Towards Complexity for Quantum Field Theory States

TL;DR

This paper proposes a new way to measure the complexity of quantum states in field theories using the Fubini-Study metric, revealing similarities with holographic complexity despite working outside gravity dual regimes.

Contribution

It introduces a complexity measure based on the Fubini-Study metric for Gaussian states in quantum field theories and analyzes minimal complexity circuits within this framework.

Findings

Complexity reduces to known geodesics on Gaussian state manifolds.

Fubini-Study metric factorizes into hyperbolic planes for these states.

Results show similarities with holographic complexity proposals.

Abstract

We investigate notions of complexity of states in continuous quantum-many body 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 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 $\mathfrak{su}(1,1)$ 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 holographic complexity proposals.