Classification of matrix product states with a local (gauge) symmetry

Abstract

Matrix Product States (MPS) are a particular type of one dimensional tensor network states, that have been applied to the study of numerous quantum many body problems. One of their key features is the possibility to describe and encode symmetries on the level of a single building block (tensor), and hence they provide a natural playground for the study of symmetric systems. In particular, recent works have proposed to use MPS (and higher dimensional tensor networks) for the study of systems with local symmetry that appear in the context of gauge theories. In this work we classify MPS which exhibit local invariance under arbitrary gauge groups. We study the respective tensors and their structure, revealing known constructions that follow known gauging procedures, as well as different, other types of possible gauge invariant states.

Simulating lattice gauge theories within quantum technologies

“…This section reviews some of the studies that appeared in the last years, covering most of the available approaches for Abelian and non-Abelian lattice gauge theories [103][104][105][108][109][110][111]. General studies of the structure and properties of PEPS with local gauge symmetries were discussed in [108,109,[111][112][113][114][115]. In addition, there has also been an effort specifically focused on classical tensor network methods with Grassmann fields [116][117][118][119][120][121], investigation of the sign problem tackled with TN and compared with Monte Carlo [122], works on the O(2) model with a purely imaginary chemical potential using TN [123,124], or on exploiting useful mappings to construct tensors and study lattice field theories [125][126][127][128][129][130][131][132].…”

Lattice quantum electrodynamics in (3+1)-dimensions at finite density with tensor networks

“…As tailored many-body quantum state ansätze, TNs are an efficient approximate entanglement-based representation of physical states, capable of efficiently describe equilibrium properties and real-time dynamics of systems described by complex actions, where Monte Carlo simulations fail to efficiently converge 22 . TN methods have proven remarkable success in simulating LGTs in (1+1) dimensions [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] , and very recently they have shown potential in (2+1) dimensions [42][43][44][45][46][47][48][49][50] . To date, due to the lack of efficient numerical algorithms to describe high-dimensional systems via TNs, no results are available regarding the realistic scenario of LGTs in three spatial dimensions.…”

Quantum computation by teleportation and symmetry

“…For instance, the spin-1 1D AKLT state [58] has global SO(3) symmetry, which supports universal QC on a single qubit. Furthermore, there are also states with both global and local symmetries, such as coupled gauge-matter system [56], the symmetry action on gauge tensor (A) and matter tensor (B) has to be consistent, see Fig. 2(c).…”