Topological code[1]
Description
A code whose codewords form the ground-state or low-energy subspace of a (typically geometrically local) code Hamiltonian realizing a topological phase. A topological phase may be bosonic or fermionic, i.e., constructed out of underlying subsystems whose operators commute or anti-commute with each other, respectively. Unless otherwise noted, the phases discussed are bosonic.
2D topological codes
The physical Hilbert space of 2D topological codes consists of \(n\) subsystems which lie on edges, vertices, or faces of a tesselation of a 2D surface \(\Sigma^2\). 2D topological order requires weight-four (four-body) Hamiltonian terms, i.e., it cannot be stabilized via weight-two (two-body) or weight-three (three-body) terms on nearly Euclidean geometries of qubits or qutrits [2–4].
For subsystems with finite local dimension, topological phases are defined by their anyons [5–8], which are local bulk excitations of the code Hamiltonian defined on a lattice; see Refs. [9–11] for more rigorous formulations.
Anyons are created in pairs by local operators, and two anyons lie in the same superselection sector if a local operator can convert one anyon into the other. Each superselection sector is assumed to be labeled by one anyon type, and local operators cannot change superselection sectors.
Anyons can braid with themselves, with their exchange statistics (a.k.a. topological spin) defined by phases \(\theta(a)\in U(1)\) obtained by exchanging two anyons of each type \(a\). They can also braid with each other, a process defined by braiding relations \(B(a,b)\) for an anyon pair \(a,b\). An anyon theory is called non-modular or pre-modular if there exists an anyon \(a\) that braids trivially with all anyons.
Anyons \(a\) and \(b\) can also fuse with each other, meaning that one considers both anyons as one anyon \(ab\) and decomposes \(ab\) into the anyons representing each superselection sector according to the anyons' fusion rules. For example, two anyons \(a,b\) may fuse to the trivial (i.e., vacuum) anyon \(1\), \(ab=1\), meaning that the composite excitation \(ab\) is indistinguishable from the case of no excitation. Anyon fusion for many anyon theories is equivalent to a truncated Kronecker product of irreps of deformed Lie groups (i.e., "quantum groups"), which is a combination of the ordinary Kronecker product and a restriction to only decomposable representations [12; Secs. 4.4-4.5].
The exchange statistics and fusion rules of anyons cannot be arbitrary and have to satisfy certain consistency relations. Admissible exchange and fusion data are characterized by a unitary braided fusion category.
Each anyon \(a\) has a quantum dimension \(d_a\) associated with it. The quantum dimensions add up to the total quantum dimension \(D\), \begin{align} \sum_{a}d_{a}^{2}=D^{2}~. \tag*{(1)}\end{align} Quantum dimensions do not correspond to dimensions of vector spaces and may not be integer-valued.
An anyon theory that does not admit gapped boundaries (when put on a manifold with boundaries) is called chiral; otherwise, it is non-chiral or gapped. Chiral topological phases admit a nonzero value of the chiral central charge \(c_{-}\). A generalization [1] of the Gauss-Milgram sum rule for an anyon theory \(A\) admitting \(|A|\) anyon types, \begin{align} \frac{1}{\sqrt{|A|}}\sum_{a\in A}d_{a}^{2}\theta_{a}=De^{i\frac{2\pi}{8}c_{-}}~, \tag*{(2)}\end{align} relates the chiral central charge (modulo 8) to the exchange statistics and quantum dimensions. Gapped anyon theories admit a Lagrangian subalgebra [13–15].
The topological entanglement entropy [16,17] of a 2D topological code is a measure of the entanglement between a region and its complement; this measure extracts the total quantum dimension \(D\) of the phase. Other functions of code states extract the topological \(S\)-matrix [18,19] and the chiral central charge \(c_-\) [20]. No observable can distinguish topological order from product states [21; Appx. I].
There is no 1D bosonic topological order at nonzero temperature [22], and no commuting-projector model with nontrivial 2D topological order at nonzero temperature [23] (see also Ref. [24]). General thermal states are separable above a sufficiently high temperature [25].
3D and higher-dimensional topological codes
The physical Hilbert space of 3D topological codes consists of \(n\) subsystems which lie on edges, vertices, or faces of a tesselation of a 3D surface. 3D topological phases can have point-like and loop-like excitations, with the latter being created in pairs by 2D operators acting on subsystems supported on a plane or, more generally, a "membrane".
In the case when all point-like excitations satisfy bosonic braiding statistics, the topological phase can be realized by a Dijkgraaf-Witten gauge theory. Such cases are thus characterized by the gauge theory's underlying data, a finite group and a cohomological cycle (i.e., cocycle) [26,27].
Phases with fermionic point-like excitations are examples of beyond-group-cohomology phases [28]. They have been classified [29], and some of them can be described by a two-gauge theory [30].
The classification of 4D [31] and higher-dimensional [32–34] topological phases is ongoing.
Protection
Geometrically local 2D commuting-projector topological code Hamiltonians satisfy the two topological quantum order (TQO) conditions, TQO-1 and TQO-2 [35–39].
TQO conditions: The TQO-1 condition states that the distance of the ground-state-subspace code is macroscopic, i.e., grows as a positive power of the lattice size [36]. The TQO-2 condition relates the ground states of restrictions of the Hamiltonian to some geometrically local region to those of the full Hamiltonian. Let \(\Pi_{N(X)}\) be the ground-state subspace projector of the Hamiltonian that includes all terms with at least some support on a geometrically local region \(X\), with \(N(X)\) consisting of the smallest region containing the support of all included terms. TQO-2 states that any operator \(O_X\) that annihilates the codespace projector \(\Pi\) also has to annihilate the local projector \(\Pi_{N(X)}\), \begin{align} O_{X}\Pi=0\quad\Rightarrow\quad O_{X}\Pi_{N(X)}=0~. \tag*{(3)}\end{align} This condition implies that any operator supported solely on \(X\) cannot distinguish the global projector from the local one [35,40].
A notion of topological order generalizing both the cleaning lemma and the TQO conditions is homogeneous topological order [41]. Related topological order definitions include equivalence under course-graining (i.e., renormalization group) [42,43]. See [41; Sec. 4] for a discussion.
Certain topological codes have nontrivial codespace complexity [44].
Rate
The logical dimension \(K\) of 2D topological codes described by unitary modular fusion categories depends on the type of manifold \(\Sigma^2\) that is tesselated to form the many-body system. For closed orientable manifolds [45,46], \begin{align} K=\sum_{a\in A}\left(d_{a}/D\right)^{\chi(\Sigma^{2})}~, \tag*{(4)}\end{align} and a generalization of the formula to the non-orientable case can be found in Ref. [47].Encoding
A depth of order \(\Omega(L)\) is necessary for a unitary circuit to initialize in a 2D topologically ordered state using geometrically local gates on an \(L\times L\) lattice [48,49], irrespective of whether the ground state admits Abelian or non-Abelian anyonic excitations. However, only a finite-depth circuit and one round of measurements is required for non-Abelian topological orders with a Lagrangian subgroup [50].Algorithm that takes in reduced density matrices and outputs a circuit preparing the global state in polynomial time [51].Gates
Ising anyon braiding and fusion were studied in a phenomenological model that was the first to study error correction with non-Abelian anyons [52].Codes with non-Abelian anyons present two complications. First, their anyons are not always represented as tensor-product unitary operators and thus cannot be corrected by such operations. Second, fusing such anyons can yield more than one outcome, complicating decoding algorithms [53].Notes
Ref. [55–60][54; Appx. F] for introductions to topological phases.See AnyonWiki for lists of categories relevant to anyons.Cousins
- Cluster-state code— There exist necessary and sufficient conditions for a family of cluster states to exhibit the TQO-1 property [61].
- String-net code— String-net codes realize 2D topological phases based on unitary fusion categories. Any 2D many-body state satisfying the entanglement bootstrap axioms can be mapped into the ground-state subspace of a string-net model via a constant-depth unitary circuit [62]. Different string-net models that have Morita-equivalent input fusion categories describing the same topological order are dual to each other [63].
- Quantum LDPC (QLDPC) code— Topological codes are not generally defined using Pauli strings or their qudit and bosonic generalizations. However, for appropriate tesselations, the codespace is the ground-state subspace of a geometrically local Hamiltonian. In this sense, topological codes are QLDPC codes. Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TQO conditions, meaning that a notion of a phase can be defined [35–39]. This notion can be extended to semi-hyperbolic manifolds [40] and non-geometrically local QLDPC codes exhibiting check soundness [64] (see also [65]).
- Approximate quantum error-correcting code (AQECC)— In the case of topological codes, the Petz infidelity is related to the topological entanglement entropy [66].
- Monitored random-circuit code— Topological order can be generated in 2D monitored random circuits [67].
- Commuting-projector Hamiltonian code— Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TQO conditions, meaning that a notion of a phase can be defined [35–39]. This notion can be extended to semi-hyperbolic manifolds [40] and non-geometrically local QLDPC codes exhibiting check soundness [64] (see also [65]). Hamiltonians admitting a Peierls condition are stable to off-diagonal perturbations [68]. 2D states admitting strict area-law entanglement necessarily have commuting-projector Hamiltonians [69]. 2D topological order on qubit manifolds requires weight-four (four-body) commuting-projector Hamiltonian terms, i.e., it cannot be stabilized via weight-two (two-body) or weight-three (three-body) terms on nearly Euclidean geometries of qubits or qutrits [2–4].
- Frustration-free Hamiltonian code— Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the TQO conditions), meaning that a notion of a phase can be defined [37,70].
- Local Haar-random circuit qubit code— Local Haar-random codewords, like topological codewords, are locally indistinguishable [71].
- Eigenstate thermalization hypothesis (ETH) code— ETH codewords, like topological codewords, are locally indistinguishable.
- Fusion-based quantum computing (FBQC) code— Arbitrary topological codes can be created using FBQC, as can topological features such as defects and boundaries, by modifying fusion measurements or adding single qubit measurements.
- Quantum expander code— Quantum expander codes realize topological quantum spin glass order [72].
- Dinur-Lin-Vidick (DLV) code— DLV codes are expected to realize topological quantum spin glass order [72].
- Kitaev honeycomb code— The Kitaev honeycomb model realizes all anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) anyon theory [54][73; Footnote 25]. This includes the (non-Abelian) Ising-anyon topological order [54] (a.k.a. \(p+ip\) superconducting phase [74]) as well as Abelian \(\mathbb{Z}_2\) topological order.
- Fracton stabilizer code— Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. Fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory with defects [75,76].
- Expander LP code— Expander lifted-product codes are expected to realize topological quantum spin glass order [72].
Primary Hierarchy
References
- [1]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [2]
- S. Bravyi and M. Vyalyi, “Commutative version of the k-local Hamiltonian problem and common eigenspace problem”, (2004) arXiv:quant-ph/0308021
- [3]
- D. Aharonov and L. Eldar, “On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems”, (2011) arXiv:1102.0770
- [4]
- D. Aharonov, O. Kenneth, and I. Vigdorovich, “On the Complexity of Two Dimensional Commuting Local Hamiltonians”, (2018) arXiv:1803.02213 DOI
- [5]
- J. M. Leinaas and J. Myrheim, “On the theory of identical particles”, Il Nuovo Cimento B Series 11 37, 1 (1977) DOI
- [6]
- G. A. Goldin, R. Menikoff, and D. H. Sharp, “Particle statistics from induced representations of a local current group”, Journal of Mathematical Physics 21, 650 (1980) DOI
- [7]
- F. Wilczek, “Quantum Mechanics of Fractional-Spin Particles”, Physical Review Letters 49, 957 (1982) DOI
- [8]
- L. Biedenharn, E. Lieb, B. Simon, and F. Wilczek, “The Ancestry of the ‘Anyon’”, Physics Today 43, 90 (1990) DOI
- [9]
- S. Doplicher, R. Haag, and J. E. Roberts, “Local observables and particle statistics I”, Communications in Mathematical Physics 23, 199 (1971) DOI
- [10]
- S. Doplicher, R. Haag, and J. E. Roberts, “Local observables and particle statistics II”, Communications in Mathematical Physics 35, 49 (1974) DOI
- [11]
- M. Cha, P. Naaijkens, and B. Nachtergaele, “On the Stability of Charges in Infinite Quantum Spin Systems”, Communications in Mathematical Physics 373, 219 (2019) arXiv:1804.03203 DOI
- [12]
- Fuchs, Jürgen. Affine Lie algebras and quantum groups: An Introduction, with applications in conformal field theory. Cambridge university press, 1995.
- [13]
- A. Davydov, M. Mueger, D. Nikshych, and V. Ostrik, “The Witt group of non-degenerate braided fusion categories”, (2011) arXiv:1009.2117
- [14]
- J. Fuchs, C. Schweigert, and A. Valentino, “Bicategories for Boundary Conditions and for Surface Defects in 3-d TFT”, Communications in Mathematical Physics 321, 543 (2013) arXiv:1203.4568 DOI
- [15]
- J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “Higher central charges and topological boundaries in 2+1-dimensional TQFTs”, SciPost Physics 13, (2022) arXiv:2107.13091 DOI
- [16]
- A. Kitaev and J. Preskill, “Topological Entanglement Entropy”, Physical Review Letters 96, (2006) arXiv:hep-th/0510092 DOI
- [17]
- M. Levin and X.-G. Wen, “Detecting Topological Order in a Ground State Wave Function”, Physical Review Letters 96, (2006) arXiv:cond-mat/0510613 DOI
- [18]
- Y. Zhang, T. Grover, A. Turner, M. Oshikawa, and A. Vishwanath, “Quasiparticle statistics and braiding from ground-state entanglement”, Physical Review B 85, (2012) arXiv:1111.2342 DOI
- [19]
- J. Haah, “An Invariant of Topologically Ordered States Under Local Unitary Transformations”, Communications in Mathematical Physics 342, 771 (2016) arXiv:1407.2926 DOI
- [20]
- I. H. Kim, B. Shi, K. Kato, and V. V. Albert, “Chiral Central Charge from a Single Bulk Wave Function”, Physical Review Letters 128, (2022) arXiv:2110.06932 DOI
- [21]
- H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, “Provably efficient machine learning for quantum many-body problems”, Science 377, (2022) arXiv:2106.12627 DOI
- [22]
- K. Kato and F. G. S. L. Brandão, “Locality of edge states and entanglement spectrum from strong subadditivity”, Physical Review B 99, (2019) arXiv:1804.05457 DOI
- [23]
- M. B. Hastings, “Topological Order at Nonzero Temperature”, Physical Review Letters 107, (2011) arXiv:1106.6026 DOI
- [24]
- M. Kargarian, “Finite temperature topological order in 2D topological color codes”, (2009) arXiv:0904.4492
- [25]
- A. Bakshi, A. Liu, A. Moitra, and E. Tang, “High-Temperature Gibbs States are Unentangled and Efficiently Preparable”, (2025) arXiv:2403.16850
- [26]
- T. Lan, L. Kong, and X.-G. Wen, “Classification of (3+1)D Bosonic Topological Orders: The Case When Pointlike Excitations Are All Bosons”, Physical Review X 8, (2018) arXiv:1704.04221 DOI
- [27]
- C. Zhu, T. Lan, and X.-G. Wen, “Topological nonlinear σ -model, higher gauge theory, and a systematic construction of 3+1D topological orders for boson systems”, Physical Review B 100, (2019) arXiv:1808.09394 DOI
- [28]
- A. Kapustin, “Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology”, (2014) arXiv:1403.1467
- [29]
- T. Lan and X.-G. Wen, “Classification of 3+1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions”, Physical Review X 9, (2019) arXiv:1801.08530 DOI
- [30]
- Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies”, SciPost Physics 14, (2023) arXiv:2110.14644 DOI
- [31]
- T. Johnson-Freyd and M. Yu, “Topological Orders in (4+1)-Dimensions”, SciPost Physics 13, (2022) arXiv:2104.04534 DOI
- [32]
- L. Kong and X.-G. Wen, “Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions”, (2014) arXiv:1405.5858
- [33]
- L. Kong, X.-G. Wen, and H. Zheng, “Boundary-bulk relation in topological orders”, Nuclear Physics B 922, 62 (2017) arXiv:1702.00673 DOI
- [34]
- T. Johnson-Freyd, “On the Classification of Topological Orders”, Communications in Mathematical Physics 393, 989 (2022) arXiv:2003.06663 DOI
- [35]
- S. Bravyi and M. B. Hastings, “A Short Proof of Stability of Topological Order under Local Perturbations”, Communications in Mathematical Physics 307, 609 (2011) arXiv:1001.4363 DOI
- [36]
- S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: Stability under local perturbations”, Journal of Mathematical Physics 51, (2010) arXiv:1001.0344 DOI
- [37]
- S. Michalakis and J. P. Zwolak, “Stability of Frustration-Free Hamiltonians”, Communications in Mathematical Physics 322, 277 (2013) arXiv:1109.1588 DOI
- [38]
- B. Nachtergaele, R. Sims, and A. Young, “Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms”, Journal of Mathematical Physics 60, (2019) arXiv:1810.02428 DOI
- [39]
- B. Nachtergaele, R. Sims, and A. Young, “Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States”, Annales Henri Poincaré 23, 393 (2021) arXiv:2010.15337 DOI
- [40]
- A. Lavasani, M. J. Gullans, V. V. Albert, and M. Barkeshli, “On stability of k-local quantum phases of matter”, (2024) arXiv:2405.19412
- [41]
- J. Haah, “A degeneracy bound for homogeneous topological order”, SciPost Physics 10, (2021) arXiv:2009.13551 DOI
- [42]
- B. Zeng and X.-G. Wen, “Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity”, Physical Review B 91, (2015) arXiv:1406.5090 DOI
- [43]
- B. Swingle and J. McGreevy, “Renormalization group constructions of topological quantum liquids and beyond”, Physical Review B 93, (2016) arXiv:1407.8203 DOI
- [44]
- J. Yi, W. Ye, D. Gottesman, and Z.-W. Liu, “Complexity and order in approximate quantum error-correcting codes”, Nature Physics (2024) arXiv:2310.04710 DOI
- [45]
- E. Witten, “Quantum field theory and the Jones polynomial”, Communications in Mathematical Physics 121, 351 (1989) DOI
- [46]
- G. Moore and N. Seiberg, “Classical and quantum conformal field theory”, Communications in Mathematical Physics 123, 177 (1989) DOI
- [47]
- M. Barkeshli, P. Bonderson, M. Cheng, C.-M. Jian, and K. Walker, “Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies”, Communications in Mathematical Physics 374, 1021 (2019) arXiv:1612.07792 DOI
- [48]
- S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order”, Physical Review Letters 97, (2006) arXiv:quant-ph/0603121 DOI
- [49]
- J. Eisert and T. J. Osborne, “General Entanglement Scaling Laws from Time Evolution”, Physical Review Letters 97, (2006) arXiv:quant-ph/0603114 DOI
- [50]
- N. Tantivasadakarn, R. Verresen, and A. Vishwanath, “Shortest Route to Non-Abelian Topological Order on a Quantum Processor”, Physical Review Letters 131, (2023) arXiv:2209.03964 DOI
- [51]
- H.-S. Kim, I. H. Kim, and D. Ranard, “Learning State Preparation Circuits for Quantum Phases of Matter”, (2024) arXiv:2410.23544
- [52]
- C. G. Brell, S. Burton, G. Dauphinais, S. T. Flammia, and D. Poulin, “Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems”, Physical Review X 4, (2014) arXiv:1311.0019 DOI
- [53]
- D. Beckman, D. Gottesman, A. Kitaev, and J. Preskill, “Measurability of Wilson loop operators”, Physical Review D 65, (2002) arXiv:hep-th/0110205 DOI
- [54]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [55]
- P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI
- [56]
- C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and topological quantum computation”, Reviews of Modern Physics 80, 1083 (2008) arXiv:0707.1889 DOI
- [57]
- B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, “Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems”, (2018) arXiv:1508.02595
- [58]
- X.-G. Wen, “Colloquium : Zoo of quantum-topological phases of matter”, Reviews of Modern Physics 89, (2017) arXiv:1610.03911 DOI
- [59]
- R. Moessner and J. E. Moore, Topological Phases of Matter (Cambridge University Press, 2021) DOI
- [60]
- L. Kong and Z.-H. Zhang, “An invitation to topological orders and category theory”, (2022) arXiv:2205.05565
- [61]
- P. Liao, B. C. Sanders, and D. L. Feder, “Topological graph states and quantum error-correction codes”, Physical Review A 105, (2022) arXiv:2112.02502 DOI
- [62]
- I. H. Kim and D. Ranard, “Classifying 2D topological phases: mapping ground states to string-nets”, (2024) arXiv:2405.17379
- [63]
- Y. Zhao and Y. Wan, “Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization”, (2025) arXiv:2408.02664
- [64]
- C. Yin and A. Lucas, “Low-density parity-check codes as stable phases of quantum matter”, (2024) arXiv:2411.01002
- [65]
- W. De Roeck, V. Khemani, Y. Li, N. O’Dea, and T. Rakovszky, “LDPC stabilizer codes as gapped quantum phases: stability under graph-local perturbations”, (2024) arXiv:2411.02384
- [66]
- Y. Hu and Y. Zou, “Petz map recovery for long-range entangled quantum many-body states”, Physical Review B 110, (2024) arXiv:2408.00857 DOI
- [67]
- A. Lavasani, Y. Alavirad, and M. Barkeshli, “Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits”, Physical Review Letters 127, (2021) arXiv:2011.06595 DOI
- [68]
- M. Raginsky, “Almost any quantum spin system with short-range interactions can support toric codes”, Physics Letters A 294, 153 (2002) arXiv:quant-ph/0111035 DOI
- [69]
- I. H. Kim, T.-C. Lin, D. Ranard, and B. Shi, “Strict area law implies commuting parent Hamiltonian”, (2024) arXiv:2404.05867
- [70]
- S. Bachmann, W. De Roeck, B. Donvil, and M. Fraas, “Stability of invertible, frustration-free ground states against large perturbations”, Quantum 6, 793 (2022) arXiv:2110.11194 DOI
- [71]
- F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, “Local Random Quantum Circuits are Approximate Polynomial-Designs”, Communications in Mathematical Physics 346, 397 (2016) DOI
- [72]
- B. Placke, T. Rakovszky, N. P. Breuckmann, and V. Khemani, “Topological Quantum Spin Glass Order and its realization in qLDPC codes”, (2024) arXiv:2412.13248
- [73]
- T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [74]
- F. J. Burnell and C. Nayak, “SU(2) slave fermion solution of the Kitaev honeycomb lattice model”, Physical Review B 84, (2011) arXiv:1104.5485 DOI
- [75]
- D. Aasen, D. Bulmash, A. Prem, K. Slagle, and D. J. Williamson, “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020) arXiv:2002.05166 DOI
- [76]
- Z. Song, A. Dua, W. Shirley, and D. J. Williamson, “Topological Defect Network Representations of Fracton Stabilizer Codes”, PRX Quantum 4, (2023) arXiv:2112.14717 DOI
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- Victor V. Albert (2022-06-05)
- Victor V. Albert (2022-01-05)
Cite as:
“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological