On Relationship Between Circuit Depth and Trainability of VQAs
Abstract
The training efficiency of Variational Quantum Algorithms (VQAs) is dictated by the geometry of their loss landscapes, which can be formally analysed by mapping these functions to random fields on manifolds. Instead of VQAs, we can then directly study the corresponding random fields, in our case, the Wishart Hypertoroidal Random Fields (WHRFs). We are mostly interested in the distribution of critical points (especially local minima) of WHRFs. For this purpose, the Kac-Rice formula is presented, reformulated, and simulated. The findings identify a phase transition in the distribution of local minima. Beyond a specific threshold, local minima concentrate near the global minimum in function value, meaning that even local minima are good approximators of the global one. The threshold is governed by the ratio between the problem Hamiltonian degrees of freedom and by the number of independent parameters in the VQA. Since the degrees of freedom parameter scales exponentially, we propose symmetry reduction operations to lower the degrees of freedom. This mathematically reduces the dimension, scaling down the required parameter threshold and enabling to solve bigger problems.
Related
- Trainability of IQP Quantum Circuit Born Machines Under Gaussian Initialization
- Krylov-Lie Algebras for Variational Quantum Algorithms: Geometric, Depth-Aware Insights into Expressivity and Trainability
- Data and code for collision-model Dicke-state preparation: depth-fidelity frontiers, circuit costs, and superconducting-processor measurements
- One-dimensional quantum Gibbs states in constant circuit depth
- Mid-circuit measurements and feedforward implement every unitary exactly in constant depth
BibTeX
@misc{michalek2026relationship,
title = {{On Relationship Between Circuit Depth and Trainability of VQAs}},
author = {Jan Michálek and Martin Friák and Petr Vašík},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27488},
eprint = {2609.27488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27488},
}
RIS
TY - UNPB
TI - On Relationship Between Circuit Depth and Trainability of VQAs
AU - Jan Michálek
AU - Martin Friák
AU - Petr Vašík
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27488
UR - https://arxiv.org/abs/2609.27488
AB - The training efficiency of Variational Quantum Algorithms (VQAs) is dictated by the geometry of their loss landscapes, which can be formally analysed by mapping these functions to random fields on manifolds. Instead of VQAs, we can then directly study the corresponding random fields, in our case, the Wishart Hypertoroidal Random Fields (WHRFs). We are mostly interested in the distribution of critical points (especially local minima) of WHRFs. For this purpose, the Kac-Rice formula is presented, reformulated, and simulated. The findings identify a phase transition in the distribution of local minima. Beyond a specific threshold, local minima concentrate near the global minimum in function value, meaning that even local minima are good approximators of the global one. The threshold is governed by the ratio between the problem Hamiltonian degrees of freedom and by the number of independent parameters in the VQA. Since the degrees of freedom parameter scales exponentially, we propose symmetry reduction operations to lower the degrees of freedom. This mathematically reduces the dimension, scaling down the required parameter threshold and enabling to solve bigger problems.
ER -