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This paper advances the theoretical framework of quantum complexity by introducing a hierarchy of cost factors to Nielsen's geometric approach, moving beyond single-penalty models to better capture non-local dynamics. By deriving modified Euler-Arnold and Jacobi equations, it reveals how distinct non-local sectors generate multiple families of conjugate points in both single-qubit and many-body SYK-type systems. These findings provide a more realistic description of geodesic evolution and complexity growth, offering deeper insights into the structural scaling of quantum circuits under varied penalty constraints.
Nielsen complexity with multiple cost factors
arXiv:2606.02817v3 Announce Type: replace 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.
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.
35 pages, 9 figures; v2: added references; v3: 39 pages, 10 figures, revised and improved Sec. 3, other minor changes, added references, published version
arXiv:2606.02817 [quant-ph]
(or arXiv:2606.02817v3 [quant-ph] for this version)
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