Chaos and multifold complexity for an inverted harmonic oscillator

TL;DR

This paper analyzes the multifold complexity and Loschmidt echo of an inverted harmonic oscillator, deriving analytic expressions and revealing universal structural properties that align with holographic results.

Contribution

It provides the first analytic expressions for multifold complexity in an inverted harmonic oscillator and proposes a universal structure for quantum complexity.

Findings

Complexity is dominated by the longest permutation in an alternating zig-zag order.

Results match holographic calculations of complexity.

Conjecture of universal complexity structure for generic quantum systems.

Abstract

We examine the multifold complexity and Loschmidt echo for an inverted harmonic oscillator. We give analytic expressions for any number of precursors, implementing multiple backward and forward time evolutions of the quantum state, at the leading order in the perturbation. We prove that complexity is dominated by the longest permutation of the given time combination in an alternating ``zig-zag'' order, the exact same result obtained with holography. We conjecture that the general structure for multifold complexity should hold true universally for generic quantum systems, in the limit of a large number of precursors.