Equivalence of maximal and generic reachability for non-universal Variational Quantum Circuits
Abstract
Employing problem-specific non-universal Variational Quantum Circuits aligned with a suitable state preparation has become a standard approach to counter the difficulty in training universal ansätze. However, due to their non-universality, diagnosing their reachability and trainability remains reference-state-specific. In this work, we establish the equivalence of maximal reachability and generic reachability over arbitrary reference states as a consequence of the \emph{principal orbit-type theorem}. Thereafter, assuming that the global minimum of the cost function is achieved on a subset that can be described as the image of a smooth function, we derive necessary and sufficient conditions for non-zero probability of reachability under generically sampled reference states. Furthermore, when the solution set is assumed to be realised through a real analytic map that embeds a solution manifold, we show that local surjectivity is generically obtained on the entire solution manifold if it is attained at a single point. As a practical design rule, the need for local surjectivity automatically translates to a necessary dimensional criterion: reachability requires the target set's topological dimension to be at least as large as the co-dimension of the generic orbit. Numerical simulations show that finite-depth optimisation consistently fails when the dimensional obstruction applies, while unobstructed cases show improved convergence.
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BibTeX
@misc{ngairangbam2026equivalence,
title = {{Equivalence of maximal and generic reachability for non-universal Variational Quantum Circuits}},
author = {Vishal S. Ngairangbam and Michael Spannowsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27053},
eprint = {2609.27053},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equivalence of maximal and generic reachability for non-universal Variational Quantum Circuits
AU - Vishal S. Ngairangbam
AU - Michael Spannowsky
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.27053
UR - https://arxiv.org/abs/2609.27053
AB - Employing problem-specific non-universal Variational Quantum Circuits aligned with a suitable state preparation has become a standard approach to counter the difficulty in training universal ansätze. However, due to their non-universality, diagnosing their reachability and trainability remains reference-state-specific. In this work, we establish the equivalence of maximal reachability and generic reachability over arbitrary reference states as a consequence of the \emph{principal orbit-type theorem}. Thereafter, assuming that the global minimum of the cost function is achieved on a subset that can be described as the image of a smooth function, we derive necessary and sufficient conditions for non-zero probability of reachability under generically sampled reference states. Furthermore, when the solution set is assumed to be realised through a real analytic map that embeds a solution manifold, we show that local surjectivity is generically obtained on the entire solution manifold if it is attained at a single point. As a practical design rule, the need for local surjectivity automatically translates to a necessary dimensional criterion: reachability requires the target set's topological dimension to be at least as large as the co-dimension of the generic orbit. Numerical simulations show that finite-depth optimisation consistently fails when the dimensional obstruction applies, while unobstructed cases show improved convergence.
ER -