Probing the spectral dimension of quantum network geometries

TL;DR

This paper introduces a method to estimate the spectral dimension of quantum network geometries by coupling an auxiliary system and analyzing normal mode frequencies, demonstrating robustness to noise and parameter variations.

Contribution

It presents a novel indirect approach to determine the spectral dimension of quantum networks through auxiliary system coupling and spectral analysis.

Findings

Spectral dimension can be estimated from low-frequency normal modes.

The method is robust to noise and parameter variations.

Coupling to a subset of nodes helps resolve normal mode frequencies.

Abstract

We consider an environment for an open quantum system described by a "Quantum Network Geometry with Flavor" (QNGF) in which the nodes are coupled quantum oscillators. The geometrical nature of QNGF is reflected in the spectral properties of the Laplacian matrix of the network which display a finite spectral dimension, determining also the frequencies of the normal modes of QNGFs. We show that an a priori unknown spectral dimension can be indirectly estimated by coupling an auxiliary open quantum system to the network and probing the normal mode frequencies in the low frequency regime. We find that the network parameters do not affect the estimate; in this sense it is a property of the network geometry, rather than the values of, e.g., oscillator bare frequencies or the constant coupling strength. Numerical evidence suggests that the estimate is also robust both to small changes in the high frequency cutoff and noisy or missing normal mode frequencies. We propose to couple the auxiliary system to a subset of network nodes with random coupling strengths to reveal and resolve a sufficiently large subset of normal mode frequencies.