## Description

A family of abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Toric code often either refers to the construction on the two-dimensional torus or is an alternative name for the general construction. The construction on surfaces with boundaries is often called the planar code [4].

The original construction can be naturally extended to arbitrary \(D\)-dimensional manifolds [5][6]. Given a cellulation, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces. Such extensions are often called the \(D\)-dimensional surface or \(D\)-dimensional toric codes.

The stabilizers of the surface code on the 2-dimensional torus are generated by star operators \(A_v\) and plaquette operators \(B_p\). Each star operator is a product of four Pauli-\(X\) operators on the edges adjacent to a vertex \(v\) of the lattice; each plaquette operator is a product of four Pauli-\(Z\) operators applied to the edges adjacent to a face, or plaquette, \(p\) of the lattice (Figure I).

The two-dimensional toric code encodes two logical qubits. We denote by \(\overline{X}_i,\overline{Z}_i\) the logical Pauli-\(X\) and Pauli-\(Z\) operator of the \(i\)-th logical qubit. They can are represented by strings of Pauli-\(X\) operators or Pauli-\(Z\) operators that wrap around the torus as shown in Figure I.

## Protection

Toric code on an \(L\times L\) torus is a \([[2L^2,2,L]]\) CSS code, and there exists a planar code with \([[L^2,1,L]]\) [7]. More generally, the code distance is related to the homology of the cellulation [8].

Coherent physical errors are expected to become incoherent logical errors after MWPM decoding; see corroborating numerical studies performed via the Majorana mapping [9] as well as analytical bounds [10].

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Realizations

## Notes

## Parents

- Calderbank-Shor-Steane (CSS) stabilizer code
- Clifford-deformed surface code (CDSC) — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
- Abelian topological code — When treated as ground states of the code Hamiltonian, the code states realize \(\mathbb{Z}_2\) topological order, a topological phase of matter that also exists in \(\mathbb{Z}_2\) lattice gauge theory [87]. Codewords correspond to ground state of the code Hamiltonian, and error operators correspond to spontaneous creation and annihilation of pairs of charges or vortices.

## Children

- Fractal surface code
- Higher-dimensional surface code
- Hyperbolic surface code
- Projective-plane surface code
- \([[4,2,2]]\) CSS code — \([[4,2,2]]\) code is the smallest toric code.

## Cousins

- Hypergraph product code — Planar (toric) code can be obtained from hypergraph product of two repetition (cyclic) codes ([88], Ex. 6).
- Quantum-double code — A quantum-double model with \(G=\mathbb{Z}_2\) is the surface code. Non-stabilizer surface-code states can be prepared by augmenting the code with a quantum double model [34].
- String-net code — String-net model reduces to the surface code when the category is the group \(\mathbb{Z}_2\).
- Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [9].
- Analog surface code — The analog surface code is an oscillator-into-oscillator version of the surface code.
- Color code — Color code is equivalent to surface code in several ways [89][90]. For example, the color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code.
- Double-semion code — There is a logical basis for the toric and double-semion codes where each codeword is a superposition of states corresponding to all noncontractible loops of a particular homotopy type. The superposition is equal for the toric code, whereas some loops appear with a \(-1\) coefficient for the double semion.
- Galois-qudit topological code — Surface code has been extended to Galois qudits.
- Haah cubic code — The energy of any partial implementation of code 1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.
- Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code.
- Honeycomb Floquet code — Measurement of each check operator of the honeycomb Floquet code involves two qubits and projects the state of the two qubits to a two-dimensional subspace, which we regard as an effective qubit. These effective qubits form a surface code on a hexagonal superlattice. Electric and magnetic operators on the embedded surface code correspond to outer logical operators of the Floquet code. In fact, outer logical operators transition back and forth from magnetic to electric surface code operators under the measurement dynamics. Inspired by this code, stabilizer measurement circuits consisting of two-body measurements have been designed for the surface code [91][92].
- Lifted-product (LP) code — A lifted product code for the ring \(R=\mathbb{F}_2[x,y]/(x^L-1,y^L-1)\) is the toric code.
- Modular-qudit surface code — The qudit surface code with \(q=2\) is the surface code.
- Raussendorf-Bravyi-Harrington (RBH) code — Without symmetry protection, one of 2D boundaries of the cubic RBH code is effectively a 2D toric code.
- Subsystem surface code
- Translationally invariant stabilizer code — Translation-invariant 2D qubit topological stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [93][94][95].

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## Cite as:

“Kitaev surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/surface