Circuit Complexity for Coherent States

TL;DR

This paper analyzes the circuit complexity of coherent states in free scalar field theory, comparing Nielsen's geometric approach with the Fubini-Study method, and finds differences in complexities and optimal circuits.

Contribution

It introduces a comparison of Nielsen's geometric approach and Fubini-Study method for calculating the circuit complexity of coherent states, highlighting differences in results.

Findings

Complexity of coherent states shares UV divergences with vacuum state.

Optimal circuits often entangle normal modes at intermediate steps.

Different complexity measures yield different results for coherent states.

Abstract

We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen's geometric approach as in [1]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen's approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.