Circuit complexity for free fermions

• Published in: The journal of high energy physics Vol. 2018; no. 7; pp. 1 - 72
• Main Authors: Hackl, Lucas, Myers, Robert C.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2018; Springer Nature B.V; SpringerOpen
• Subjects: AdS-CFT Correspondence; Circuits; Classical and Quantum Gravitation; Complexity; Effective Field Theories; Eigenvectors; Elementary Particles; Fermions; High energy physics; Holography; Lie groups; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; AdS-CFT Correspondence; Effective Field Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP07(2018)139

A bstract 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 energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.