Spread complexity as classical dilaton solutions

TL;DR

This paper establishes a geometric connection between different quantum complexity measures by relating Nielsen's circuit complexity and Krylov complexity through classical dilaton solutions in a gravity model, revealing new insights into quantum state geometry.

Contribution

It introduces a geometric framework linking Nielsen's and Krylov complexities via classical dilaton solutions in a gravity model, unifying two complexity notions.

Findings

Dilaton can be interpreted as quantum expectation values of symmetry generators.

State space geometry preserves metric and symplectic forms under unitary transformations.

A classical gravity model (JT gravity) describes the geometric structure of quantum states.

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 K\"ahler 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.