Here is a list of non-lattice qubit QLDPC codes. Lattice qubit QLDPC codes are listed in the list of lattice qubit stabilizer codes.

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Code Description
2D hyperbolic surface code Hyperbolic surface codes based on a tessellation of a closed 2D manifold with a hyperbolic geometry (i.e., non-Euclidean geometry, e.g., saddle surfaces when defined on a 2D plane).
Ball code A distance-two color code defined on a colorable \(D\)-ball, equivalently on a \(D\)-colex with boundary [1; Appx. A]. In the morphing construction of Ref. [1], ball codes arise as the child codes associated with the morphed ball-like regions. This family includes hypercube codes (defined on balls constructed from hyperoctahedra) and 3D ball codes (defined on duals of certain Archimedean solids).
Campbell double homological product code A multi-dimensional HGP code derived from two applications of the hypergraph product to a classical code, resulting in a length-\(4\) chain complex. The construction method allows for the use of two different classical codes as inputs, with Ref. [2] assuming identical input codes for simplicity.
Classical-product code A QLDPC qubit CSS code constructed by separately constructing the \(X\) and \(Z\) check matrices using product constructions from classical codes. A particular \([[512,174,8]]\) code performed well [3] against erasure and depolarizing noise when compared to other notable CSS codes, such as the asymptotically good quantum Tanner codes. These codes have been generalized to the intersecting subset code family [4].
Color code Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs.
Dinur-Hsieh-Lin-Vidick (DHLV) code A family of asymptotically good QLDPC codes which are related to expander LP codes in that the roles of the check operators and physical qubits are exchanged.
Dinur-Lin-Vidick (DLV) code Member of a family of codes constructed using cubical chain complexes, which are \(t\)-order extensions of the complexes underlying expander codes (\(t=1\)) and expander lifted-product codes (\(t=2\)).
Fiber-bundle code A CSS code constructed by combining one code as the base and another as the fiber of a fiber bundle. In particular, taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance of order \(\Omega( n^{3/5}/\text{polylog}(n) )\) and rate of order \(\Omega( n^{-2/5}/\text{polylog}(n) )\).
Finite-geometry (FG) qubit QLDPC code CSS code constructed from linear binary codes whose parity-check or generator matrices are incidence matrices of points, hyperplanes, or other structures in finite geometries. These codes can be interpreted as quantum versions of FG-LDPC codes, but some of them [5,6] are not strictly QLDPC.
Fractal surface code Kitaev surface code on a fractal geometry, which is obtained by removing qubits from the surface code on a cubic lattice. A related construction, the fractal product code, is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [7]. The underlying classical codes form classical self-correcting memories [8–10].
Freedman-Meyer-Luo code Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [11]. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries [12]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(\mathbb{Z}_2\)-homology.
Generalized homological-product qubit CSS code A qubit CSS code whose properties are determined from an underlying chain complex via the qubit CSS-to-homology correspondence. This complex often consists of some type of product of other chain complexes.
Generalized quantum Tanner code An extension of quantum Tanner codes to codes constructed from two commuting regular graphs with the same vertex set. This allows for code construction using finite sets and Schreier graphs, yielding a broader family of square complexes.
Golden code Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations of 4-dimensional hyperbolic space.
Guth-Lubotzky code Homological linear-rate code based on cellulations of certain 4D hyperbolic manifolds with particular homology and systolic properties.
Hemicubic code Homological code constructed out of cubes in high dimensions. The hemicubic code family has asymptotically diminishing soundness that scales as order \(\Omega(1/\log n)\), locality of stabilizer generators scaling as order \(O(\log n)\), and distance of order \(\Theta(\sqrt{n})\).
Hierarchical code Member of a family of \([[n,k,d]]\) qubit stabilizer codes resulting from a concatenation of a constant-rate QLDPC code with a rotated surface code. Concatenation allows for syndrome extraction to be performed on a 2D geometry while maintaining a threshold at the expense of a logarithmically vanishing rate. The growing syndrome extraction circuit depth allows known bounds in the literature to be weakened [13,14].
High-dimensional expander (HDX) code CSS code obtained by applying the generalized distance-balancing/product construction of Ref. [15] to a Ramanujan-complex quantum code and an asymptotically good classical LDPC code.
Higher-dimensional homological product code A qubit CSS code formulated using a tensor product of two or more chain complexes, each of length one or greater. The number of chain complexes participating in the product is the dimension of the code. When all chain complexes are length-one, meaning that they correspond to classical codes, the code is called a higher-dimensional HGP code (a.k.a. multi-sector HGP code or iterative HGP code).
Homological code A CSS extension of the Kitaev surface code to arbitrary manifolds. The version on a Euclidean manifold of some fixed dimension is called the \(D\)-dimensional “surface” or \(D\)-dimensional toric code.
Homological product code CSS code formulated using the tensor product of two chain complexes of length one or greater (see Qubit CSS-to-homology correspondence).
Hurwitz surface code Homological code constructed on triangulations of Hurwitz surfaces.
Hyperbolic color code An extension of the color code construction to hyperbolic manifolds. As opposed to there being only three types of uniform three-valent and three-colorable lattice tilings in the 2D Euclidean plane, there is an infinite number of admissible hyperbolic tilings in the 2D hyperbolic plane [16]. Certain double covers of hyperbolic tilings also yield admissible tilings [17]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [18]; see also a construction based on the more general quantum pin codes [19].
Hyperbolic surface code An extension of the Kitaev surface code construction to hyperbolic manifolds. Given a cellulation of a hyperbolic manifold of arbitrary dimension, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces.
Hypersphere product code Homological code based on products of hyperspheres. The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance of order \(\Theta(\sqrt{n})\).
Layer code Member of a family of qubit QLDPC CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface codes according to the Tanner graph of a QLDPC code (or a more general qubit stabilizer code). Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares is fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.
Lift-connected surface (LCS) code Member of one of several families of lifted-product codes that consist of sparsely interconnected stacks of surface codes.
Lossless expander balanced-product code QLDPC code constructed by taking the balanced product of lossless expander graphs. Using one part of a quantum-code chain complex constructed with one-sided loss expanders [20] yields a \(c^3\)-LTC [21]. Using two-sided expanders [22] yields an asymptotically good QLDPC code family [23].
Quantum Tanner code Member of a family of QLDPC codes based on two compatible classical Tanner codes defined on a two-dimensional Cayley complex, a complex constructed from Cayley graphs of groups. For certain choices of codes and complex, the resulting codes have asymptotically good parameters. See Ref. [24] for explicit instances based on dihedral groups. This construction has been generalized to Schreier graphs [25].
Quasi-hyperbolic color code An extension of the color code construction to quasi-hyperbolic 3-manifolds, e.g., a product of a 2D hyperbolic surface and a circle.
Qubit QLDPC code Member of a family of \([[n,k,d]]\) qubit 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 \(w\) as \(n\to\infty\). The code can be denoted by \([[n,k,d,w]]\). Sometimes, the two parameters are explicitly stated: each site of an \((l,w)\)-regular qubit QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\).
Spacetime circuit code Qubit stabilizer code constructed from a Clifford circuit, i.e., a circuit made up of Clifford gates and Pauli measurements, in order to detect and correct circuit faults. The code utilizes redundancy in the measurement outcomes of a circuit to correct circuit faults, which correspond to Pauli errors of the code.
Square homological product code Homological product code whose underlying quantum-code boundary operators are square matrices (see Qubit CSS-to-homology correspondence).
Tensor-product HDX code A code constructed in a similar way as the HDX code, but utilizing iterated homological products of multiple Ramanujan complexes and then applying distance balancing. For any fixed tensor-power parameter \(c\), these yield explicit QLDPC codes with distance scaling as \(\sqrt{n}\log^{c} n\), improving on the original HDX construction by replacing a single logarithmic enhancement with arbitrarily high fixed polylogarithmic enhancement. The utility of such tensor products comes from the fact that one of the Ramanujan complexes is a collective cosystolic expander as opposed to just a cosystolic expander.
\([[2^D,D,2]]\) hypercube quantum code Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. The measured physical bit string can be post-processed into both a logical output string and stabilizer checks, enabling end-of-circuit error detection directly from classical samples [26]. Puncturing the \([[2^D,D,2]]\) hypercube quantum code yields the \([[2^D-1,D,2]]\) punctured-hypercube family.
\([[2^r-1,1,3]]\) simplex code Member of a color-code family constructed from a punctured first-order RM\((1,m=r)\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [27,28]. Each code is a color code defined on a simplex in \(r-1\) dimensions [29,30], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself.
\([[2m,2m-2,2]]\) error-detecting code Self-complementary and self-dual CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [31; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [32].
\([[30,8,3]]\) Bring code A \([[30,8,3]]\) hyperbolic surface code on a quotient of the \(\{5,5\}\) hyperbolic tiling called Bring’s curve. Its qubits and stabilizer generators lie on the vertices of the small stellated dodecahedron. It admits a set of weight-five stabilizer generators.
\([[8,2,2]]\) hyperbolic color code An \([[8,2,2]]\) hyperbolic color code defined on the projective plane. It is a self-dual CSS code.

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