Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning

Thu, 15 May 2025 00:00:00 +0000

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 ( 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 ( 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.

Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning

Mon, 02 Oct 2023 00:00:00 +0000

Quantum machine learning 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 quantum circuits. These circuits use gates that preserve subspaces of the Hilbert space, spanned by basis states with fixed Hamming weight $k$. They are good candidates for mimicking neural networks, by both loading classical data and performing trainable layers. 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 a n-qubit quantum circuit. Then, we analyze more generally the trainability of Hamming weight preserving circuits, and show that the variance of their gradients is bounded according to the size of the preserved subspace. This proves the conditions of existence 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.

Romain Kukla | LIP6 - Équipe QI

Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning

Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning