Here is a list of non-qubit QLDPC codes. For the list of qubit QLDPC codes, see Qubit QLDPC codes.
| Code | Description |
|---|---|
| 1D lattice stabilizer code | Lattice stabilizer code in one Euclidean dimension, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality. |
| 2D lattice stabilizer code | Lattice stabilizer code in two Euclidean dimensions, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality. |
| 3D lattice stabilizer code | Lattice stabilizer code in three Euclidean dimensions, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality. |
| 4D lattice stabilizer code | Lattice stabilizer code in four Euclidean dimensions, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality. |
| Abelian LP code | A lifted-product code whose lift group \(G\) is Abelian. The case of \(G\) being a cyclic group is a GB code (a.k.a. a quasi-cyclic LP code) [1; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with parameters \([[n,k=\Theta(\log n),d=\Theta(n/\log n)]]\) [1]. |
| Abelian TQD stabilizer code | Modular-qudit stabilizer code whose codewords realize a 2D Abelian twisted-quantum-double topological order on composite-dimensional qudits. For every finite Abelian group \(G=\prod_i \mathbb{Z}_{N_i}\) and every product of Type-I and Type-II cocycles, there is a Pauli stabilizer Hamiltonian realizing the corresponding Abelian TQD [2]. Equivalently, these codes exhaust the 2D Abelian topological orders that admit gapped boundaries [2,3]. |
| Abelian quantum-double stabilizer code | Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. The corresponding anyon theory is defined by an Abelian group. The \(G=\mathbb{Z}_2\) instance on a torus is the toric code, and cyclic-group instances reduce to modular-qudit surface codes. All such codes can be realized by a stack of modular-qudit surface codes because all finite Abelian groups are direct products of cyclic groups. |
| Analog repetition code | An \([[n,1]]_{\mathbb{R}}\) analog stabilizer version of the quantum repetition code, encoding the position states of one mode into an odd number \(n\) of modes. |
| Analog surface code | An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory. |
| Balanced product (BP) code | Family of CSS quantum codes obtained from two classical-code chain complexes that share a common group symmetry. The balanced product can be understood as taking the usual tensor or hypergraph product and then quotienting by the shared symmetry action. This can reduce the overall number of physical qubits \(n\) while, in favorable cases, preserving the number of encoded qubits and the code distance, thereby improving the encoding rate \(k/n\) and normalized distance \(d/n\) compared to the underlying tensor or hypergraph product. |
| Chiral semion Walker-Wang model code | A 3D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) whose low-energy excitations on boundaries realize the chiral semion topological order. The model admits 2D chiral semion topological order at one of its surfaces [4,5]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [6]. |
| Compactified \(\mathbb{R}\) gauge theory code | An integer-homology bosonic CSS code realizing 2D \(U(1)\) gauge theory on bosonic modes. The code can be obtained from the analog surface code by condensing certain anyons [7]. This results in a pinning of each mode to the space of periodic functions, which is the Hilbert space of a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code. |
| Distance-balanced code | Galois-qudit CSS code obtained from a CSS code by increasing the smaller of the \(X\)- and \(Z\)-distances using a homological-product-based balancing step or one of its generalizations. The initial code is said to be unbalanced, i.e., tailored to noise biased toward either bit- or phase-flip errors, and the procedure can result in a code that treats both types of errors on a more equal footing. |
| Double-semion stabilizer code | A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that realizes the 2D double semion topological phase. The code can be obtained from a \(\mathbb{Z}_4\) toric-code ground state by condensing the emergent boson \(e^2 m^2\); in the stabilizer construction this condensation is implemented by two-body measurements [2,8]. Its ground-state subspace can be mapped to that of the double-semion string-net model by a finite-depth quantum circuit with ancillas [2]. |
| Expander LP code | Family of \(G\)-lifted product codes constructed using two classical expander codes, equivalently two regular Tanner codes defined on the same expander graph [9]. For certain parameters, this construction yields the first asymptotically good QLDPC codes. Classical codes resulting from the same lifted-product complexes are one of the first two families of \(c^3\)-LTCs [10]. |
| Fracton stabilizer code | A 3D modular-qudit stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase. Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. |
| GKP-surface code | A concatenated code whose outer code is a GKP code and whose inner code is a surface code, including toric surface-code variants [11,12], rotated surface codes [13–16], and XZZX surface codes [17]. |
| Galois-qudit HGP code | A member of a family of Galois-qudit CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear \(q\)-ary codes. |
| Galois-qudit color code | Extension of the color code to 2D lattices of Galois qudits. |
| Galois-qudit expander code | Galois-qudit CSS code obtained from tensor products of chain complexes associated with an explicit family of expander codes with Reed-Solomon local checks. |
| Galois-qudit surface code | Extension of the surface code to 2D lattices of Galois qudits. |
| Generalized bicycle (GB) code | A quasi-cyclic Galois-qudit CSS code constructed using a generalized version of the bicycle ansatz [18] from a pair of equivalent index-two quasi-cyclic linear codes. Equivalently, the codes can be constructed via the lifted-product construction for \(G\) being a cyclic group [1; Sec. III.E]. |
| Generalized homological-product CSS code | CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. |
| Generalized homological-product code | Stabilizer code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. The Qubit CSS-to-homology correspondence yields an interpretation of codes in terms of chain complexes, thus allowing for the use of various products from homology in constructing codes. |
| Good QLDPC code | Also called asymptotically good QLDPC codes. A family of QLDPC codes \([[n_i,k_i,d_i]]\) whose asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive. |
| Integer-homology bosonic CSS code | A bosonic stabilizer code whose physical modes have been restricted, via a single GKP stabilizer, from the space of functions on the real line to the space of periodic functions. This restriction effectively realizes a rotor on each physical mode, allowing one to construct homological rotor codes out of displacement stabilizer groups. The stabilizer group is continuous, but contains discrete components in the form of the single-mode GKP stabilizers. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension. |
| Kitaev current-mirror qubit code | Member of the family of \([[2n,(0,2),(2,n)]]_{\mathbb{Z}}\) homological rotor codes storing a logical qubit on a thin Möbius strip. The ideal code can be obtained from a Josephson-junction [19] system [20]. |
| Lattice stabilizer code | A geometrically local stabilizer code with sites organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\), using either the ordinary block notion of locality or the fermionic/Majorana notion of locality. On an infinite lattice, its stabilizer group is generated by few-site Pauli-type operators and their translations, in which case the code is called translationally invariant stabilizer code. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions. Lattice defects and boundaries between different codes can also be introduced. |
| Lifted-product (LP) code | Galois-qudit code that utilizes the notion of a lifted product in its construction. Lifted products of certain classical Tanner codes are the first (asymptotically) good QLDPC codes. |
| Modular-qudit 3D surface code | A generalization of the 3D surface code to modular qudits. Qudits are placed on edges, \(Z\)-type stabilizer generators are placed on square plaquettes oriented in all three directions, and \(X\)-type stabilizers are placed on the six edges neighboring every vertex [21]. |
| Modular-qudit lattice color code | Extension of the color code to lattices of modular qudits. Codes are defined analogously to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizers commute. This can be done by puncturing a hyperspherical lattice [22] or constructing a star-bipartition; see [23; Sec. III]. Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and/or any twist defects present. |
| Modular-qudit surface code | Extension of the surface code to prime-dimensional [24,25] and more general modular qudits. Stabilizer generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. |
| QLDPC code | Member of a family of stabilizer codes for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant as \(n\to\infty\). Sometimes, the two parameters are explicitly stated: each site of an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\). |
| Quasi-cyclic QLDPC (QC-QLDPC) code | A QLDPC code such that cyclic shifts of the subsystems by a fixed \(\ell\geq 1\) leave the codespace invariant. Stabilizer generator matrices of such codes can be put into block form, where each nonzero block is a circulant matrix [26,27]. |
| Qudit X-cube model code | Generalization of the X-cube model code to modular qudits. |
| Qudit cubic code | Generalization of the Haah cubic code to modular qudits. |
| Two-block group-algebra (2BGA) codes | 2BGA codes are the one-by-one, or smallest, LP codes: \(LP(a,b)\) is defined by a pair of group algebra elements \(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group. If \(|G|=\ell\), then the code has length \(n=2\ell\). |
| Type-II fractal spin-liquid code | A type-II fracton prime-qudit CSS code defined on a cubic lattice [28; Eqs. (D9-D10)]. |
| \(D_4\) hyper-diamond GKP code | Two-mode GKP qubit-into-oscillator code based on the \(D_4\) hyper-diamond lattice [29]. |
| \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code | A non-CSS multimode GKP code defined on a 2D mode lattice that encodes a qudit logical space and whose excitations are characterized by the \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons theory. The code can be obtained from the analog surface code by condensing certain anyons [7]. |
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