Restricting Trainable Lie-Algebra Growth in Equivariant Quantum Networks via Hierarchical Ancilla-Controlled Subspace Projections
Abstract
Equivariant quantum networks encode symmetry as an inductive bias, which can improve generalization and may also favor optimization convergence. Equivariance alone, however, does not constrain the noncommuting closure of trainable generators, and this closure can still grow rapidly in symmetry-preserving variational circuits. We introduce a hierarchical ancilla-controlled architecture that addresses this Lie-algebra-growth mechanism. Commuting invariant-sector projectors on the data register select parameterized operations on a shared ancilla register, where the noncommuting trainable dynamics is confined. The trainable circuit decomposes into compatible joint sectors, giving a sector-probability-weighted ancilla response and an explicit view of parameter sharing across hierarchical paths. For an ancilla dimension dA=2m and Kℓ retained layer-wise control modes, we prove the group-independent bound dim(g)≤ (dA2−1)∏ℓ=1L(Kℓ+1). The bound is polynomial in the number of data qubits when m and Kℓ remain constant along a logarithmic-depth hierarchy. Particle-number and parity projectors illustrate the general construction, while a fixed Clebsch--Gordan coupling tree supplies a concrete SU(2) realization with rotation-invariant scalar outputs. Finite-size state-vector simulations exhibit slower gradient-variance decay and larger initialization gradients than generic and conventional rotationally equivariant circuits over the studied system sizes. The same realization fits sparse rotation-invariant classification tasks and geometry-dependent Heisenberg ground-state energies. These results support restricted trainable Lie-algebra growth as a structural strategy for initialization trainability in the regimes considered here.
Related
- Trainable, self-verifying variational quantum eigensolver using adiabatic warm starts
- Quantum Coherence in Guest-Controlled Endohedral Fullerenes: Hierarchical Equations of Motion and a Neural-Network Model.
- Quantum Fokker-Planck solvers as trainable generative models
- Measuring trainable degrees of freedom in materials graph neural networks: a random-subspace intrinsic dimension analysis
- Fast Trainable Multilinear Bases for Image Compression
BibTeX
@misc{li2026restricting,
title = {{Restricting Trainable Lie-Algebra Growth in Equivariant Quantum Networks via Hierarchical Ancilla-Controlled Subspace Projections}},
author = {Ting Li and Zhiming Xiao and Qibiao Tang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30283},
eprint = {2609.30283},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30283},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Restricting Trainable Lie-Algebra Growth in Equivariant Quantum Networks via Hierarchical Ancilla-Controlled Subspace Projections
AU - Ting Li
AU - Zhiming Xiao
AU - Qibiao Tang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.30283
UR - https://arxiv.org/abs/2609.30283
AB - Equivariant quantum networks encode symmetry as an inductive bias, which can improve generalization and may also favor optimization convergence. Equivariance alone, however, does not constrain the noncommuting closure of trainable generators, and this closure can still grow rapidly in symmetry-preserving variational circuits. We introduce a hierarchical ancilla-controlled architecture that addresses this Lie-algebra-growth mechanism. Commuting invariant-sector projectors on the data register select parameterized operations on a shared ancilla register, where the noncommuting trainable dynamics is confined. The trainable circuit decomposes into compatible joint sectors, giving a sector-probability-weighted ancilla response and an explicit view of parameter sharing across hierarchical paths. For an ancilla dimension d_A=2^m and K_ℓ retained layer-wise control modes, we prove the group-independent bound dim(g)≤ (d_A^2−1)∏_ℓ=1^L(K_ℓ+1). The bound is polynomial in the number of data qubits when m and K_ℓ remain constant along a logarithmic-depth hierarchy. Particle-number and parity projectors illustrate the general construction, while a fixed Clebsch--Gordan coupling tree supplies a concrete SU(2) realization with rotation-invariant scalar outputs. Finite-size state-vector simulations exhibit slower gradient-variance decay and larger initialization gradients than generic and conventional rotationally equivariant circuits over the studied system sizes. The same realization fits sparse rotation-invariant classification tasks and geometry-dependent Heisenberg ground-state energies. These results support restricted trainable Lie-algebra growth as a structural strategy for initialization trainability in the regimes considered here.
ER -