Spread complexity as classical dilaton solutions

TL;DR

The paper builds a geometric bridge between Nielsen circuit complexity and Krylov/spread complexity by endowing the quantum state space of low-rank algebras with a Kähler structure based on the Fubini-Study metric. It couples these state manifolds to a matter-free JT dilaton gravity model and shows the dilaton field $\eta$ coincides with the quantum expectation values of symmetry generators and with the spread complexity in a Krylov basis, thereby unifying two notions of complexity within a classical gravitational framework. For low-rank algebras like $su(1,1)$, $su(2)$, and the Heisenberg-Weyl group, the FS metric and spread complexity emerge as classical solutions of the JT action, with explicit relations between the reference state, cost-function coefficients, and the dilaton profile. The results point to a broader geometric program wherein complexity measures may be encoded in dilaton gravity backgrounds, offering a path to generalizations to higher-rank algebras and possible connections to conformal field theories and Liouville gravity.

Abstract

We demonstrate a relation between Nielsen's approach towards circuit complexity and Krylov complexity through a particular construction of quantum state space geometry. We start by associating Kähler structures on the full projective Hilbert space of low rank algebras. This geometric structure of the states in the Hilbert space ensures that every unitary transformation of the associated algebras leave the metric and the symplectic forms invariant. We further associate a classical matter free Jackiw-Teitelboim (JT) gravity model with these state manifolds and show that the dilaton can be interpreted as the quantum mechanical expectation values of the symmetry generators. On the other hand we identify the dilaton with the spread complexity over a Krylov basis thereby proposing a geometric perspective connecting two different notions of complexity.