Toric code

Toric code[1,2]

Description

Version of the Kitaev surface code on a square lattice with periodic boundary conditions, 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 (Fig. 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 Fig. 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 in the toric code are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed by embedding each physical qubit into two fermions via the tetron code [5] as well as deriving analytical bounds [6]. More generally, there is a tensor-network routine that calculates the effective logical channel [7]

Encoding

Lindbladian-based dissipative encoding for the toric code [8] that does not give a speedup relative to circuit-based encoders [9].

Transversal Gates

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

Gates

Logical \(CX\) gate for the \([[12,2,3]]\) twisted toric code [10].

Code Capacity Threshold

Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding [11] (see also Ref. [12]), \(9.9\%\) under BP-OSD decoding [13], and \(8.9\%\) under GBP decoding [14]. 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 [15,16], calculated to be \(10.94 \pm 0.02\%\) in Ref. [17], \(10.93(2)\%\) in Ref. [18], \(10.9187\%\) in Ref. [19], \(10.917(3)\%\) in Ref. [20], \(10.939(6)\%\) in Ref. [21], and estimated to be between \(10.9\%\) and \(11\%\) in Ref. [12]. The Bravyi-Suchara-Vargo (BSV) tensor network decoder [12] 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 [22]. ML decoding [16] is \(\#P\)-hard in general for the surface code [23]. Above values are for one type of noise only, and the ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\) [14; Table 1]. Thresholds for various lattices have been obtained in Refs. [24,25]. Depolarizing noise: between \(17\%\) and \(18.5\%\) under BSV tensor-network decoding [12], \(14\%\) under GBP decoding [14], \(16.5\%\) under recursive MWPM [26], between \(16\%\) and \(17.5\%\) under AMBP4 (depending on whether surface or toric code is considered) [27], and between \(15\%\) and \(16\%\) under RG [28], Markov-chain [29], or MWPM [30] decoding. The threshold under ML decoding corresponds to the value of a critical point of the disordered eight-vertex Ising model, calculated to be \(18.9(3)\%\) [31] (see also APS Physics viewpoint [32]).Erasure noise: \(50\%\) for square tiling [33,34]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [35].AD noise: \(39\%\) [36].Correlated noise: the threshold under ML decoding corresponds to the value of a critical point of a particular random-bond Ising model (RBIM) [37,38]. A threshold of \(10.04(6)\%\) under mildly correlated bit-flip noise is obtained in Ref. [39].The toric code has a measurement threshold of one [40].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 [41,42]. Another statistical mechanical mapping has been studied for \(X\)-type noise channels interpolating between coherent and incoherent noise [43].Threshold under real-time geometrically local decoder based on introducing confining interactions between anyons [44].

Threshold

The threshold under ML decoding with measurement errors corresponds to the value of a critical point of a three-dimensional random plaquette model [11,16].\(0.133\%\) for independent \(X,Z\) noise and faulty syndrome measurements using a local automaton decoder [45].Toric-code thresholds for post-selected QEC can be studied with statistical mechanical models [46].

Realizations

Neutral atom arrays: One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes [47].

Cousins

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 [48; Appx. B].
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 [49; Exam. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [49; Exam. 5].
Tetron code— Coherent physical errors in the toric code are expected to become incoherent logical errors under syndrome measurement; see corroborating numerical studies performed by embedding each physical qubit into two fermions via the tetron code [5] as well as deriving analytical bounds [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 [50].
GKP-surface code— GKP codes have been concatenated with toric codes [51].

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 Hamiltonian-based QECC Quantum

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

Homological code

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

Parents

Kitaev surface code

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

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

The toric code can be obtained from a hypergraph product of two repetition codes [49; Exam. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [49; Exam. 5].