# Quantum Complexity of Time Evolution with Chaotic Hamiltonians

## TL;DR

This paper investigates the quantum complexity growth in chaotic systems like the SYK model, demonstrating linear growth over exponential times and its relation to geodesics on the unitary manifold, with implications for quantum computational complexity.

## Contribution

It introduces the Eigenstate Complexity Hypothesis (ECH) and shows that in large-N chaotic systems, complexity grows linearly for exponential times, supported by numerical evidence.

## Key findings

- Large-N SYK model satisfies ECH and exhibits linear complexity growth.
- Integrable N=2 fermion case shows initial linear growth followed by oscillations.
- Complexity growth relates to spectral properties and energy eigenstates of physical theories.

## Abstract

We study the quantum complexity of time evolution in large-$N$ chaotic systems, with the SYK model as our main example. This complexity is expected to increase linearly for exponential time prior to saturating at its maximum value, and is related to the length of minimal geodesics on the manifold of unitary operators that act on Hilbert space. Using the Euler-Arnold formalism, we demonstrate that there is always a geodesic between the identity and the time evolution operator $e^{-iHt}$ whose length grows linearly with time. This geodesic is minimal until there is an obstruction to its minimality, after which it can fail to be a minimum either locally or globally. We identify a criterion - the Eigenstate Complexity Hypothesis (ECH) - which bounds the overlap between off-diagonal energy eigenstate projectors and the $k$-local operators of the theory, and use it to show that the linear geodesic will at least be a local minimum for exponential time. We show numerically that the large-$N$ SYK model (which is chaotic) satisfies ECH and thus has no local obstructions to linear growth of complexity for exponential time, as expected from holographic duality. In contrast, we also study the case with $N=2$ fermions (which is integrable) and find short-time linear complexity growth followed by oscillations. Our analysis relates complexity to familiar properties of physical theories like their spectra and the structure of energy eigenstates and has implications for the hypothesized computational complexity class separations PSPACE $\nsubseteq$ BQP/poly and PSPACE $\nsubseteq$ BQSUBEXP/subexp, and the "fast-forwarding" of quantum Hamiltonians.