Tensor network methods for quantum-inspired image processing and classical optics

TL;DR

Tensor networks are leveraged to bridge quantum-inspired computation and classical imaging and optics. By encoding natural images with MPS/TTN representations and formulating optical propagators as low-rank MPOs, the authors achieve fast, scalable operations in high dimensional spaces while preserving essential correlations. Key contributions include quantics and FRQI based image encodings, finite size compression scaling with $epsilon(L)=L^{-kappa}\phi(\xi)$ and $\phi(\xi) ~ \xi^{-\alpha}$ with $\alpha\approx0.75$, and the construction of exact diagonal MPOs for Fresnel and angular spectrum propagators via Pauli string decompositions and Bessel expansions with exponential convergence. The work further extends to relativistic-like propagators and demonstrates practical speedups in image formation and sampling tasks, offering a route to efficient phase retrieval and related optical inverse problems in large-scale applications.

Abstract

Tensor network methods strike a middle ground between fully-fledged quantum computing and classical computing, as they take inspiration from quantum systems to significantly speed up certain classical operations. Their strength lies in their compressive power and the wide variety of efficient algorithms that operate within this compressed space. In this work, we focus on applying these methods to fundamental problems in image processing and classical optics such as wave-front propagation and optical image formation, by using directly or indirectly parallels with quantum mechanics and computation. These quantum-inspired methods are expected to yield faster algorithms with applications ranging from astronomy and earth observation to microscopy and classical imaging more broadly.

Figures (24)

• Figure 1: A black and white picture and its height field representation. This picture will be used throughout this paper.
• Figure 2: Matrix to vector reshaping. Below the image, the corresponding vector noted $x_i$ with $i=0...2^{2n-1}$.
• Figure 3: Quantics representation of the linear grid $L=8$. Let us denote $h_{ijk}$ the resulting tensor. In this example, we have $x_1\rightarrow h_{000}$, $x_2\rightarrow h_{100}$...,$x_7\rightarrow h_{111}$ and $x_8\rightarrow h_{001}$.
• Figure 4: Tensor network compression of an image. Here, a generic tensor network is showed for the sake of generality. The approximate symbol will quantify the precision of the compression.
• Figure 5: Illustration of the decomposition of high-order tensor into a matrix product state. The arrow represents a sequential singular value decomposition.