Can today's quantum hardware calculate the energy of a real molecule? VQE is the algorithm we use to find out. It runs on every platform we test — IBM, Tuna-9, and emulator — connecting the Hamiltonian (the problem) to the ansatz (the circuit). This page explains how the algorithm works, then shows what happens when we run it on real hardware.
VQE repeats four steps until the energy converges. Drag the slider to adjust θ and watch the energy update in real time. The loop highlights which step is active.
Parameterized circuit (ansatz) creates a trial quantum state |ψ(θ)⟩
Run circuit on hardware, measure in Z, X, and Y bases
Classical computer calculates ⟨ψ|H|ψ⟩ from measurements
Classical optimizer (COBYLA, SPSA) adjusts parameters θ
Before VQE can run, the molecule must be translated into a quantum circuit. This pipeline turns atomic coordinates into something a quantum computer can execute.
How molecular Hamiltonians are built, compressed via Jordan-Wigner, and how Pauli terms change as you stretch the bond.
Circuit architectures from 4 papers, mapped to 3 quantum processors. Hardware-efficient vs chemically-inspired tradeoffs.
The energy E(θ) for H₂ at equilibrium (R = 0.735 Å). Click or drag on the chart to explore. The green band shows chemical accuracy (±1 kcal/mol around FCI). VQE's job is to find the minimum.
We ran VQE at 14 bond distances from 0.3 to 3.0 Å (65,536 shots each) and compared against exact (FCI) and classical (Hartree-Fock) references. The green dots are real emulator measurements — not theory. Hover to see values.
Theory is clean; hardware is noisy. These are real measurements from our H₂ VQE experiment at equilibrium, ranked from best to worst. The dashed line marks chemical accuracy (1 kcal/mol) — the threshold where quantum chemistry becomes useful for predicting reactions.
0.22 kcal/mol
TREX readout mitigation on IBM Torino. Within chemical accuracy.
80% readout noise
Most error is readout, not gates. Mitigation recovers nearly all lost accuracy.
0.28 kcal/mol avg
Shot noise floor with 65k shots. No hardware errors, just statistical sampling.
VQE isn't just theory — we replicated three landmark VQE papers on real hardware and compared our results to the originals. Each replication tests specific claims from the paper and reports pass/fail against our measurements.
First-ever variational quantum eigensolver on a photonic processor. We replicated HeH⁺ bond sweep.
Error mitigation via parity symmetry post-selection on a superconducting transmon.
Full H₂ dissociation curve with hardware-efficient ansatz on IBM. The paper that defined modern VQE circuits.
See the full glossary for more definitions.
The atomic unit of energy. 1 Ha = 627.509 kcal/mol. Molecular energies are typically between -0.5 and -150 Ha. H₂ ground state: -1.1373 Ha.
1 kcal/mol (1.6 mHa) — the precision needed to predict chemical reaction outcomes. Our best hardware VQE achieves 0.22 kcal/mol via TREX mitigation.
The parameterized quantum circuit that prepares trial states. Choice of ansatz determines expressiveness, depth, and hardware compatibility. See the Ansatz Explorer.
The operator H = ∑ gᵢ Pᵢ encoding a molecule’s energy as a sum of Pauli strings. VQE measures ⟨ψ|H|ψ⟩ to estimate the energy.
⟨ψ|H|ψ⟩ — the average energy from many circuit runs. Each Pauli term needs separate measurements. More shots = lower statistical noise.
The algorithm adjusting θ each iteration. COBYLA (gradient-free) and SPSA (stochastic gradient) are common. The optimizer never touches the quantum computer directly.
The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the ground-state energy of molecules. The quantum processor prepares a trial state using a parameterized circuit (ansatz), measures the energy, and a classical optimizer adjusts the parameters to minimize it. This loop repeats until convergence.
VQE is designed for near-term noisy hardware: the circuits are short enough to run before decoherence destroys the signal, and the classical optimizer handles the noise by treating energy measurements as a stochastic objective function. It was first demonstrated by Peruzzo et al. in 2014 on a photonic processor.
This page shows real VQE results from IBM Quantum and Quantum Inspire Tuna-9 for hydrogen (H₂) and lithium hydride (LiH), compared against exact classical solutions. The gap between hardware results and the exact answer reveals how noise, ansatz choice, and error mitigation affect accuracy.