Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits
Abstract
Angle-encoded variational quantum circuits implement truncated Fourier series. For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget E can; we find that this exchange of width for depth is not free in first-order training. Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call \emph{phase-locking}. We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most 2E+2 tangent directions, one more than its 2E+1 Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least 6E+6. The cost in training is measured against this control, as the number of runs out of 100 that reach the optimum within a fixed step budget. On an exact loss with flat-spectrum targets the gap grows with E; at E=32 the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100. On three real-world series, at matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam and in 32 to 93 under Levenberg--Marquardt, the control in at least 98 and in all 100; its held-out error is up to about $1.5$ times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.
Related
- Space Phase-Locking Theory (SPLT) Part II.5: Mass-Lifetime Duality of the Hadron Spectrum and Geometric Decay Deduction
- Classical Pre-Training of Large Shallow Parametrized Quantum Circuits through Pauli Propagation and deployment on Quantum Hardware
- Fully quantum training of parametric circuits with Quantum Hamiltonian Descent
- Re-uploading quantum data: a universal function approximator for quantum inputs
- A Tutorial on Quantum Amplitude Embedding with Data Re-Uploading
BibTeX
@misc{poppel2026same,
title = {{Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits}},
author = {Michael Poppel and David Bucher and Maximilian Zorn and Markus Baumann and Sebastian Wölckert and Claudia Linnhoff-Popien and Philipp Altmann and Jonas Stein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21778},
eprint = {2610.21778},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21778},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits
AU - Michael Poppel
AU - David Bucher
AU - Maximilian Zorn
AU - Markus Baumann
AU - Sebastian Wölckert
AU - Claudia Linnhoff-Popien
AU - Philipp Altmann
AU - Jonas Stein
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21778
UR - https://arxiv.org/abs/2610.21778
AB - Angle-encoded variational quantum circuits implement truncated Fourier series. For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget E can; we find that this exchange of width for depth is not free in first-order training. Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call \emph{phase-locking}. We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most 2E+2 tangent directions, one more than its 2E+1 Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least 6E+6. The cost in training is measured against this control, as the number of runs out of 100 that reach the optimum within a fixed step budget. On an exact loss with flat-spectrum targets the gap grows with E; at E=32 the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100. On three real-world series, at matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam and in 32 to 93 under Levenberg--Marquardt, the control in at least 98 and in all 100; its held-out error is up to about $1.5$ times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.
ER -