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].

There exist functions of code states that extract the total quantum dimension \(D\) [16,17], 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]. General thermal states are separable above a sufficiently high temperature [24].

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) [25,26].

Phases with fermionic point-like excitations are examples of beyond-group-cohomology phases [27]. They have been classified [28], and some of them can be described by a two-gauge theory [29].

The classification of 4D [30] and higher-dimensional [31–33] 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 [34–38].

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 [35]. 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_{A}\Pi=0\quad\Rightarrow\quad O_{A}\Pi_{N(A)}=0~. \tag*{(3)}\end{align} This condition implies that any operator supported solely on \(A\) cannot distinguish the global projector from the local one [34,39].

A notion of topological order generalizing both the cleaning lemma and the TQO conditions is homogeneous topological order [40]. Related topological order definitions include equivalence under course-graining (i.e., renormalization group) [41,42]. See [40; Sec. 4] for a discussion.

Certain topological codes have nontrivial codespace complexity [43].

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 [44,45], \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. [46].

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 [47,48], 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 [49].Algorithm that takes in reduced density matrices and outputs a circuit preparing the global state in polynomial time [50].

Gates

Ising anyon braiding and fusion were studied in a phenomenological model that was the first to study error correction with non-Abelian anyons [51].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 [52].

Notes

Ref. [54–59][53; Appx. F] for introductions to topological phases.See AnyonWiki for lists of categories relevant to anyons.

Parents

  • Block quantum code — Topological codes are block codes because an infinite family of tensor-product Hilbert spaces is required to formally define a phase of matter.
  • Hamiltonian-based code — Codespace of a topological code is typically the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase. Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes). 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 [36,60].

Children

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].
  • Approximate quantum error-correcting code (AQECC) — In the case of topological codes, the Petz infidelity is related to the topological entanglement entropy [64].
  • Monitored random-circuit code — Topological order can be generated in 2D monitored random circuits [65].
  • 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 [34–38]. This notion can be extended to semi-hyperbolic manifolds [39] and non-geometrically local QLDPC codes exhibiting check soundness [66] (see also [67]).
  • 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 [36,60].
  • 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 [34–38]. This notion can be extended to semi-hyperbolic manifolds [39] and non-geometrically local QLDPC codes exhibiting check soundness [66] (see also [67]).
  • Local Haar-random circuit qubit code — Local Haar-random codewords, like topological codewords, are locally indistinguishable [68].
  • 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 [69].
  • Dinur-Lin-Vidick (DLV) code — DLV codes are expected to realize topological quantum spin glass order [69].
  • 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 [53][70; Footnote 25]. This includes the (non-Abelian) Ising-anyon topological order [53] (a.k.a. \(p+ip\) superconducting phase [71]) 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 [72,73].
  • Expander LP code — Expander lifted-product codes are expected to realize topological quantum spin glass order [69].

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“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological
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@incollection{eczoo_topological, title={Topological code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/topological} }
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