Fourier-Geometric Circuit Design for Gate and Entanglement Placement in Quantum Neural Networks
Abstract
The output of a parameterized quantum circuit (PQC) can be expressed as a finite Fourier series whose accessible frequencies are fixed by the data-encoding gates. While the encoder determines which frequencies can appear, the corresponding Fourier coefficients depend on how the trainable and entangling gates are arranged. Although existing studies provide metrics for characterizing how gate structure affects Fourier coefficients, they do not translate these analyses into an explicit design criterion specifying how gates should be arranged to make a target coefficient reachable. In this paper, we provide such a criterion. Using the adjoint action of the encoding generator, we decompose operator space into two-dimensional invariant planes indexed by frequency and show that each Fourier coefficient is exactly a sum of bilinear projections of the effective state and observable onto the planes at that frequency. Because, for Pauli encodings, high-frequency planes are spanned by mixed multi-qubit Pauli strings, a target coefficient can contribute to the output only when local rotations and entangling layers are arranged so that both the effective state and observable acquire support on one of its planes. For Pauli readouts and commuting two-qubit entanglers, this yields a circuit design rule that specifies, for a given interaction graph, a placement of local rotations and entangling layers that makes a target Fourier coefficient reachable. Using a single encoding layer, we validate the proposed design through a placement ablation, regression tasks from PDEBench and physics-informed Maxwell field modeling, and further demonstrate its robustness to moderate simulated gate noise and damping.
Related
- Towards an Optimally Distributed Quantum Fourier Transform Circuit
- Feasibility Ordering of Entanglement-Source Placement for Qubit Channels
- Square-root growth of operator entanglement in an integrable brickwork circuit
- Generation of entanglement statistics with a large-scale integrated photonic-electronic circuit
- Entanglement-Aware Quantum Circuit Optimization through Globally Optimal Partitioning
BibTeX
@misc{oh2026fouriergeometric,
title = {{Fourier-Geometric Circuit Design for Gate and Entanglement Placement in Quantum Neural Networks}},
author = {Seungcheol Oh and Chaemoon Im and Daeyeun Kim and Soohyun Park and Vaneet Aggarwal and Mohsen Heidari and Joongheon Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35489},
eprint = {2609.35489},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35489},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fourier-Geometric Circuit Design for Gate and Entanglement Placement in Quantum Neural Networks
AU - Seungcheol Oh
AU - Chaemoon Im
AU - Daeyeun Kim
AU - Soohyun Park
AU - Vaneet Aggarwal
AU - Mohsen Heidari
AU - Joongheon Kim
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35489
UR - https://arxiv.org/abs/2609.35489
AB - The output of a parameterized quantum circuit (PQC) can be expressed as a finite Fourier series whose accessible frequencies are fixed by the data-encoding gates. While the encoder determines which frequencies can appear, the corresponding Fourier coefficients depend on how the trainable and entangling gates are arranged. Although existing studies provide metrics for characterizing how gate structure affects Fourier coefficients, they do not translate these analyses into an explicit design criterion specifying how gates should be arranged to make a target coefficient reachable. In this paper, we provide such a criterion. Using the adjoint action of the encoding generator, we decompose operator space into two-dimensional invariant planes indexed by frequency and show that each Fourier coefficient is exactly a sum of bilinear projections of the effective state and observable onto the planes at that frequency. Because, for Pauli encodings, high-frequency planes are spanned by mixed multi-qubit Pauli strings, a target coefficient can contribute to the output only when local rotations and entangling layers are arranged so that both the effective state and observable acquire support on one of its planes. For Pauli readouts and commuting two-qubit entanglers, this yields a circuit design rule that specifies, for a given interaction graph, a placement of local rotations and entangling layers that makes a target Fourier coefficient reachable. Using a single encoding layer, we validate the proposed design through a placement ablation, regression tasks from PDEBench and physics-informed Maxwell field modeling, and further demonstrate its robustness to moderate simulated gate noise and damping.
ER -