On Quantum Complexity

TL;DR

This work proposes a pole-structure criterion in the energy-space matrix elements of an observable as a candidate signature of quantum complexity in chaotic systems. By defining an $A$-function $A(E_1,E_2)=\langle E_1|{\cal O}|E_2\rangle$ and showing that a double pole $A(\varepsilon,\omega)\sim -a(\varepsilon)/\omega^2$ as $\omega\to0$ induces a linear-in-time growth of the late-time observable ${\cal A}_{\cal O}(\beta,t)$, the authors link complexity to non-local, pole-driven dynamics. They illustrate this with an explicit one-dimensional model where the choice $f(x,x')=\delta(x-x')\,x$ yields the predicted linear growth, interpretable as a geodesic length in JT gravity, and discuss connections to random-matrix observables and Krylov complexity. The discussion further addresses saturation at very late times via connected terms and universal sine-kernel corrections, suggesting a broad landscape of operators that realize double-pole structures and hence complexity-like growth, with implications for holography and quantum chaos.

Abstract

The ETH ansatz for matrix elements of a given operator in the energy eigenstate basis results in a notion of thermalization for a chaotic system. In this context for a certain quantity - to be found for a given model - one may impose a particular condition on its matrix elements in the energy eigenstate basis so that the corresponding quantity exhibit linear growth at late times. The condition is to do with a possible pole structure the corresponding matrix elements may have. Based on the general expectation of complexity one may want to think of this quantity as a possible candidate for the quantum complexity. We note, however, that for the explicit examples we have considered in this paper, there are infinitely many quantities exhibiting similar behavior.