Explicit Connections Between Krylov and Nielsen Complexity
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912750643445760 |
|---|---|
| author | Craps, Ben Pascuzzi, Gabriele Pedraza, Juan F. Qu, Le-Chen Ruan, Shan-Ming |
| author_facet | Craps, Ben Pascuzzi, Gabriele Pedraza, Juan F. Qu, Le-Chen Ruan, Shan-Ming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Connections Between Krylov and Nielsen Complexity Craps, Ben Pascuzzi, Gabriele Pedraza, Juan F. Qu, Le-Chen Ruan, Shan-Ming High Energy Physics - Theory Quantum Physics 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. |
| title | Explicit Connections Between Krylov and Nielsen Complexity |
| topic | High Energy Physics - Theory Quantum Physics |
| url | https://arxiv.org/abs/2511.15799 |