Toric code[1,2]  


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).

Figure I: Stabilizer generators and logical operators of the toric code. The star operators \(A_v\) and the plaquette operators \(B_p\) generate the stabilizer group. The logical operators are strings that wrap around the torus.

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.


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].


Lindbladian-based dissipative encoding for the toric code [6] that does not give a speedup relative to circuit-based encoders [7].

Transversal Gates

Transversal logical Pauli gates correspond to Pauli strings on non-trivial loops of the torus.

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [8] (see also Ref. [9]), \(9.9\%\) under BP-OSD decoding [10], and \(8.9\%\) under GBP decoding [11]. The threshold under ML decoding corresponds to the value of a critical point of the two-dimensional random-bond Ising model (RBIM) on the Nishimori line [12,13], calculated to be \(10.94 \pm 0.02\%\) in Ref. [14], \(10.93(2)\%\) in Ref. [15], \(10.9187\%\) in Ref. [16], \(10.917(3)\%\) in Ref. [17], \(10.939(6)\%\) in Ref. [18], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [9]. Above values are for one type of noise only, and the ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\).Depolarizing noise: between \(17\%\) and \(18.5\%\) under tensor-network decoding [9], \(14\%\) under GBP decoding [11], \(16.5\%\) under recursive MWPM [19], between \(16\%\) and \(17.5\%\) under AMBP4 (depending on whether surface or toric code is considered) [20], and between \(15\%\) and \(16\%\) under RG [21], Markov-chain [22], or MWPM [23] decoding. The threshold under ML decoding corresponds to the value of a critical point of the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [24] (see also APS Physics viewpoint [25]).Erasure noise: \(50\%\) for square tiling [26]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [27].Correlated noise: \(10.04(6)\%\) under mildly correlated bit-flip noise [28].The toric code has a measurement threshold of one [29].


\(0.133\%\) for independent \(X,Z\) noise and faulty syndrome measurements using a cellular automaton decoder [30].


One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [31].



  • \([[4,2,2]]\) CSS 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 [34].


  • Balanced product (BP) code — Twisted toric codes can be obtained from balanced products of cyclic graphs over a cyclic group [3; Fig. 8].
  • Double-semion stabilizer code — The double semion phase also has a realization in terms of qubits [35] 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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“Toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
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“Toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.