## 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). Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.

The construction on closed surfaces (surfaces with boundaries) is called the toric code (planar code [4,5]). There are two types of stabilizers one can put on edges, yielding open (a.k.a. rough) and closed (a.k.a. smooth) boundaries. A mixed boundary consists of an interleaving of the two stabilizer types [6].

## Protection

## Rate

## Encoding

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Threshold

## Realizations

## Notes

## Parents

- Twist-defect surface code — Twist-defect surface codes reduce to surface codes when there are no defects.
- Clifford-deformed surface code (CDSC) — CDSC codes are deformations of the surface code via constant-depth Clifford circuits that may not be CSS.
- Homological code — The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.
- Lift-connected surface (LCS) code — LCS codes consist of sparsely interconnected stacks of surface codes.
- Modular-qudit surface code — The modular-qudit surface code for \(q=2\) reduces to the surface code.
- Galois-qudit topological code — The Galois-qudit surface code for \(q=2\) reduces to the surface code.

## Children

- Rotated surface code — The lattice of the rotated surface code can be obtained by taking the medial graph of the surface code lattice (treated as a graph) and applying a similar procedure to construct the check operators [6,130][131; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [132,133].
- Toric code — The toric code is the surface code on a 2D torus.
- 2D hyperbolic surface code

## Cousins

- Layer code — Layer codes are combinations of constant-rate QLDPC codes with surface codes built using lattice surgery.
- Long-range enhanced surface code (LRESC) — LRESCs reduce to planar surface codes when a trivial LDPC code is used in the hypergraph product.
- La-cross code — La-cross codes at \(k=1\) yield the toric (planar surface) code and periodic (open) boundary conditions.
- Majorana stabilizer code — The Majorana mapping can be used to construct efficient algorithms for simulating rounds of error correction for the surface code [134].
- 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 [35].
- Hamiltonian-based code — While codewords of the surface code form ground states of the code's stabilizer Hamiltonian, they can also be ground states of other gapless Hamiltonians [135].
- Self-correcting quantum code — Various candidates for self-correcting quantum memories have been constructed by coupling neighboring anyons so as to prevent them from spreading [73,75,136–138]
- Asymmetric quantum code — The surface code on a hexagonal lattice is an asymmetric CSS code [139].
- 2D lattice stabilizer code — Translation-invariant 2D qubit lattice stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [140–142]. There exists an algorithm with which one can determine the fusion and braiding rules of a 2D translationally invariant qubit code, and decompose the given code into copies of the surface code [143].
- Dynamical automorphism (DA) code — One of the instantaneous stabilizer codes of the 2D DA color code are stacks of surface codes
- Floquet color code — The ISG of the Floquet color code is the stabilizer group of one of nine realizations of the \(\mathbb{Z}_2\) 2D surface code.
- X-cube Floquet code — The ISG of the X-cube Floquet code can be that of the X-cube model code or that of several decoupled surface codes.
- 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 the honeycomb Floquet code, various weight-two measurement schemes have been designed [108–110].
- Spacetime circuit code — Stabilizer generators of a spacetime code are called detectors in Refs. [144,145].
- Majorana surface code — The Majorana surface code is a Majorana stabilizer analogue of the surface code.
- Neural network code — Reinforcement learners can be used to optimize the geometry of the surface code to be more suited to a noise channel [146].
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH state encodes the temporal gate operations on the surface code into a third spatial dimension [115,147]. In addition, one of possible 2D boundaries of the RBH code is effectively a 2D toric code.
- Bivariate bicycle (BB) code — Bivariate bicycle codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators.
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code — Quantum Hamming codes can be concatened with surface codes [148].
- 2D color code — The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit [149–151]. This process can be viewed as an ungauging [152–154,154] of certain symmetries. 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 triangular directions to the stabilizer group and then taking the center of the resulting nonabelian group [155]. Both the surface and 2D color codes can be constructed from two distinct types of lattices, namely, 4-valent and 3-valent 3-colorable lattices, respectively [156].
- Generalized five-squares code — Decoding of five-squares codes leads to a mapping of these codes to two copies of the surface code [157,158].
- Heavy-hexagon code — Surface code stabilizers are used to measure the Z-type stabilizers of the code. There are various ways to embed the surface code into the heavy-hex lattice [159].
- 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 \(em\) [160; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [160; Fig. 12].
- 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 \(e_2m_2\) [160; Sec. 7.4].
- Subsystem surface code
- Fracton stabilizer code — Foliated type-I fracton phase codes can be grown by foliation, i.e., stacking copies of the 2D surface code; see [161; Eq. (D32)].

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## Page edit log

- Victor V. Albert (2023-03-29) — most recent
- Marcus P da Silva (2023-03-20)
- Victor V. Albert (2022-09-20)
- Victor V. Albert (2022-06-15)
- Tony Lau (2022-04-02)
- Hassan Shapourian (2022-04-01)
- Victor V. Albert (2022-02-15)
- Philippe Faist (2022-02-11)
- Victor V. Albert (2021-11-05)
- Philippe Faist (2021-11-03)
- Michael Vasmer (2021-11-02)

## Cite as:

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