Circuit complexity in quantum field theory

TL;DR

This paper develops a quantum circuit model for Gaussian states in free scalar quantum field theory, applying Nielsen's geometric approach to analyze complexity and comparing it with holographic complexity results.

Contribution

It introduces a quantum circuit model for Gaussian states in quantum field theory and applies geometric methods to analyze their complexity, linking it to holographic complexity studies.

Findings

Complexity of ground states is characterized by shortest geodesics in circuit space.

Surprising similarities found between quantum field theory complexity and holographic complexity.

Provides a framework for comparing quantum circuit complexity with holographic models.

Abstract

Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.