Nielsen complexity with multiple cost factors
Abstract
We investigate Nielsen's geometric approach to quantum complexity in the presence of multiple cost factors, extending the standard framework where a single penalty distinguishes easy from hard directions of the group manifold. By introducing a hierarchy of penalties associated with different degrees of non-locality, we develop a generalized right-invariant complexity geometry and analyze its implications for geodesic evolution. We derive the modified Euler-Arnold and Jacobi equations and study how multiple cost factors reshape the structure and scaling of conjugate points, where geodesic optimality breaks down. The formalism is illustrated in two settings: a single-qubit system with two cost factors, where we develop a semi-analytic approach to investigate complexity growth and its dependence on penalty hierarchies, and SYK-type models, where we analyze both free and chaotic regimes. In these many-body systems, we show that distinct non-local sectors generate multiple families of conjugate points whose occurrence depends on both the cost hierarchy and the system size. Our results highlight how refining the penalty structure provides a richer and more realistic description of quantum complexity and its dynamical behavior.
Versions
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BibTeX
@misc{ribeiro2026nielsen,
title = {{Nielsen complexity with multiple cost factors}},
author = {Marcos Rios Ribeiro and Diego Trancanelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.02817},
eprint = {2606.02817},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.02817},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nielsen complexity with multiple cost factors
AU - Marcos Rios Ribeiro
AU - Diego Trancanelli
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2606.02817
UR - https://arxiv.org/abs/2606.02817
AB - We investigate Nielsen's geometric approach to quantum complexity in the presence of multiple cost factors, extending the standard framework where a single penalty distinguishes easy from hard directions of the group manifold. By introducing a hierarchy of penalties associated with different degrees of non-locality, we develop a generalized right-invariant complexity geometry and analyze its implications for geodesic evolution. We derive the modified Euler-Arnold and Jacobi equations and study how multiple cost factors reshape the structure and scaling of conjugate points, where geodesic optimality breaks down. The formalism is illustrated in two settings: a single-qubit system with two cost factors, where we develop a semi-analytic approach to investigate complexity growth and its dependence on penalty hierarchies, and SYK-type models, where we analyze both free and chaotic regimes. In these many-body systems, we show that distinct non-local sectors generate multiple families of conjugate points whose occurrence depends on both the cost hierarchy and the system size. Our results highlight how refining the penalty structure provides a richer and more realistic description of quantum complexity and its dynamical behavior.
ER -