# Toward a Definition of Complexity for Quantum Field Theory States

Chapman, Shira (Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada) ; Heller, Michal P. (Max Planck Institute for Gravitational Physics, Potsdam-Golm D-14476, Germany) ; Marrochio, Hugo (Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada) (Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada) ; Pastawski, Fernando (Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Berlin D-14195, Germany) (Max Planck Institute for Gravitational Physics, Potsdam-Golm D-14476, Germany)

22 March 2018

Abstract: We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form $su\left(1,1\right)$ algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals.

Published in: Phys. Rev. Lett. 120 (2018)