Growth and collapse of subsystem complexity under random unitary circuits

TL;DR

This paper investigates the growth and collapse of subsystem complexity in chaotic quantum systems modeled by random unitary circuits, revealing linear growth followed by a sudden collapse in certain regimes.

Contribution

It provides rigorous proofs for the behavior of subsystem complexity over time and offers holographic evidence for a sudden decay in complexity.

Findings

Subsystem complexity grows linearly up to a certain time.

Complexity of smaller subsystems collapses to zero after a specific time.

Holographic methods suggest a linear growth followed by abrupt decay.

Abstract

For chaotic quantum dynamics modeled by random unitary circuits, we study the complexity of reduced density matrices of subsystems as a function of evolution time where the initial global state is a product pure state. The state complexity is defined as the minimum number of local quantum channels to generate a given state from a product state to a good approximation. In $1+1$d, we prove that the complexity of subsystems of length $\ell$ smaller than half grows linearly in time $T$ at least up to $T = \ell / 4$ but becomes zero after time $T = \ell /2$ in the limit of a large local dimension, while the complexity of the complementary subsystem of length larger than half grows linearly in time up to exponentially late times. Using holographic correspondence, we give some evidence that the state complexity of the smaller subsystem should actually grow linearly up to time $T = \ell/2$ and then abruptly decay to zero.