Explicit Connections Between Krylov and Nielsen Complexity

• URL: http://arxiv.org/abs/2511.15799v1
• Date: Wed, 19 Nov 2025 19:00:07 GMT
• Title: Explicit Connections Between Krylov and Nielsen Complexity
• Authors: Ben Craps, Gabriele Pascuzzi, Juan F. Pedraza, Le-Chen Qu, Shan-Ming Ruan,
• Abstract summary: We establish a direct correspondence between Krylov and Nielsen complexity by choosing the Krylov basis to be part of the elementary gate set of Nielsen geometry.<n>Up to normalization, the Krylov complexity of a Hermitian operator equals the length squared of a straight-line trajectory on the manifold of unitaries.<n>The corresponding length provides an upper bound on Nielsen complexity that saturates whenever the straight line is a minimal geodesic.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We establish a direct correspondence between Krylov and Nielsen complexity by choosing the Krylov basis to be part of the elementary gate set of Nielsen geometry and selecting a Nielsen complexity metric compatible with the Krylov metric. Up to normalization, the Krylov complexity of a Hermitian operator then equals the length squared of a straight-line trajectory on the manifold of unitaries that connects the identity operator with a precursor operator. The corresponding length provides an upper bound on Nielsen complexity that saturates whenever the straight line is a minimal geodesic. While for general systems we can only establish saturation in the limit of small precursors, we provide evidence that in the Sachdev-Ye-Kitaev (SYK) model there is a precise correspondence between Krylov complexity and (the square of) Nielsen complexity for a finite range of precursors.

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