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 derive approximate analytic solutions for the 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.

arXiv:2606.02817 [quant-ph]

(or arXiv:2606.02817v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.02817 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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