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
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 BSV 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 [26,27]. There is an inverse relationship between coordination number of the syndrome graph, with the threshold corresponding to a percolation transition [28].Correlated noise: \(10.04(6)\%\) under mildly correlated bit-flip noise [29].The toric code has a measurement threshold of one [30].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 [31,32].
Threshold
\(0.133\%\) for independent \(X,Z\) noise and faulty syndrome measurements using a cellular automaton decoder [33].
Realizations
One cycle of syndrome readout on 19-qubit planar and 24-qubit toric codes realized in two-dimensional Rydberg atomic arrays [34].
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 [35; Ex. 6]. Other hypergraph products of two repetition codes yield the related \([[2d^2-2d+1,1,d]]\) CSS code family [35; 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 [36; 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 [37].
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 [38].
- GKP-surface code — GKP codes have been concatenated with toric codes [39].
- Double-semion stabilizer code — The double semion phase also has a realization in terms of qubits [40] 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.
References
- [1]
- A. Y. Kitaev, “Quantum computations: algorithms and error correction”, Russian Mathematical Surveys 52, 1191 (1997) DOI
- [2]
- A. Yu. Kitaev, “Quantum Error Correction with Imperfect Gates”, Quantum Communication, Computing, and Measurement 181 (1997) DOI
- [3]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
- [4]
- S. Bravyi et al., “Correcting coherent errors with surface codes”, npj Quantum Information 4, (2018) arXiv:1710.02270 DOI
- [5]
- J. K. Iverson and J. Preskill, “Coherence in logical quantum channels”, New Journal of Physics 22, 073066 (2020) arXiv:1912.04319 DOI
- [6]
- J. Dengis, R. König, and F. Pastawski, “An optimal dissipative encoder for the toric code”, New Journal of Physics 16, 013023 (2014) arXiv:1310.1036 DOI
- [7]
- R. König and F. Pastawski, “Generating topological order: No speedup by dissipation”, Physical Review B 90, (2014) arXiv:1310.1037 DOI
- [8]
- C. Wang, J. Harrington, and J. Preskill, “Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory”, Annals of Physics 303, 31 (2003) arXiv:quant-ph/0207088 DOI
- [9]
- S. Bravyi, M. Suchara, and A. Vargo, “Efficient algorithms for maximum likelihood decoding in the surface code”, Physical Review A 90, (2014) arXiv:1405.4883 DOI
- [10]
- J. Roffe et al., “Decoding across the quantum low-density parity-check code landscape”, Physical Review Research 2, (2020) arXiv:2005.07016 DOI
- [11]
- J. Old and M. Rispler, “Generalized Belief Propagation Algorithms for Decoding of Surface Codes”, Quantum 7, 1037 (2023) arXiv:2212.03214 DOI
- [12]
- H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
- [13]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [14]
- A. Honecker, M. Picco, and P. Pujol, “Universality Class of the Nishimori Point in the 2D±JRandom-Bond Ising Model”, Physical Review Letters 87, (2001) arXiv:cond-mat/0010143 DOI
- [15]
- F. Merz and J. T. Chalker, “Two-dimensional random-bond Ising model, free fermions, and the network model”, Physical Review B 65, (2002) arXiv:cond-mat/0106023 DOI
- [16]
- M. Ohzeki, “Locations of multicritical points for spin glasses on regular lattices”, Physical Review E 79, (2009) arXiv:0811.0464 DOI
- [17]
- F. Parisen Toldin, A. Pelissetto, and E. Vicari, “Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model”, Journal of Statistical Physics 135, 1039 (2009) arXiv:0811.2101 DOI
- [18]
- S. L. A. de Queiroz, “Location and properties of the multicritical point in the Gaussian and±JIsing spin glasses”, Physical Review B 79, (2009) arXiv:0902.4153 DOI
- [19]
- A. deMarti iOlius et al., “Performance enhancement of surface codes via recursive minimum-weight perfect-match decoding”, Physical Review A 108, (2023) arXiv:2212.11632 DOI
- [20]
- K.-Y. Kuo and C.-Y. Lai, “Exploiting degeneracy in belief propagation decoding of quantum codes”, npj Quantum Information 8, (2022) arXiv:2104.13659 DOI
- [21]
- G. Duclos-Cianci and D. Poulin, “Fast Decoders for Topological Quantum Codes”, Physical Review Letters 104, (2010) arXiv:0911.0581 DOI
- [22]
- A. Hutter, J. R. Wootton, and D. Loss, “Efficient Markov chain Monte Carlo algorithm for the surface code”, Physical Review A 89, (2014) arXiv:1302.2669 DOI
- [23]
- D. S. Wang et al., “Threshold error rates for the toric and surface codes”, (2009) arXiv:0905.0531
- [24]
- H. Bombin et al., “Strong Resilience of Topological Codes to Depolarization”, Physical Review X 2, (2012) arXiv:1202.1852 DOI
- [25]
- D. Gottesman, “Keeping One Step Ahead of Errors”, Physics 5, (2012) DOI
- [26]
- T. M. Stace, S. D. Barrett, and A. C. Doherty, “Thresholds for Topological Codes in the Presence of Loss”, Physical Review Letters 102, (2009) arXiv:0904.3556 DOI
- [27]
- T. M. Stace and S. D. Barrett, “Error correction and degeneracy in surface codes suffering loss”, Physical Review A 81, (2010) arXiv:0912.1159 DOI
- [28]
- N. Nickerson and H. Bombín, “Measurement based fault tolerance beyond foliation”, (2018) arXiv:1810.09621
- [29]
- C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021) arXiv:1809.10704 DOI
- [30]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [31]
- J. Behrends and B. Béri, “Statistical mechanical mapping and maximum-likelihood thresholds for the surface code under generic single-qubit coherent errors”, (2024) arXiv:2410.22436
- [32]
- Y. Bao and S. Anand, “Phases of decodability in the surface code with unitary errors”, (2024) arXiv:2411.05785
- [33]
- N. P. Breuckmann et al., “Local Decoders for the 2D and 4D Toric Code”, (2016) arXiv:1609.00510
- [34]
- D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI
- [35]
- A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1202.0928 DOI
- [36]
- P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
- [37]
- B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
- [38]
- J. P. Hansen, “Toric Codes, Multiplicative Structure and Decoding”, (2017) arXiv:1702.06569
- [39]
- C. Vuillot et al., “Quantum error correction with the toric Gottesman-Kitaev-Preskill code”, Physical Review A 99, (2019) arXiv:1810.00047 DOI
- [40]
- M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
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