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arXiv:2212.01021

Groupoid Toric Codes

Pramod Padmanabhan, Indrajit Jana

Published Dec 2, 2022• 0 citations
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Abstract

The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an inverse. We show that a consistent system can be constructed for an arbitrary groupoid and analyze the simplest example that can be seen as the analog of the Abelian $\mathbb{Z}_2$ toric code. We find several exactly solvable models that have fracton-like features which include an extensive ground state degeneracy and excitations that are either immobile or have restricted mobility. Among the possibilities we study in detail the one where the ground state degeneracy scales as $2\times 2^{N_v}$, where $N_v$ is the number of vertices in the lattice. The origin of this degeneracy can be traced to loop operators supported on both contractible and non-contractible loops. In particular, different non-contractible loops, along the same direction on a torus, result in different ground states. This is an exponential increase in the number of logical qubits that can be encoded in this code. Moreover the face excitations in this system are deconfined, free to move without an energy cost along certain directions of the lattice, whereas in certain other directions their movement incurs an energy cost. This places a restriction on the types of loop operators that contribute to the ground state degeneracy. The vertex excitations are immobile. The results are also extended to the groupoid analogs of Abelian $\mathbb{Z}_N$ toric codes.

DOI
10.48550/arxiv.2212.01021

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The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an inverse. We show that a consistent system can be constructed for an arbitrary groupoid and analyze the simplest example that can be seen as the analog of the Abelian Z2\mathbb{Z}_2Z2​ toric code. We find several exactly solvable models that have fracton-like features which include an extensive ground state degeneracy and excitations that are either immobile or have restricted mobility. Among the possibilities we study in detail the one where the ground state degeneracy scales as 2×2Nv2\times 2^{N_v}2×2Nv​, where NvN_vNv​ is the number of vertices in the lattice. The origin of this degeneracy can be traced to loop operators supported on both contractible and non-contractible loops. In particular, different non-contractible loops, along the same direction on a torus, result in different ground states. This is an exponential increase in the number of logical qubits that can be encoded in this code. Moreover the face excitations in this system are deconfined, free to move without an energy cost along certain directions of the lattice, whereas in certain other directions their movement incurs an energy cost. This places a restriction on the types of loop operators that contribute to the ground state degeneracy. The vertex excitations are immobile. The results are also extended to the groupoid analogs of Abelian ZN\mathbb{Z}_NZN​ toric codes.
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FINDING: The toric code is a stabilizer Hamiltonian on a 2D square lattice whose ground-state degeneracy and logical operators are governed by the homology of the torus — specifically, non-contractible loops of X and Z Pauli operators, with the p4m wallpaper group (square lattice, order-8 dihedral symmetry) as the underlying spatial symmetry. | MATH: Hamiltonian H=−∑vAv−∑pBp H = -\sum_v A_v - \sum_p B_p H=−∑v​Av​−∑p​Bp​, where Av=∏i∈vXi A_v = \prod_{i \in v} X_i Av​=∏i∈v​Xi​, Bp=∏i∈pZi B_p = \prod_{i \in p} Z_i Bp​=∏i∈p​Zi​. Ground-state degeneracy on genus-ggg surface = 4g4^g4g (for torus, g=1g=1g=1, degeneracy = 4). Logical operators: Xˉ1,Zˉ1,Xˉ2,Zˉ2 \bar{X}_1, \bar{Z}_1, \bar{X}_2, \bar{Z}_2 Xˉ1​,Zˉ1​,Xˉ2​,Zˉ2​ — non-contractible loops along the two independent cycles of the torus. These satisfy XˉiZˉj=(−1)δijZˉjXˉi \bar{X}_i \bar{Z}_j = (-1)^{\delta_{ij}} \bar{Z}_j \bar{X}_i Xˉi​Zˉj​=(−1)δij​Zˉj​Xˉi​. Anyon excitations: e e e (electric, from AvA_vAv​ violation) and m m m (magnetic, from BpB_pBp​ violation), with mutual braiding phase eiπ=−1 e^{i\pi} = -1 eiπ=−1. The code distance d=L d = L d=L (linear size of lattic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We propose a lattice spin model on a cubic lattice that shares many of the properties of the 3D toric code and the X-cube fracton model. The model, made of Z3{\mathbb{Z}}_{3}Z3​ degrees of freedom at the links, has the vertex, the cube, and the plaquette terms. Being a stabilizer code the ground states are exactly solved. With only the vertex and the cube terms present, we show that the ground state degeneracy (GSD) is ${3}^{{L}^{3}+3L\ensuremath{-}1}$, where LLL is the linear dimension of the cubic lattice. In addition to fractons, there are free vertex excitations we call the freeons. With the addition of the plaquette terms, GSD is vastly reduced to 33{3}^{3}33, with fracton, fluxon, and freeon excitations, among which only the freeons are deconfined. The model is called the AB model if only the vertex (Av)({A}_{v})(Av​) and the cube (Bc)({B}_{c})(Bc​) terms are present and the ABC model if in addition the plaquette terms (Cp)({C}_{p})(Cp​) are included. The AC model consisting of vertex and plaquette terms is the Z3{\mathbb{Z}}_{3}Z3​ 3D toric code. The extensive GSD of the AB model derives from the existence of both local and nonlocal logical operators that connect different ground states. The latter operators are identical to the logical operators of the Z3{\mathbb{Z}}_{3}Z3​ X-cube model. Fracton excitations are immobile and accompanied by the creation of fluxons---plaquettes having nonzero flux. In the ABC model, such fluxon creation costs energy and ends up confining the fractons. Unlike past models of fractons, vertex excitations are free to move in any direction and pick up a nontrivial statistical phase when passing through a fluxon or a fracton cluster.
We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group GGG, a choice of an Abelian normal subgroup NNN, and a choice of foliation structure. These hybrid fracton orders, examples of which were introduced in a previous paper [Tantivasadakarn et al., Phys. Rev. B 103, 245136 (2021)], can also host immobile, pointlike excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate between a conventional, three-dimensional GGG gauge theory and a pure fracton order, by varying the choice of normal subgroup NNN. We demonstrate that certain universal data of the topological excitations and their mobilities are directly related to the choice of GGG and NNN, and also present complementary perspectives on these orders: certain orders may be obtained by gauging a global symmetry which enriches a particular fracton order, by either fractionalizing on or permuting the excitations with restricted mobility, while certain hybrid orders can be obtained by condensing excitations in a stack of initially decoupled, two-dimensional topological orders.
FINDING: The toric code is a topological quantum error-correcting code whose logical operators are non-contractible loops on a torus, with ground-state degeneracy determined by the genus of the surface and the group cohomology of the underlying lattice symmetry (p4m wallpaper group). | MATH: The toric code Hamiltonian H=−∑vAv−∑pBp H = -\sum_v A_v - \sum_p B_p H=−∑v​Av​−∑p​Bp​, where Av=∏i∈vσix A_v = \prod_{i \in v} \sigma^x_i Av​=∏i∈v​σix​ and Bp=∏i∈pσiz B_p = \prod_{i \in p} \sigma^z_i Bp​=∏i∈p​σiz​. Logical operators XL,ZL X_L, Z_L XL​,ZL​ are non-contractible loops; on a genus-ggg surface, degeneracy =4g = 4^g =4g. For the p4m wallpaper group (square lattice with reflection/rotation symmetries), the code space is invariant under the group action; the anyonic excitations (e, m, ε) obey Z2×Z2 \mathbb{Z}_2 \times \mathbb{Z}_2 Z2​×Z2​ fusion rules, and the braiding phase is eiπ=−1 e^{i\pi} = -1 eiπ=−1. | CONNECTION: The p4m group is the full symmetry group of the square lattice — its point group is D4 D_4 D4​ (order 8), with generators r r r (90° rotation) and s s s ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We construct two dimensional lattice models from transfer matrices of lattice gauge theories with discrete gauge groups. These transfer matrices are built out of local operators acting on links, vertices and plaquettes and are parametrized by the center of the gauge group algebra and its dual. They contain the familiar 2D quantum double models for a particular choice of parameters, which includes the well studied example of the toric code. However for more general choices of parameters the transfer matrix contains operators acting on links which can also be thought of as perturbations to the quantum double model driving it out of its topological phase towards a paramagnetic phase. These perturbations can be thought of as magnetic fields added to the system which destroy the exact solvability of the quantum double model. Nevertheless from the same transfer matrix with perturbations we exhibit exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. The Abelian cases are shown to be in the quantum double phase whereas the non-Abelian phases are shown to be in a modified phase of the corresponding quantum double phase. This is shown by working with the groups Zn and S3 for the Abelian and non-Abelian cases respectively. The quantum phases are found by studying the excitations of these systems. The fusion rules and the statistics of these anyons indicate the quantum phases of these models. The implementation of these models can possibly improve the use of quantum double models for fault tolerant quantum computation. We then construct theories which arise from transfer matrices that are not the transfer matrices of lattice gauge theories. In particular we show that for the Z2 case this contains the double semion model. More generally for other discrete groups these transfer matrices contain the twisted quantum double models. These transfer matrices can be thought of as being obtained by introducing extra parameters into the transfer matrix of lattice gauge theories. These parameters are central elements belonging to the tensor products of the algebra and its dual and are associated to vertices and volumes of the three dimensional lattice. As in the case of the lattice gauge theories we construct the operators creating the excitations in this case and study their braiding and fusion properties.