Entangling power and quantum circuit complexity

TL;DR

This paper explores the relationship between entanglement and quantum circuit complexity, providing bounds that relate entanglement growth to complexity growth, with implications for quantum simulation and insights into short-term complexity dynamics.

Contribution

It introduces a new bound linking entanglement and circuit complexity for small values, enhancing understanding of complexity growth and comparing quantum simulation methods.

Findings

Entanglement growth linearly implies complexity growth.

Provides a continuous-variable small incremental entangling bound.

Offers insights into short-time complexity dynamics.

Abstract

Notions of circuit complexity and cost play a key role in quantum computing and simulation where they capture the (weighted) minimal number of gates that is required to implement a unitary. Similar notions also become increasingly prominent in high energy physics in the study of holography. While notions of entanglement have in general little implications for the quantum circuit complexity and the cost of a unitary, in this note, we discuss a simple such relationship when both the entanglement of a state and the cost of a unitary take small values, building on ideas on how values of entangling power of quantum gates add up. This bound implies that if entanglement entropies grow linearly in time, so does the cost. The implications are two-fold: It provides insights into complexity growth for short times. In the context of quantum simulation, it allows to compare digital and analog quantum simulators. The main technical contribution is a continuous-variable small incremental entangling bound.