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.

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 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].

Threshold

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

Realizations

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

Parents

Child

  • \([[4,2,2]]\) 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].

Cousins

  • 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.

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]
N. Nickerson and H. Bombín, “Measurement based fault tolerance beyond foliation”, (2018) arXiv:1810.09621
[28]
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
[29]
D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[30]
N. P. Breuckmann et al., “Local Decoders for the 2D and 4D Toric Code”, (2016) arXiv:1609.00510
[31]
D. Bluvstein et al., “A quantum processor based on coherent transport of entangled atom arrays”, Nature 604, 451 (2022) arXiv:2112.03923 DOI
[32]
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
[33]
P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
[34]
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
[35]
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

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: toric

Cite as:
“Toric code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/toric
BibTeX:
@incollection{eczoo_toric, title={Toric code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/toric} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/toric

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml.