Conformal field theory complexity from Euler-Arnold equations

Abstract

Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.

Subsystem complexity after a global quantum quench

“…Many results have been obtained for the complexity of quantum circuits made by pure states constructed through lattice models [17][18][19][20][21][22][23][24][25] and in the gravitational side of the holographic correspondence. Some proposals have been done also to study the circuit complexity in quantum fields theories [26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Holographic and QFT complexity with angular momentum

“…1 The CV and CA conjectures stimulated an extensive effort aimed at investigating properties of these new gravitational observables and at testing the validity of the proposals . In parallel, various approaches have been explored to define and understand the complexity of states in quantum field theory, e.g., following Nielsen's geometric approach [81][82][83][84][85] which we review in section 4, the Fubini-Study metric approach for the space of states [86], path integral optimization [87][88][89][90][91][92][93][94][95], or CFT notions of complexity [96][97][98][99][100][101][102].…”

Holographic complexity of LST and single trace $$ T\overline{T} $$

“…Several attempts have been made to define complexity in the continuum limit (see e.g. [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] for an incomplete but representative list). However, it is fair to say that up to now there exists neither any universal and/or unanimous definition of complexity in the continuum limit nor an exhaustive study of its possible universality classes.…”