(Quantum) Fractional Brownian Motion and Multifractal Processes under the Loop of a Tensor Networks

TL;DR

This paper introduces a network-based framework for constructing fractional Brownian motion and multifractal processes, utilizing renormalization-inspired methods and tensor networks, with improved sampling efficiency via MERA.

Contribution

It presents new representations of fractional Brownian motion using Gaussian conditional probability networks and demonstrates the application of MERA for efficient sampling of multifractal processes.

Findings

Derived new representations of fractional Brownian motion.

Extended constructions to include multifractal properties.

MERA-based sampling scales as O(N log N), outperforming traditional methods.

Abstract

We derive fractional Brownian motion and stochastic processes with multifractal properties using a framework of network of Gaussian conditional probabilities. This leads to the derivation of new representations of fractional Brownian motion. These constructions are inspired from renormalization. The main result of this paper consists of constructing each increment of the process from two-dimensional gaussian noise inside the light-cone of each seperate increment. Not only does this allows us to derive fractional Brownian motion, we can introduce extensions with multifractal flavour. In another part of this paper, we discuss the use of the multi-scale entanglement renormalization ansatz (MERA), introduced in the study critical systems in quantum spin lattices, as a method for sampling integrals with respect to such multifractal processes. After proper calibration, a MERA promises the generation of a sample of size $N$ of a multifractal process in the order of $O(N\log(N))$, an improvement over the known methods, such as the Cholesky decomposition and the circulant methods, which scale between $O(N^2)$ and $O(N^3)$.