From $SU(2)$ holonomies to holographic duality via tensor networks

TL;DR

This paper develops tensor network methods for $SU(2)$ spin networks in Loop Quantum Gravity, enhancing quantum circuit efficiency and establishing connections to holographic duality.

Contribution

It introduces a tensor network representation for $SU(2)$ spin networks, improving circuit construction and linking LQG with holographic duality.

Findings

Reduces qubit count in quantum circuit models of spin networks

Facilitates implementation of holographic states in LQG

Strengthens the connection between LQG and holography

Abstract

Tensor networks provide a powerful tool for studying many-body quantum systems, particularly making quantum simulations more efficient. In this article, we construct a tensor network representation of the spin network states, which correspond to $SU(2)$ gauge-invariant discrete field theories. Importantly, the spin network states play a central role in the Loop Quantum Gravity (LQG) approach to the Planck scale physics. Our focus is on the Ising-type spin networks, which provide a basic model of quantum space in LQG. It is shown that the tensor network approach improves the previously introduced methods of constructing quantum circuits for the Ising spin networks by reducing the number of qubits involved. It is also shown that the tensor network approach is convenient for implementing holographic states involving the bulk-boundary conjecture, which contributes to establishing a link between LQG and holographic duality.