A Geometric View on Quantum Tensor Networks

Alexander Tsirulev,Gh Adam,J Buša,M Hnatič

doi:10.1051/epjconf/202022602022

Abstract

Tensor network states and algorithms play a key role in understanding the structure of complex quantum systems and their entanglement properties. This report is devoted to the problem of the construction of an arbitrary quantum state using the differential geometric scheme of covariant series in Riemann normal coordinates. The building blocks of the scheme are polynomials in the Pauli operators with the coefficients specified by the curvature, torsion, and their covariant derivatives on some base manifold. The problem of measuring the entanglement of multipartite mixed states is shortly discussed.

Highlights

The quantum tensor networks, which will be considered below in a wide sense, give us a useful method for the approximation and high-performance processing of quantum states [1–3]
Tensor network states and algorithms play a key role in understanding the structure of complex quantum systems and their entanglement properties
This report is devoted to the problem of the construction of an arbitrary quantum state using the differential geometric scheme of covariant series in Riemann normal coordinates

Summary

Introduction

The quantum tensor networks, which will be considered below in a wide sense, give us a useful method for the approximation and high-performance processing of quantum states [1–3]. Geometric ideas and geometric tools, including the geometry of quantum tensor networks [4], are quite common in quantum computing and quantum information theory, especially in connection with the study of the entanglement [5, 6] and the emergence of gravity [7] This short report presents an outline of a new geometric approach to the simulation of quantum states by using quantum tensor networks with a relatively small number of independent parameters. This approach is based on covariant series in normal coordinates on the direct product of k (k n) four-dimensional manifolds equipped with suitable linear connections and the corresponding curvatures and/or torsions; we will consider only the case k = 1, but the generalization to an arbitrary k > 1 is obvious.

Quantum states in the Pauli basis

Simulation of the state space by covariant expansions

Conclusion