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Krylov complexity is not a measure of distance between states or operators (2311.04093v2)
Published 7 Nov 2023 in hep-th and quant-ph
Abstract: We ask whether Krylov complexity is mutually compatible with the circuit and Nielsen definitions of complexity. We show that the Krylov complexities between three states fail to satisfy the triangle inequality and so cannot be a measure of distance: there is no possible metric for which Krylov complexity is the length of the shortest path to the target state or operator. We show this explicitly in the simplest example, a single qubit, and in general.
- M. A. Nielsen, Quantum Info. Comput. 6, 213–262 (2006).
- M. R. Dowling and M. A. Nielsen, Quantum Information & Computation 8, 861 (2008).
- P. Caputa and S. Datta, JHEP 12, 188 (2021), [Erratum: JHEP 09, 113 (2022)], arXiv:2110.10519 [hep-th] .
- G. Fubini, Sulle metriche definite da una forma hermitiana: nota (Office graf. C. Ferrari, 1904).
- E. Study, Mathematische Annalen 60, 321 (1905).
- M. Alishahiha and S. Banerjee, SciPost Phys. 15, 080 (2023), arXiv:2212.10583 [hep-th] .
