Toward a Definition of Complexity for Quantum Field Theory States.

Shira Chapman,Hugo Marrochio,Fernando Pastawski,Michal P Heller

doi:10.1103/physrevlett.120.121602

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(1,1) 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.

Highlights

Ryu-Takayanagi surfaces are often unable to access the whole holographic geometry [5,6,7]. This observation has led to significant interest in novel, from the point of view of quantum gravity, codimension-1 and codimension-0 bulk quantities, whose behavior suggests conjecturing a link with the information theory notion of quantum state complexity [8,9,10,11,12,13,14]
In the context of holography, the organization of discrete tensor networks has been suggested to give a qualitative picture of how quantum states give rise to emergent geometries [20]. This heuristic analysis was applied to the multiscale entanglement renormalization ansatz (MERA) [21], employed to find ground states of critical physical theories presenting a tensor network structure reminiscent of an anti–de Sitter (AdS) time slice
While the FS prescription is quite general, our choices for (a), (b), and (d) render the necessary calculations tractable. Some of these choices are inspired by the continuous MERA approach to free quantum field theory (QFT) [24,25,26], which we briefly review in Sec

Summary

Toward a Definition of Complexity for Quantum Field Theory States

Ryu-Takayanagi surfaces are often unable to access the whole holographic geometry [5,6,7] This observation has led to significant interest in novel, from the point of view of quantum gravity, codimension-1 (volume) and codimension-0 (action) bulk quantities, whose behavior suggests conjecturing a link with the information theory notion of quantum state complexity [8,9,10,11,12,13,14]. In the context of holography, the organization of discrete tensor networks (seen as a quantum circuit U) has been suggested to give a qualitative picture of how quantum states give rise to emergent geometries [20] This heuristic analysis was applied to the multiscale entanglement renormalization ansatz (MERA) [21], employed to find ground states of critical physical theories presenting a tensor network structure reminiscent of an anti–de Sitter (AdS) time slice.

Published by the American Physical Society

The leading divergence in the complexity measures CðnÞ is proportional to