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For a bosonic Abelian anyon theory, a Lagrangian subgroup of Abelian anyons is a subgroup of types of anyons closed under anyon fusion, with trivial topological spin for every member and trivial mutual braiding among members. Every anyon outside the subgroup must braid nontrivially with at least one member; pairwise compatibility alone is insufficient.

These conditions describe a maximal condensable collection for an appropriate fully gapped boundary. In the toric code, the vacuum together with the electric anyon gives one choice, and the vacuum together with the magnetic anyon gives another. Electric and magnetic anyons cannot both belong to the same collection because their mutual braiding is nontrivial. Additional physical context, including a nonchiral bulk and any imposed symmetry, matters when using this data to construct a boundary.^[ref]

Use the standard toric code conventionAn open $Z$ string anticommutes with $A_v$ at its endpoints and creates the electric particles $e$ there; an open dual-lattice $X$ string creates magnetic particles $m$ at its endpoint plaquettes. Joining two strings cancels every shared Pauli because $X_j^2=Z_j^2=1$. Bringing two identical endpoints together therefore annihilates them:

Taking $e$ around $m$ makes its closed $Z$ string cross the $X$ string ending on $m$ once. At that crossing $ZX=-XZ$, so the state acquires $-1$. The same argument with the roles reversed givesIdentical $Z$ strings commute with one another, as do identical $X$ strings. Their exchanges can be deformed without a crossing between anticommuting operators, so $e^{i{\theta }_{ee}}=e^{i{\theta }_{mm}}=1$. Thus $e$ and $m$ are bosons and are mutual semions, as summarized by the surface-code anyon model.

On the square lattice the PEPO intertwines the local operators asIts image also obeys the automatic zero-flux constraints $B_p={\prod }_{e\in \partial p}{\tilde{Z}}_e=1$. Thus the transverse-field Ising Hamiltonian maps to a ${\mathbb{Z}}_2$ lattice-gauge Hamiltonian generated by edge fields and stars in the zero-flux sector.

At the commuting-projector fixed point, promoting the image constraints to energetic terms givesthe toric code Hamiltonian. Conversely, within a simply connected zero-flux sector one may solve ${\tilde{Z}}_{uv}=X_u{X}_v$ and recover vertex Ising variables. On a torus, the remaining noncontractible loop eigenvalues correspond to the different Ising boundary twists, which accounts for the toric code's topological ground-state sectors.

Let $Z_e$ denote the dual edge variable. The PEPO implements the domain-wall mapBecause edge domain walls arise from vertex spins, their product around every plaquette is constrained:Thus the transverse-field Ising operators map to the ${\mathbb{Z}}_2$ lattice-gauge operators, while the image is projected into the positive eigenspace of all $B_p$. At the commuting-projector fixed point, adjoining these automatic projectors givesthe toric code Hamiltonian. Conversely, solving the zero-flux constraint writes $Z_e=Z_u{Z}_v$ locally and recovers the Ising variables, establishing the duality on the supported symmetry sectors.

The original Ising symmetry is the global spin flipThe dual system has a one-form symmetry generated by products of edge Pauli operators along closed noncontractible loops; equivalently, closed Wilson loops label its electric and magnetic topological sectors.

The PEPO annihilates the nonsymmetric Ising sector because each virtual index is summed with the global parity constraint. On the dual side it has support only on configurations obeying all contractible zero-flux constraints and, for a fixed untwisted PEPO, one choice of noncontractible loop eigenvalues. Therefore the duality is invertible only after restricting both Hilbert spaces to corresponding symmetry sectors.

On a torus the toric code has four ground states distinguished by two independent noncontractible loop eigenvalues. A single untwisted duality operator reaches only one of them; inserting the two possible Ising twists supplies the other three. This explains why the global-symmetry Ising description and topologically degenerate toric-code description do not contradict each other.

A topological stabilizer code is a quantum stabilizer code whose local commuting checks are arranged on a spatial lattice and whose logical information is encoded in global structure. Local errors create detectable syndrome patterns; a nontrivial logical operator typically requires a sufficiently extended chain or surface.

The toric code is a central example: products of Pauli operators around vertices and faces define the stabilizers, while noncontractible loops act on encoded qubits. Protection depends on code distance and an appropriate decoder, not on topology alone guaranteeing error-free storage.

The toric code is a stabilizer code with a qubit on every edge of a two-dimensional lattice on a torus. A vertex check multiplies Pauli operators $X$ on its incident edges, while a face check multiplies Pauli operators $Z$ around its boundary:The checks commute because a vertex and face share either zero or two edges. The code is their common $+1$ eigenspace, equivalently the space of ground states of

On the torus, ${\prod }_s{A}_s={\prod }_p{B}_p=I$ gives two redundant checks. The remaining independent checks leave a four-dimensional code space, encoding two logical qubits. Products of Pauli operators around noncontractible loops act on that information. ^[ref]

A violated vertex or face check is an excitation. Open strings create pairs of excitations at their endpoints; closed noncontractible strings can change a logical state without creating local excitations. The electric and magnetic excitations are bosons with mutual full-braiding phase $-1$, while their composite is a fermion. ^[ref]

For an invertible K-matrix, an Abelian Chern--Simons theory hastopological ground states on a spatial torus. On a closed orientable surface of genus of a surface $g$, the corresponding count is $|\det K{|}^g$. ^[ref]

The determinant counts the anyon classes in ${\mathbb{Z}}^N/K{\mathbb{Z}}^N$: the fundamental cell of the sublattice $K{\mathbb{Z}}^N$ has volume $|\det K|$ relative to the unit integer lattice. Transport around intersecting torus cycles gives the noncommuting Wilson-loop algebra of an Abelian Chern--Simons theory, which distinguishes these states.

For the toric code matrixthe quotient has four classes, represented by $(0,0),(1,0),(0,1),(1,1)$, so the torus count is four. These represent different global sectors rather than four locally distinguishable arrangements. In a finite microscopic realization, local perturbations can produce small finite-size splitting; the topological count describes the protected low-energy sector in the large-system limit.