Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning
1Laboratoire d’Informatique de Paris 6, CNRS, Sorbonne Université, 4 Place Jussieu, 75005 Paris, France
2CEMIS, Direction Technique, Naval Group, 83190 Ollioules, France
3School of Informatics, University of Edinburgh, 10 Crichton Street, Edinburgh, United Kingdom
4QC Ware, Palo Alto, USA and Paris, France
| Published: | 2025-05-15, volume 9, page 1745 |
| Editor: | Tongyang Li |
| Eprint: | arXiv:2309.15547v4 |
| Doi: | https://doi.org/10.22331/q-2025-05-15-1745 |
| Citation: | Quantum 9, 1745 (2025). |
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Abstract
Quantum machine learning (QML) has become a promising area for real world applications of quantum computers, but near-term methods and their scalability are still important research topics. In this context, we analyze the trainability and controllability of specific Hamming weight preserving variational quantum circuits (VQCs). These circuits use qubit gates that preserve subspaces of the Hilbert space, spanned by basis states with fixed Hamming weight $k$. In this work, we first design and prove the feasibility of new heuristic data loaders, performing quantum amplitude encoding of $\binom{n}{k}$-dimensional vectors by training an $n$-qubit quantum circuit. These data loaders are obtained using controllability arguments, by checking the Quantum Fisher Information Matrix (QFIM)'s rank. Second, we provide a theoretical justification for the fact that the rank of the QFIM of any VQC state is almost-everywhere constant, which is of separate interest. Lastly, we analyze the trainability of Hamming weight preserving circuits, and show that the variance of the $l_2$ cost function gradient is bounded according to the dimension $\binom{n}{k}$ of the subspace. This proves conditions of existence/lack of Barren Plateaus for these circuits, and highlights a setting where a recent conjecture on the link between controllability and trainability of variational quantum circuits does not apply.
Featured image: Representation of the unitary and output state spaces. The Dynamical Lie Algebra is the tangent space of the unitary space. The possible directions for the evolution of the output state are given by the Quantum Fisher Information Matrix eigenvectors. In this work, we study the controllability of both the unitary dynamics and the output state in Hamming-weight preserving quantum circuits.
Popular summary
In this work, we show theoretical guarantees on their expressivity and trainability. First, we show how to design and prove the feasibility of new heuristic data loaders that are keen for Machine Learning algorithms. To do so, we study the controllability of the equivalent unitary representation and of the resulting quantum state. Then, we analyse how the behavior of the training of a Hamming-Weight preserving quantum circuits, and we show that a proposed connection between the controllability and trainability of quantum circuits breaks down, offering a counterexample.
Our results offer new insights into how structured quantum circuits can be effectively used for machine learning, particularly in regimes accessible to current and near-term quantum devices.
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