Toric code

Toric code[1,2]

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).

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.

Protection

Toric code on an \(L\times L\) torus is a \([[2L^2,2,L]]\) CSS code. The number of error patterns can be used to bound the ground-state energy of a \(\pm J\) Ising model [4].

Coherent physical errors are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed via the Majorana mapping [5] as well as analytical bounds [6].

Encoding

Lindbladian-based dissipative encoding for the toric code [7] that does not give a speedup relative to circuit-based encoders [8].

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 [9] (see also Ref. [10]), \(9.9\%\) under BP-OSD decoding [11], and \(8.9\%\) under GBP decoding [12]. The threshold under ML decoding corresponds to the value of a critical point of a two-dimensional random-bond Ising model (RBIM) on the Nishimori line [13,14], calculated to be \(10.94 \pm 0.02\%\) in Ref. [15], \(10.93(2)\%\) in Ref. [16], \(10.9187\%\) in Ref. [17], \(10.917(3)\%\) in Ref. [18], \(10.939(6)\%\) in Ref. [19], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [10]. The Bravyi-Suchara-Vargo (BSV) tensor network decoder [10] exactly solves the ML decoding problem under independent \(X,Z\) noise for the surface code and has complexity of order \(O(n^2)\); the decoder provides an efficient tensor-network contraction for the partition function resulting from the statistical mechanical mapping, which is known to be solvable for an Ising model on a planar graph [20]. ML decoding [14] is \(\#P\)-hard in general for the surface code [21]. Above values are for one type of noise only, and the ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\) [12; Table 1]. Thresholds for various lattices have been obtained in Refs. [22,23]. Depolarizing noise: between \(17\%\) and \(18.5\%\) under BSV tensor-network decoding [10], \(14\%\) under GBP decoding [12], \(16.5\%\) under recursive MWPM [24], between \(16\%\) and \(17.5\%\) under AMBP4 (depending on whether surface or toric code is considered) [25], and between \(15\%\) and \(16\%\) under RG [26], Markov-chain [27], or MWPM [28] decoding. The threshold under ML decoding corresponds to the value of a critical point of the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [29] (see also APS Physics viewpoint [30]).Erasure noise: \(50\%\) for square tiling [31,32]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [33].AD noise: \(39\%\) [34].Correlated noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) [35,36]. A threshold of \(10.04(6)\%\) under mildly correlated bit-flip noise is obtained in Ref. [37].The toric code has a measurement threshold of one [38].Coherent noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) called the complex-coupled Ashkin-Teller model [39,40]. Another statistical mechanical mapping has been studied for \(X\)-type noise channels interpolating between coherent and incoherent noise [41].

Threshold

The threshold under ML decoding with measurement errors corresponds to the value of a critical point of a three-dimensional random plaquette model [9,14].\(0.133\%\) for independent \(X,Z\) noise and faulty syndrome measurements using a local automaton decoder [42].Toric-code thresholds for post-selected QEC can be studied with statistical mechanical models [43].

Realizations

One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [44].

Cousins

Balanced product (BP) code— Twisted toric codes can be obtained from balanced products of cyclic graphs over a cyclic group [3; Fig. 8].
Repetition code— The toric code can be obtained from a hypergraph product of two repetition codes [45; Exam. 6].
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 [46].
GKP-surface code— GKP codes have been concatenated with toric codes [47].
Double-semion stabilizer code— The double semion phase also has a realization in terms of qubits [48] 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.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Homological code

Kitaev surface codeCDSC Twist-defect surface Lattice stabilizer Generalized homological-product QLDPC CSS Stabilizer Qubit Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

Kitaev surface code

The toric code is the surface code on a 2D torus.

\(D\)-dimensional twisted toric codeHomological Generalized homological-product QLDPC CSS Stabilizer Qubit Hamiltonian-based QECC Quantum

The \(D\)-dimensional twisted toric code reduces to the toric code for \(D=2\) and a square lattice.

Hypergraph product (HGP) codeGeneralized homological-product Lattice stabilizer QLDPC CSS Stabilizer Qubit Hamiltonian-based QECC Quantum

The toric code can be obtained from a hypergraph product of two repetition codes [45; Exam. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [45; Exam. 5].

Lifted-product (LP) codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

A lifted-product code for the ring \(R=\mathbb{F}_2[x,y]/(x^L-1,y^L-1)\) is the toric code [49; Appx. B].