Algorithms · Variational quantum algorithm
Variational quantum eigensolver for ground-state energy
A hybrid chemistry workflow that compares a quantum expectation loop with classical eigensolvers.
Strong empiricalUpdated
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3 steps across 3 wires. Hover a step, or press play to go through them in order.
Expected outcomes
- Energy92%
- Steps3
- Wires3
- QubitsProblem mapped
- LoopQuantum + classical
- OutputEnergy estimate
- Papers2
VQE combines a parameterized state, repeated measurements of a Hamiltonian, and a classical optimizer. It remains an important NISQ-era chemistry pattern because it trades deep coherent evolution for repeated shallow experiments.
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
VQE estimates a ground-state energy using the variational principle: for any normalized trial state and Hamiltonian , the Rayleigh quotient obeys , the true ground energy, with equality only when is an exact ground eigenstate. The circuit prepares with a parameterized ansatz , measures each Pauli term of to build , and hands that scalar to a classical optimizer that updates to push the energy down — a hybrid quantum/classical loop repeated to convergence. This record checks that expectation-value loop for a small Pauli Hamiltonian against exact diagonalization, matching its own caveat: the variational bound is guaranteed by the math above, but reaching a chemically useful energy still depends on ansatz expressivity, optimizer convergence, and shot noise this toy fixture does not certify.
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
- Qubits
- Problem mapped
- Loop
- Quantum + classical
- Output
- Energy estimate
- Hamiltonian
- Pauli sum
- Optimizer
- Classical
- Reference
- Exact diagonalization
- Main risk
- Convergence + shots
U(θ)on ansatz state⟨H⟩on ansatz state, Pauli measurementsθ ← opton Pauli measurements
Quantum vs classical
Classical baseline
Exact diagonalization for small Hamiltonians; coupled-cluster, DMRG, tensor networks, or other electronic-structure methods at scale.
Quantum claim
VQE can estimate a variational energy using shallow circuits plus a classical optimizer, not an automatic advantage over mature chemistry software.
How to compare
Compare energy error, total shots, circuit depth, optimizer calls, basis/mapping choices, and classical wall-clock time.
Nobody has reviewed this record for gaps yet.
import pennylane as qml
from pennylane import numpy as np
dev = qml.device("default.qubit", wires=2)
H = qml.Hamiltonian([0.5, 0.5], [qml.PauliZ(0), qml.PauliZ(1)])
@qml.qnode(dev)
def energy(theta):
qml.RY(theta[0], wires=0)
qml.CNOT(wires=[0, 1])
return qml.expval(H)
FINAL_CIRCUIT = energyThe fixture demonstrates the hybrid loop shape; it is not a molecular chemistry benchmark.
Where this sits
This record is named by the layer graph at:
- Variational quantum eigensolver Method
Takes A Hermitian reachable as a sum of terms, as sparse-access oracles or as a block-encoding; a way to prepare trial states, and — for the methods that need it — a trial state whose overlap with the ground state is not negligible; a target additive error and a confidence . Returns A scalar estimate of the lowest eigenvalue with a stated additive-error guarantee, plus the run or query budget it consumed. Whether that estimate is also a rigorous upper bound is a property of the method and is not promised by the slot.
Related entries
- Quantum phase estimation
Algorithms
- GHZ state preparation
Basic circuits
- Quantum amplitude estimation
Algorithms
References
- A variational eigenvalue solver on a quantum processor
Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, Jeremy L. O'Brien · 2013
Original VQE proposal and photonic demonstration for a small chemistry problem.
- Optimization of the Variational Quantum Eigensolver for Quantum Chemistry Applications
R. J. P. T. de Keijzer, V. E. Colussi, B. Škorić, S. J. J. M. F. Kokkelmans · 2021
Surveys optimizer, ansatz, and measurement choices that affect chemistry-oriented VQE runs.