The variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are two foundational hybrid quantum–classical methods developed before 2022 to tackle chemistry and optimization problems on NISQ hardware. These approaches use parameterized quantum circuits whose parameters are updated by a classical optimizer, enabling useful computation despite hardware noise [1].

Variational Quantum Eigensolver

VQE employs a parameterized ansatz circuit to prepare trial states |ψ(θ)⟩ and measures the expectation value ⟨ψ(θ)|H|ψ(θ)⟩, which a classical optimizer then minimizes by adjusting θ [2, 1]. The algorithm was first demonstrated on a photonic processor to compute the ground state energy of the HeH⁺ molecule with chemical accuracy [2]. Hardware-efficient ansätze, such as those tailoring circuit structure to native gate sets, reduce depth and are well suited to superconducting qubits and trapped ions [3]. VQE has since been applied to molecules like H₂, LiH, and BeH₂, and extended to quantum magnets, demonstrating ground-state energy estimation on up to six qubits [3]. Key theoretical foundations and optimization strategies were formalized in New Journal of Physics (2016) by McClean et al. [4].

Quantum Approximate Optimization Algorithm

QAOA alternates between applying a problem Hamiltonian H_P and a mixing Hamiltonian H_M, each for times parameterized by angles (γ, β), repeated p times to approximate solutions of combinatorial problems [5]. Farhi et al. showed that for p=1 on 3‑regular graphs, QAOA guarantees a cut of size ≥0.6924 times optimal on MaxCut instances [5]. Subsequent work derived analytic performance formulas in the infinite-size limit for spin-glass models [6] and demonstrated provable improvements for bounded‑occurrence constraint problems [7]. QAOA circuits remain shallow for small p, facilitating implementation on existing superconducting processors and photonic devices [5].

Hybrid Approach and Applications

Both VQE and QAOA follow a variational workflow: the quantum device evaluates an objective function and the classical routine updates parameters to optimize it. This hybrid paradigm underpins broad NISQ-era applications, from quantum chemistry with VQE to portfolio optimization and scheduling with QAOA [1].

Outlook

By early 2022, VQE and QAOA formed the backbone of hybrid quantum algorithms on NISQ hardware, guiding research toward improved ansätze, efficient parameter optimization, and integration of error mitigation techniques to push performance beyond current noise limits [1].

References

[1] Wikipedia contributors. (2022). Variational quantum eigensolver. Wikipedia. https://en.wikipedia.org/wiki/Variational_quantum_eigensolver

[2] Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P. J., Aspuru-Guzik, A., & O’Brien, J. L. (2014). A variational eigenvalue solver on a quantum processor. Nature Communications, 5, 4213. https://arxiv.org/abs/1304.3061

[3] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J. M., & Gambetta, J. M. (2017). Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549(7671), 242–246. https://arxiv.org/abs/1704.05018

[4] McClean, J. R., Romero, J., Babbush, R., & Aspuru-Guzik, A. (2016). The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18(2), 023023.

[5] Farhi, E., Goldstone, J., & Gutmann, S. (2014). A Quantum Approximate Optimization Algorithm. arXiv:1411.4028. https://arxiv.org/abs/1411.4028

[6] Farhi, E., Goldstone, J., Gutmann, S., & Zhou, L. (2019). The Quantum Approximate Optimization Algorithm and the Sherrington–Kirkpatrick Model at Infinite Size. Quantum, 6, 759. https://quantum-journal.org/papers/q-2022-07-07-759/

[7] Farhi, E., Goldstone, J., & Gutmann, S. (2014). A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem. arXiv:1412.6062. https://arxiv.org/abs/1412.6062