The prospect of an exponential quantum speed-up in chemistry ultimately depends on fault-tolerant quantum computers, machines that are still beyond current engineering capabilities. While algorithms designed for fully error-corrected hardware remain largely theoretical, a new class of hybrid or heuristic approaches has emerged.
By off-loading part of the workload to classical processors and reducing quantum depth, these methods aim to run on today’s noisy, intermediate-scale quantum (NISQ) devices, whose qubit counts are low and coherence times limited. In quantum chemistry, the Variational Quantum Eigensolver (VQE) is the most popular NISQ-era algorithm for estimating a molecule’s ground-state energy from its Hamiltonian. Like other variational methods, standard VQE can suffer from barren plateaus—regions where cost-function gradients vanish, making optimisation in large qubit systems intractable.
The ADAPT-VQE variant tackles this issue by building the ansatz iteratively: at each step it adds the operator whose gradient has the largest magnitude. This quasi-greedy, gradient-driven construction has been shown to bypass barren plateaus issues and yields more reliable convergence. However, ADAPT-VQE requires an impractical number of circuit evaluations, both to select the next operator and to perform the global optimisation at each step. As a result, it has not been implemented on hardware and suffers severe accuracy loss under realistic shot noise in simulations.
In this study, we give ADAPT-VQE a practical makeover by using a simple trick.
In this study, we give ADAPT-VQE a practical makeover by using a simple trick.
Upon the addition of a new operator to the circuit, the energy expectation value is a simple trigonometric function of the rotation angle of the operator. Because this function can be fully determined by extrapolation from just a few measurements, we can easily locate its minimum. Consequently, we propose selecting both the next operator and its optimal angle in one step, a process we call Greedy Gradient-free Adaptive VQE (GGA-VQE):
- For each candidate operator, take a few of shots to fit the theoretical energy curve.
- Find the angle where that curve is minimal.
- Pick the operator whose lowest energy is the very lowest of them all.
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Fix that angle in the circuit and move on, never changing already-chosen parameters.
By building the ansatz one local update at a time, GGA-VQE sidesteps the costly, noise-sensitive, and shot-intensive optimisation loops of standard ADAPT-VQE. The result is a somewhat less flexible circuit, but one that is far more compatible with today’s NISQ hardware. This procedure has been tested on a 25-qubit trapped-ion quantum computer, tackling the ground state of a trasverse field ising model.
This experiment addresses a key question: “How close are today’s quantum devices to useful chemistry computations?” While the 25-qubit processor already spans a Hilbert space that challenges naïve classical simulation, noise and shot limitations currently confine us to proof-of-principle benchmarks, such as a 25-spin Ising model. Nevertheless, this research brings key progress towards the full implementation of variational methods for quantum chemistry and proposes a first - real life - converged computation on an actual NISQ quantum computer. These demonstrations remain also essential for benchmarking actual hardware and guiding algorithm development toward ever more efficient use of quantum resources.
For a deeper dive into the algorithm’s implementation and our experimental results, refer to the original article.
Greedy gradient-free adaptive variational quantum algorithms on a noisy intermediate scale quantum computer. C. Feniou, M. Hassan, B. Claudon, A. Courtat, O. Adjoua, Y. Maday, J.-P. Piquemal, Scientific Reports volume 15, Article number: 18689 (2025)
https://www.nature.com/articles/s41598-025-99962-1