Circuit complexity for free fermions

TL;DR

This paper develops a mathematical framework to compute circuit complexity for free fermionic field theories, applying it to the Dirac ground state and excited states, with implications for holography and quantum information.

Contribution

It introduces a new approach to fermionic circuit complexity using geodesic distances on the Lie group of orthogonal transformations, extending previous bosonic methods.

Findings

Computed complexity of Dirac ground state and excited states

Established connections to holography and alternative complexity measures

Provided a comprehensive mathematical framework for fermionic Gaussian states

Abstract

We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited states. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.