Variational Quantum Eigensolver (VQE)

Overview

The variational quantum eigensolver, or VQE, is a hybrid algorithm for finding the lowest-energy state of a quantum system, which is the central problem of quantum chemistry. A quantum computer prepares a trial state controlled by a small set of adjustable angles and measures its energy, and a classical optimizer adjusts the angles to lower that energy, repeating until it converges.

VQE was designed for the noisy devices of today. It uses short circuits that fit within the limited coherence time of current hardware, and it replaces the deep circuits of quantum phase estimation with many repeated shallow runs.

The problem it solves

Finding the ground-state energy of a quantum system (like a molecule), which is exponentially hard to compute classically as system size grows.

History and background

VQE was introduced in 2014 by Peruzzo and colleagues in Nature Communications, who ran it on a two-qubit photonic processor to compute the ground-state energy of the helium hydride ion. McClean and collaborators generalized the approach in 2016, and the same year researchers at Google, including O'Malley, ran it on a superconducting chip to compute the energy of molecular hydrogen across bond lengths.

In 2017, IBM's Kandala and colleagues used a hardware-efficient ansatz to compute energies of hydrogen, lithium hydride, and beryllium hydride on up to six qubits. Later work added adaptive ansatz construction such as ADAPT-VQE in 2019, and in 2018 McClean and colleagues identified barren plateaus, regions where gradients vanish exponentially, which remain a central obstacle. In 2020, Google ran Hartree-Fock-based calculations on 12 qubits.

How it works

A quantum computer prepares trial states and measures their energy, while a classical optimizer iteratively adjusts parameters to minimize that energy — splitting the work between quantum and classical hardware.

VQE rests on the variational principle from quantum mechanics: the energy of any trial state is at least the true ground-state energy. So if you search over a family of states and minimize the energy, the best energy found is an upper bound on the true value, and a better ansatz gets closer.

The quantum computer's job is to prepare states from a parameterized family and estimate the energy, which is a sum of many simple terms called Pauli strings. Each term is measured separately by rotating the basis and counting outcomes. The classical computer sums the terms, updates the parameters, and repeats. The quantum computer is used for what classical computers struggle with, representing entangled states, while the classical optimizer handles the rest.

The design of the ansatz involves a trade-off. Chemically motivated ansatzes, such as unitary coupled cluster, are accurate but deep. Hardware-efficient ansatzes use gates native to the device and are shallow but may not represent the right state and can suffer from barren plateaus.

Step by step

1. Write the molecular Hamiltonian in the second-quantized form and map fermions to qubits with the Jordan-Wigner or Bravyi-Kitaev transformation, producing a sum of Pauli strings.
2. Choose an ansatz circuit with adjustable parameters, such as unitary coupled cluster or a hardware-efficient layered circuit, and pick initial parameters, often from Hartree-Fock theory.
3. Run the circuit on the quantum computer and measure each group of commuting Pauli terms many times to estimate their expectation values.
4. Combine the expectation values with their coefficients to obtain the energy estimate.
5. Pass the energy to a classical optimizer such as COBYLA, SPSA, or gradient descent, which proposes new parameters.
6. Repeat until the energy converges, and report the final energy and parameters.

A worked example

For molecular hydrogen in a minimal basis, the problem can be reduced to two qubits with a Hamiltonian containing about five Pauli terms. A simple ansatz applies a single rotation with one adjustable angle. Scanning that angle, the energy forms a curve with a minimum near minus 1.137 hartree at the equilibrium bond length of about 0.74 angstroms, which matches the exact answer.

Larger molecules require many more terms. A molecule with 12 qubits can have thousands of Pauli terms, and each must be measured with enough shots to keep statistical error below chemical accuracy, which is about 1.6 millihartree, or one kilocalorie per mole. This measurement overhead is one of the main practical difficulties.

Complexity and resources

What hardware it needs

• A small to moderate number of qubits with gate errors low enough to keep the circuit useful, plus error mitigation.
• Fast repeated execution, since thousands to millions of circuit runs are needed per energy evaluation.
• A classical computer for optimization and for Hamiltonian preparation.

Applications

• Computing ground-state energies of small molecules and model systems such as the Hubbard model.
• Exploring catalysts, battery materials, and drug fragments as a research tool.
• Testing hybrid quantum-classical workflows and error mitigation techniques.
• Serving as a benchmark and teaching example for variational algorithms.

Limitations and caveats

• No quantum advantage over classical chemistry methods has been shown for VQE so far, and the molecules treated are small and classically solvable.
• Barren plateaus can make training exponentially hard for deep or random ansatzes.
• Noise limits accuracy, and measurement counts grow quickly with system size.
• The optimizer can get stuck in local minima, and results depend on the starting point.

Common misconceptions

• VQE does not guarantee the exact ground state, only an upper bound on its energy.
• It does not make quantum chemistry easy today. Classical methods like coupled cluster and DMRG remain better for the molecules that devices can handle.
• A small error in measured energy does not mean the state is correct, since energy errors can hide state errors.

How it compares

• Quantum phase estimation offers accuracy guarantees but needs fault-tolerant hardware, while VQE runs on noisy devices without guarantees.
• QAOA uses a similar hybrid loop for optimization problems instead of chemistry.
• Quantum imaginary time evolution is an alternative that avoids a classical optimizer at the cost of deeper circuits.

Real-world impact

One of the most promising NISQ-era applications, actively used in early quantum chemistry and materials science research, including drug discovery exploration.

Why it matters

VQE launched the era of hybrid quantum-classical computing and showed that useful-looking experiments are possible without error correction. It remains the standard workflow in quantum chemistry software such as Qiskit Nature, PennyLane, and OpenFermion.

It is also a lesson in limits. Careful studies of VQE have clarified how noise, optimization, and measurement cost can eliminate the potential advantage, which has pushed the field toward fault-tolerant algorithms for chemistry.

Frequently asked questions

Keep learning

• Read the full deep-dive in our Learning Center
• Related algorithm: Quantum Phase Estimation (QPE)
• Related algorithm: Quantum Approximate Optimization Algorithm (QAOA)
• Related algorithm: Quantum Natural Gradient Descent
• Related algorithm: Quantum Imaginary Time Evolution (QITE)

This page is written for educational purposes. Complexity statements describe the best known results at the time of review (September 30, 2026), and research in this area moves quickly, so check recent papers before relying on a specific bound.

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