Algorithm Zoo · Optimization
VQE
Also known as: Variational Quantum Eigensolver
First described: Peruzzo, McClean, Shadbolt, Yung, Zhou, Love, Aspuru-Guzik, O'Brien, 2014
The problem
Estimate the ground-state energy of a Hamiltonian H using a parameterized ansatz |ψ(θ)⟩ and classical optimization.
Prepare |ψ(θ)⟩ on quantum hardware, measure ⟨ψ(θ)|H|ψ(θ)⟩ via sampling, classically minimize over θ. Hybrid algorithm — quantum hardware as an expectation-value oracle for a classical optimizer.
Best classical
DMRG (1D, weakly entangled 2D), CCSD(T) (small molecules), tensor networks, neural-network states.
Quantum complexity
Depends on ansatz; UCCSD scales as O(N^4) parameters with N orbitals.
Our verdict
Pedagogically essential, practically unproven. No VQE experiment has produced a chemistry result that wasn't reachable by CCSD(T) or DMRG with the same accuracy at the same molecule size. Barren plateaus and shot noise are foundational obstacles, not engineering ones — they don't go away with bigger hardware.
When to use it
- Research into ansatz design and quantum-classical optimization.
- Small chemistry benchmarks where you want to learn the workflow.
- Co-design experiments where the hardware-ansatz pairing is the topic of study.
When not to use it
- Production chemistry. CCSD(T) beats every VQE result for small molecules; DMRG beats every result for 1D strongly correlated systems.
- Anything past ~12 qubits — barren plateaus, noise, and shot budget kill it.
Classical baseline
CCSD(T) is gold-standard for small molecules and feasible up to ~50 orbitals on a workstation. DMRG handles 1D systems and weakly entangled 2D. Neural network quantum states (NNQS, Carleo & Troyer 2017+) have eaten several 'VQE-only' benchmarks.
Hardware cost
Hardware-efficient ansatz: O(L·N) gates for L layers, N qubits. UCCSD: O(N^5) gates — feasible only post-FTQC.
Key papers
- A variational eigenvalue solver on a photonic quantum processor ↗
Peruzzo et al. · 2014 · Nature Communications
- Hartree-Fock on a Superconducting Qubit Quantum Computer ↗
Google AI Quantum and Collaborators · 2020 · Science
Deep-dive tutorials
Last verified: 2026-05-24