize parameters.

The Goal

Find the ground state energy of a Hamiltonian $H$:

$$E_0=\underset{|\psi ⟩}{\min }⟨\psi |H|\psi ⟩$$

This is important for quantum chemistry (molecular ground states) and materials science.

How It Works

The Variational Principle

For any trial state $|\psi (\theta )⟩$: $$⟨\psi (\theta )|H|\psi (\theta )⟩\ge E_0$$

The energy of any guess is always ≥ true ground state energy. So minimize!

The Algorithm

┌─────────────────────────────────────────────────────┐
│                  Classical Computer                  │
│  ┌─────────────┐                    ┌────────────┐  │
│  │  Optimizer  │◄──── Energy ◄─────│   Measure  │  │
│  │   (θ→θ')    │                    │ expectation│  │
│  └──────┬──────┘                    └─────▲──────┘  │
│         │ New θ                           │         │
│         ▼                                 │         │
│  ┌──────────────────────────────────────────────┐  │
│  │              Quantum Computer                 │  │
│  │                                               │  │
│  │  |0⟩ ──[Ansatz(θ)]──── Measure               │  │
│  │                                               │  │
│  └──────────────────────────────────────────────┘  │
└─────────────────────────────────────────────────────┘

1. Prepare: Create parameterized trial state $|\psi (\theta )⟩$ on quantum computer
2. Measure: Estimate $⟨H⟩$ via measurements
3. Optimize: Classical optimizer updates $\theta$ to minimize energy
4. Repeat: Until convergence

The Ansatz

The ansatz is the parameterized circuit structure:

|0⟩ ──Ry(θ₁)──●──Ry(θ₃)──...
              │
|0⟩ ──Ry(θ₂)──⊕──Ry(θ₄)──...

Common ansatzes:

• UCCSD: Unitary coupled cluster (chemistry-inspired)
• Hardware-efficient: Native gates, easy to run
• Problem-specific: Tailored to symmetries

Measuring the Hamiltonian

Hamiltonians are decomposed into Pauli strings:

$$H=\sum _i{c}_i{P}_i$$

where $P_i$ are Pauli operators (e.g., $X\otimes Z\otimes I$).

Each term measured separately, results combined classically.

Challenges

Applications

• Quantum chemistry: Molecular energies, reaction pathways
• Materials science: Band structures, superconductivity
• Optimization: Via encoding problems as Hamiltonians

Relation to Other Algorithms

• QPE: More accurate but needs fault tolerance
• QAOA: Similar variational structure, for optimization
• VQA: VQE is a specific type of variational quantum algorithm

See also: Variational Quantum Algorithm, QAOA, Hamiltonian, NISQ