Embedding of Tree Tensor Networks into Shallow Quantum Circuits

TL;DR

This paper introduces a method to embed Tree Tensor Networks into shallow quantum circuits, enhancing the capability of Variational Quantum Algorithms to handle complex, two-dimensional, and long-range correlated systems on NISQ devices.

Contribution

The study presents a novel approach for embedding TTNs into shallow quantum circuits, overcoming limitations of previous methods that focused only on MPSs.

Findings

Embedding TTNs yields better initial quantum circuits than MPS.

The method has practical computational complexity.

Extends VQAs applicability to 2D and long-range correlated systems.

Abstract

Variational Quantum Algorithms (VQAs) are being highlighted as key quantum algorithms for demonstrating quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices, which are limited to executing shallow quantum circuits because of noise. However, the barren plateau problem, where the gradient of the loss function becomes exponentially small with system size, hinders this goal. Recent studies suggest that embedding tensor networks into quantum circuits and initializing the parameters can avoid the barren plateau. Yet, embedding tensor networks into quantum circuits is generally difficult, and methods have been limited to the simplest structure, Matrix Product States (MPSs). This study proposes a method to embed Tree Tensor Networks (TTNs), characterized by their hierarchical structure, into shallow quantum circuits. TTNs are suitable for representing two-dimensional systems and systems with long-range correlations, which MPSs are inadequate for representing. Our numerical results show that embedding TTNs provides better initial quantum circuits than MPS. Additionally, our method has a practical computational complexity, making it applicable to a wide range of TTNs. This study is expected to extend the application of VQAs to two-dimensional systems and those with long-range correlations, which have been challenging to utilize.