A bulk manifestation of Krylov complexity

TL;DR

This paper establishes a precise bulk dual description of Krylov complexity in the AdS/CFT correspondence, specifically relating it to wormhole length in JT gravity within the SYK model framework.

Contribution

It provides the first explicit bulk dual for Krylov complexity in a holographic setting using the double-scaled SYK model and JT gravity.

Findings

Krylov complexity of the thermofield double state corresponds to wormhole length in JT gravity.

Krylov basis elements map to length eigenstates in the bulk.

Chord diagram techniques connect boundary Krylov basis to bulk gravitational states.

Abstract

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.