## Description

Version of the Kitaev surface code on the two-dimensional torus, encoding two logical qubits. Being the first manifestation of the surface code, "toric code" is often an alternative name for the general construction. Twisted toric code [3; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions.

The stabilizers of the toric code 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).

We denote by \(\overline{X}_i,\overline{Z}_i\) the logical Pauli-\(X\) and Pauli-\(Z\) operator of the \(i\)-th logical qubit (with \(i\in\{1,2\}\)). They 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.

Coherent physical errors are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed via the Majorana mapping [4] as well as analytical bounds [5].

## Encoding

## Transversal Gates

## Code Capacity Threshold

## Threshold

## Realizations

## Parents

- Kitaev surface code — The toric code is the surface code on a 2D torus.
- \(D\)-dimensional twisted toric code — The \(D\)-dimensional twisted toric code reduces to the toric code for \(D=2\) and a square lattice.
- Hypergraph product (HGP) code — The toric code can be obtained from a hypergraph product of two repetition codes [33; Ex. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [33; Exam. 5].
- 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 [34; Appx. B].

## Child

- \([[4,2,2]]\) Four-qubit code — The \([[4,2,2]]\) code is the smallest toric code. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [35].

## Cousins

- Balanced product (BP) code — Twisted toric codes can be obtained from balanced products of cyclic graphs over a cyclic group [3; Fig. 8].
- Hansen toric code — The toric code is not to be confused with the CSS code constructed from a polynomial evaluation code on a toric variety [36].
- GKP-surface code — GKP codes have been concatenated with toric codes [37].
- Double-semion stabilizer code — The double semion phase also has a realization in terms of qubits [38] that can be compared to the toric 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 an odd number of loops appear with a \(-1\) coefficient for the double semion.

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## Page edit log

- Victor V. Albert (2024-05-05) — most recent

## Cite as:

“Toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/toric