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Krylov complexity is not a measure of distance between states or operators
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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.
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Cited by 5 Pith papers
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In the brickwall model of a BTZ black hole, hand-tuned Gaussian randomness at a stretched horizon reproduces random-matrix-theory spectral statistics and Krylov complexity peaks for scalar and fermionic probes.
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Quasinormal modes and complexity in saddle-dominated SU(N) spin systems
A family of SU(2) and SU(3) Lipkin-Meshkov-Glick-type Hamiltonians reproduces de Sitter quasinormal-mode densities of states, and late-time probes reveal integrability beneath saddle-dominated scrambling.
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Statistics and Complexity of Wavefunction Spreading in Quantum Dynamical Systems
The moments of the spreading-operator measurement distribution are generalized spread complexities, which for GUE Hamiltonians peak more sharply at higher order and obey a norm bound.
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