Towards Complexity for Quantum Field Theory States (1707.08582v3)

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.

• The paper introduces a novel FS metric framework to quantify state complexity by studying geodesic paths between Gaussian states in QFT.
• It employs momentum-preserving quadratic generators within the su(1,1) algebra to achieve minimal transformation paths, analogous to hyperbolic geodesics.
• The findings align with holographic complexity proposals, offering fresh insights into quantum state preparation and renormalization in continuous QFTs.

Towards Complexity for Quantum Field Theory States: An Expert Overview

The paper "Towards Complexity for Quantum Field Theory States" by Chapman, Heller, Marrochio, and Pastawski provides a sophisticated exploration of state complexity within the framework of continuous quantum systems. The authors focus primarily on Gaussian states emerging from free quantum field theories (QFTs), as well as their approximations identified in the context of the continuous Multiscale Entanglement Renormalization Ansatz (cMERA).

Complexity Metric and Methodology

The central theme of this paper is the proposal of a novel metric to quantify state complexity using the Fubini-Study (FS) metric. This approach allows for the measurement of complexity by considering the transformation path length between an uncorrelated reference Gaussian state and a target Gaussian state, which reflects ground-state correlations up to a cutoff scale in momentum space. The transformation involves momentum-preserving quadratic generators that form $\mathfrak{su}(1,1)$ algebras, extending the gates beyond typical discrete settings.

By employing the FS metric, the authors minimize state complexity through the examination of geodesics on the manifold of Gaussian states. Such geodesics are found to effectively reduce the transformation path to known minimal routes, which offer illuminating parallels to proposals in holographic complexity, despite not being within the gravitational dual regime.

Numerical Insights and Strong Claims

The paper outlines that complexity evaluations that utilize a single $\mathfrak{su}(1,1)$ generator per pair of opposite momenta yield minimal paths consistent with geodesic behavior on hyperbolic planes. It postulates that the FS-computed complexity scales as an $L^2$ or other $L^n$ norm of squeezing parameters related to the entanglement in Gaussian states. A particularly significant element involves the agreement of certain scaling divergences in these complexities with those anticipated in holographic complexity computations, aligning well with both the "complexity equals volume" (CV) and "complexity equals action" (CA) proposals in AdS/CFT contexts.

Implications and Future Developments

The proposed framework for measuring and understanding state complexity in continuous QFTs offers compelling new insights into the role of quantum information concepts in high-energy physics and quantum gravity domains. Practically, this methodology could refine how state preparation and transformation are envisioned in quantum computational applications of QFTs.

Theoretically, the work establishes a foundation for further exploration into the nature of complexity within quantum systems, inviting investigations into the relation between complexity and phenomenons such as RG flows and quantum criticality. Future work might extend to fermionic fields, investigate potential relativistic symmetry implications, or further unravel the links between UV divergences in field theory and geometric quantities in dual gravitational descriptions.

In conclusion, the authors provide a framework that promises to enrich the field's understanding of quantum state complexity, balancing rigorous mathematical treatment with expansive implications across multiple domains of theoretical physics.