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Variational quantum eigensolver (VQE)

Algorithms and benchmarks / Chemistry and materials

A hybrid method that estimates ground-state energies of molecules and magnets with short circuits whose parameters a classical optimiser tunes. It was designed for noisy devices without error correction.

Era
Noisy intermediate-scale (NISQ)
Problem class
Chemistry and materials
Speedup
None proven; the hope is to reach useful accuracy sooner than fault-tolerant methods.

How it works

A parameterised circuit prepares a trial state; the quantum computer measures the energy term by term and a classical optimiser updates the parameters until the energy stops decreasing.

Typical use cases

  • Small-molecule energies for chemistry and drug discovery.
  • Ground states of quantum magnets.

Documented use cases on related problems

From the use case directory; the sources of these cases may use other methods than this one.

Resource estimates

No resource estimate with stated assumptions was verified for this family in this release.

Demonstrations

  • 2014: Ground-state energy of He-H+ on a small photonic quantum processor combined with a conventional computer. (A. Peruzzo et al. 2014)
  • 2017: Hardware-efficient VQE for up to six-qubit Hamiltonians, the largest molecule BeH2. (A. Kandala et al. 2017)

Hardware platforms that fit

  • Photonic qubits: The first VQE ran on a small photonic processor with a conventional computer, for the helium hydride ion (Peruzzo et al. 2014).
  • Superconducting qubits: VQE computed the energy surface of molecular hydrogen on a programmable array of superconducting qubits (O’Malley et al. 2016).

What you need to run it

Classical simulation
A state-vector simulator holds 2^n complex amplitudes, so memory doubles with every qubit: about 30 qubits fit in 16 GiB at double precision. Circuits with little entanglement can go further with tensor-network methods. See the guide for the calculator.
Noisy hardware (NISQ)
Tens of qubits with shallow circuits, many shots per energy estimate, and a classical optimiser close to the quantum computer (an HPC-coupled system helps).
Fault-tolerant machine
Not applicable for this family.
Annealer or analogue simulator
Not applicable for this family.

Calculate the memory of a simulation

Where to run it in Europe

Photonic qubits

Superconducting qubits

The full infrastructure catalogue · The infrastructure deployment tracker

Software

European tools in the software directory

Other open-source frameworks

  • PennyLane (Xanadu (Canada); non-European)
  • Qiskit (IBM (United States); non-European)
  • OpenFermion (Google Quantum AI (United States); non-European)

A listing is not an endorsement; check each tool for the features you need.

Limitations and open questions

  • Barren plateaus: for many parameterised circuits the gradient vanishes exponentially with the number of qubits, which makes training impractical beyond a few qubits (McClean et al. 2018).
  • Many measurements are needed for each energy estimate, and noise biases the result.

In the Quantum Algorithm Zoo

No matching Zoo entry was found on 23 October 2026; the Zoo lists algorithms with a proven or conjectured speedup, not this family.

Index: the Quantum Algorithm Zoo by Stephen P. Jordan, last updated 2026-09-10, checked 2026-10-23. The portal links to it and copies no description from it. Browse the full Quantum Algorithm Zoo

Named in European procurements

As the tender documents state; how a benchmark is weighted against others is described in each procurement.

Sources

  1. A. Peruzzo et al. (2014). A variational eigenvalue solver on a quantum processor. Nature Communications 5, 4213 (2014); arXiv:1304.3061. Read: abstract page, retrieved 2026-10-23.
  2. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, H. Neven (2018). Barren plateaus in quantum neural network training landscapes. Nature Communications 9, 4812 (2018); arXiv:1803.11173. Read: abstract page, retrieved 2026-10-23.
  3. P. J. J. O'Malley et al. (2016). Scalable Quantum Simulation of Molecular Energies. Phys. Rev. X 6, 031007 (2016); arXiv:1512.06860. Read: abstract page, retrieved 2026-10-23.
  4. A. Kandala et al. (2017). Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets. Nature 549, 242 (2017); arXiv:1704.05018. Read: abstract page, retrieved 2026-10-23.

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