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

TL;DR

This paper investigates the trainability and expressivity of Hamming-weight preserving quantum circuits in quantum machine learning, introducing new data loaders, analyzing the Quantum Fisher Information Matrix, and discussing conditions for avoiding barren plateaus.

Contribution

It introduces heuristic data loaders for quantum amplitude encoding and provides theoretical insights into the trainability and controllability of Hamming-weight preserving quantum circuits.

Findings

Feasibility of new quantum data loaders using controllability arguments

The rank of the Quantum Fisher Information Matrix is almost-everywhere constant

Variance bounds for gradients indicate conditions to avoid barren plateaus

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.