Description
Variational Quantum Eigensolvers (VQEs) have risen as a leading paradigm for solving the many-body problem, but their success has been hindered by emerging results on trainability and noise issues. By exploiting symmetries, these issues may be mitigated, and generic frameworks have been proposed to incorporate them by means of equivariant layers [1, 2]. These strategies have been formalized in the field of Geometric Quantum Machine Learning, which draws inspiration from its classical counterpart and studies how to embed inductive biases encoding the symmetries of problems [3].
We apply these tools to the paradigmatic task of finding the ground state of the 1D Transverse Field Ising Model. An equivariant VQE, designed to incorporate the symmetries, is used to solve the problem and is compared to a non-equivariant implementation. For the required depths to find the ground state, the equivariant implementation achieves lower energies in fewer iterations and demonstrating good trainability metrics such as the absence of barren plateaus. The limitations of this approach and the future avenues it opens are also discussed, in the form of a trade-off between trainability and classical simulability of quantum circuits [4]. Finally, the minimization routine is also fully implemented in an actual quantum device, albeit with very noisy results.
[1] J. J. Meyer, et al. PRX Quantum 4, 010328 (2023)
[2] Q. T. Nguyen, et al. PRX Quantum 5, 020328 (2024)
[3] M. Larocca, F. Sauvage, F. M. Sbahi, et al. PRX Quantum 3, 030341 (2022)
[4] M. Cerezo, M. Larocca, D. García-Martín et al. Nat Commun 16, 7907 (2025)