  There are various definitions of the concept of complexity in Quantum Field
Theory as well as for finite quantum systems. For several of them there are
conjectured holographic bulk duals. In this work we establish an entry in the
AdS/CFT dictionary for one such class of complexity, namely Krylov or
K-complexity. For this purpose we work in the double-scaled SYK model which is
dual in a certain limit to JT gravity, a theory of gravity in AdS$_2$. In
particular, states on the boundary have a clear geometrical definition in the
bulk. We use this result to show that Krylov complexity of the
infinite-temperature thermofield double state on the boundary of AdS$_2$ has a
precise bulk description in JT gravity, namely the length of the two-sided
wormhole. We do this by showing that the Krylov basis elements, which are
eigenstates of the Krylov complexity operator, are mapped to length eigenstates
in the bulk theory by subjecting K-complexity to the bulk-boundary map
identifying the bulk/boundary Hilbert spaces. Our result makes extensive use of
chord diagram techniques and identifies the Krylov basis of the boundary
quantum system with fixed chord number states building the bulk gravitational
Hilbert space.
