Circuit Complexity in Fermionic Field Theory

TL;DR

This paper develops a framework for measuring the complexity of free fermionic quantum field theories using Nielsen's geometric approach, exploring both discretized and continuum methods, and analyzing fermionic cMERA tensor networks.

Contribution

It introduces a novel method to compute complexity in fermionic quantum field theories through geometric and tensor network perspectives, addressing cut-off ambiguities.

Findings

Computed complexity measures for fermionic QFTs in 1+1 and 3+1 dimensions.

Analyzed the role of Bogoliubov-Valatin transformations and squeezing operators.

Explored the connection between circuit complexity and fermionic cMERA tensor networks.

Abstract

We define and calculate versions of complexity for free fermionic quantum field theories in 1+1 and 3+1 dimensions, adopting Nielsen's geodesic perspective in the space of circuits. We do this both by discretizing and identifying appropriate classes of Bogoliubov-Valatin transformations, and also directly in the continuum by defining squeezing operators and their generalizations. As a closely related problem, we consider cMERA tensor networks for fermions: viewing them as paths in circuit space, we compute their path lengths. Certain ambiguities that arise in some of these results because of cut-off dependence are discussed.