## 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 [4–6]. Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.

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. The original planar code on a square-lattice patch with different boundary conditions on the vertical and horizontal edges is a \([[L^2+(L-1)^2,1,L]]\) CSS code [4]. Code size \(k = 2^{2g}\) for a torus of genus \(g\), and such higher genus surfaces have been investigated [7].

Coherent physical errors are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed via the Majorana mapping [8] as well as analytical bounds [9].

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Realizations

## Notes

## Parents

- Generalized surface code — The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.
- Clifford-deformed surface code (CDSC) — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
- Hypergraph product (HGP) code — Planar (toric) code can be obtained from hypergraph product of two repetition (cyclic) codes [112; Ex. 6].
- 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 [113; Appx. B].
- Modular-qudit surface code — The modular-qudit surface code for \(q=2\) reduces to the surface code.
- Galois-qudit topological code — The surface code has been extended to Galois qudits.

## Children

- \([[4,2,2]]\) CSS code — \([[4,2,2]]\) code is the smallest toric code.
- Rotated surface code — Rotated surface codes can be obtained using the same procedure as for the original surface codes but considering slightly different combinatorial surfaces [6,114] than those considered in the original proposal.

## Cousins

- Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [8].
- 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 surface code with a quantum double model [41].
- 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 [115–117].
- Dynamical automorphism (DA) code — One of the instantaneous stabilizer codes of the 2D DA color code are stacks of toric/surface codes
- Floquet color code — The ISG of the Floquet color code is the stabilizer group of one of three realizations of the \(\mathbb{Z}_2\) 2D surface 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 [85,86].
- Spacetime circuit code — Stabilizer generators of a spacetime code are called detectors in Refs. [118,119].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [120,121]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
- Color code — The 3D color code is equivalent to multiple decoupled copies of the 2D surface code [122–124]. Conversely, the 2D color code can condense to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three hexagonal directions to the stabilizer group and then taking the center of the resulting nonabelian group [125].
- Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code.
- Kitaev honeycomb code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code. This code can be obtained from the square-lattice surface code by gauging out the anyon \(em\) [126; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [126; Fig. 12].
- Subsystem surface code
- Three-fermion (3F) subsystem code — One version of the 3F subsystem code can be obtained from two copies of the square-lattice surface code by gauging out the anyons \(e_1m_1e_2\) and \(e_2m_2\) [126; Sec. 7.4].
- Double-semion stabilizer code — The double semion phase also has a realization in terms of qubits [127] that can be compared to the qubit surface code. There is a logical basis for both 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.

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