Variational Quantum Eigensolver
Variational Quantum Eigensolver (VQE) · Also known as: VQE, hybrid quantum-classical
The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm designed to find the lowest eigenvalue (ground state energy) of a quantum Hamiltonian. Introduced by Peruzzo et al. in 2014, it exploits the variational principle to combine the power of quantum circuits with classical optimization to solve chemistry and materials science problems on near-term quantum devices.
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When to use it
VQE is ideal for finding ground state energies of molecular systems, materials, and other quantum Hamiltonians on near-term quantum devices (NISQ era). It is preferred when the quantum circuit depth is limited and classical resources are sufficient for optimization. VQE assumes access to a quantum processor that can implement the ansatz and perform reliable measurements.
Strengths & limitations
- Requires only shallow quantum circuits, practical for current noisy quantum hardware.
- Suitable for near-term quantum devices (NISQ) with limited qubit coherence.
- Flexible ansatz design allows incorporation of domain knowledge.
- Hybrid approach leverages both quantum and classical computing strengths.
- Has demonstrated quantum advantage for certain chemistry problems.
- Convergence depends on ansatz expressivity and classical optimizer quality.
- Requires many measurements to estimate energies accurately, increasing circuit evaluations.
- Ansatz design is problem-specific and often requires expert knowledge.
- Barren plateaus in the cost landscape can hinder optimization for large systems.
- Noise and decoherence in quantum processors affect measurement accuracy.
Frequently asked
What is an ansatz and how do I choose one?
An ansatz is a parameterized quantum circuit whose form is chosen to represent states relevant to your problem. For chemistry, the Unitary Coupled Cluster (UCC) ansatz is popular because it approximates the wave function expansion used in classical chemistry. You choose an ansatz based on physical intuition and the number of qubits available.
Why does VQE work better than other algorithms on NISQ devices?
VQE uses shallow quantum circuits with fewer gates, reducing the impact of noise and decoherence. It offloads the heavy computational work to classical optimizers, which are robust and reliable, making the algorithm tolerant of quantum hardware imperfections.
How many measurements do I need to estimate the energy accurately?
The number of shots (measurement repetitions) scales with the desired precision. Empirically, thousands to tens of thousands of shots per parameter update are common. Noise mitigation techniques can reduce this overhead.
What is a barren plateau and how can I avoid it?
A barren plateau is a region of the parameter space where all gradients are exponentially small, making optimization difficult. Avoid it by using warm-start initialization, choosing expressive ansatzes carefully, or using gradient-free optimizers like COBYLA in early iterations.
Can VQE find excited states or just the ground state?
Standard VQE finds the ground state. To find excited states, you can use extensions like VQE with energy penalties, subspace expansion, or the Orthogonal VQE (OVQE) algorithm that enforces orthogonality constraints.
Sources
- Peruzzo, A., McClean, J., Shadbolt, P., et al. (2014). A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5, 4213. DOI: 10.1038/ncomms5213 ↗
- McClean, J. R., Romero, J., Aspuru-Guzik, A. (2016). The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18, 023023. DOI: 10.1088/1367-2630/18/2/023023 ↗
- Cao, Y., Romero, J., Aspuru-Guzik, A. (2021). Potential of quantum computing for drug discovery. IBM Journal of Research and Development, 62, 6:1-6:20. DOI: 10.1147/JRD.2018.2888987 ↗
How to cite this page
ScholarGate. (2026, June 3). Variational Quantum Eigensolver (VQE). ScholarGate. https://scholargate.app/en/quantum-computing/variational-quantum-eigensolver
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