On tensor network representations of the (3+1)d toric code

Delcamp, Clement; Schuch, Norbert

doi:10.22331/q-2021-12-16-604

Cited by 13 publications

(13 citation statements)

References 63 publications

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“…This offers an alternative derivation of the 3D toric and fracton code projected entangled pair states (PEPS) [90,91] obtained in Refs. [92,93].…”

Section: Kramers-wannier As Sptmentioning

confidence: 99%

Long-range entanglement from measuring symmetry-protected topological phases

Tantivasadakarn¹,

Thorngren²,

Vishwanath³

et al. 2021

Preprint

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A fundamental distinction between many-body quantum states are those with short-and longrange entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality.Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here we establish that LRE appears upon performing measurements on symmetry protected topological (SPT) phases-of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation, by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the Z2 Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by G×H symmetry gives rise to anomalous LRE upon measuring G-charges. This introduces a new practical tool for using SPT phases as resources for creating LRE, and uncovers the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with singlesite measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way.

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“…This offers an alternative derivation of the 3D toric and fracton code projected entangled pair states (PEPS) [90,91] obtained in Refs. [92,93].…”

Section: Kramers-wannier As Sptmentioning

confidence: 99%

Long-range entanglement from measuring symmetry-protected topological phases

Tantivasadakarn¹,

Thorngren²,

Vishwanath³

et al. 2021

Preprint

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“…Let us now revisit the results obtained in [28] in light of the previous derivations. Given the (3+1)d toric code Hamiltonian, the ground state subspace can be conveniently constructed by enforcing either the X or Z stabiliser constraints via a choice of product state in the relevant basis, and then project onto the subspace of the remaining stabiliser constraints.…”

Section: Three-dimensional Toric Codementioning

confidence: 80%

“…Topological order manifests itself in tensor network states through the existence of (non-trivial) virtual symmetry conditions, i.e. symmetries with respect to non-local operators acting solely along virtual indices [18][19][20][21][22][23][24][25][26][27][28]67]. In particular, closed versions of these operators along non-contractible cycles of the manifold may be employed to map ground states from one another.…”

Section: Virtual Symmetry Operatorsmentioning

confidence: 99%

“…In this context, the electromagnetic duality of the (3+1)d toric code is reflected into the existence of two dual tensor network representations of the ground state subspace [27,28]. These two representations are obtained by initially imposing either family of stabiliser constraints and are characterised by virtual symmetries with respect to of string-and membrane-like operators, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…These two representations are obtained by initially imposing either family of stabiliser constraints and are characterised by virtual symmetries with respect to of string-and membrane-like operators, respectively. The symmetry breaking patterns of the boundary theories are then recovered in the fixed point sectors of the corresponding transfer matrices [28]. Interestingly, we can choose two different tensor network representations over distinct submanifolds via the introduction of an intertwining tensor network that only possesses virtual indices.…”

Section: Introductionmentioning

confidence: 99%

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Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

Delcamp¹

2021

Preprint

Self Cite

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Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group G, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded into module 2-categories over the input spherical fusion 2-category. Individual tensors are characterised by symmetry conditions with respect to non-local operators acting on entanglement degrees of freedom. In the case of Dirichlet and Neumann boundary conditions, we show that the symmetry operators form the fusion 2-categories 2Vec G of G-graded 2vector spaces and 2Rep(G) of 2-representations of G, respectively. In virtue of the Morita equivalence between 2Vec G and 2Rep(G)-which we explicitly establish-the topological order can be realised as the Drinfel'd centre of either 2-category of operators; this is a realisation of the electromagnetic duality of the theory. Specialising to the case G = Z 2 , we recover tensor network representations that were recently introduced, as well as the relation between the electromagnetic duality of a pure Z 2 gauge theory and the Kramers-Wannier duality of a boundary Ising model.

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Boundary and domain wall theories of 2d generalized quantum double model

Jia¹,

Kaszlikowski²,

Sheng³

2023

J. High Energ. Phys.

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The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. The ribbon operators and the classification of topological excitations based on the representations of the quantum double of Hopf algebras are discussed. To generalize the model to a 2d surface with boundaries and surface defects, we present a systematic construction of the boundary Hamiltonian and domain wall Hamiltonian. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. The ribbon operator realization of boundary-bulk duality is also discussed. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.

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On tensor network representations of the (3+1)d toric code

Cited by 13 publications

References 63 publications

Long-range entanglement from measuring symmetry-protected topological phases

Long-range entanglement from measuring symmetry-protected topological phases

Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

Boundary and domain wall theories of 2d generalized quantum double model

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