Here is a list of non-lattice qubit QLDPC codes. Lattice qubit QLDPC codes are listed in the list of lattice qubit stabilizer codes.
| Code | Description |
|---|---|
| 2D hyperbolic surface code | Hyperbolic surface codes are 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). |
| Actively orthogonal CSS code | Qubit CSS code whose stabilizer generator matrices are the first \(J\) block rows, the active rows, of a pair of block-circulant parent matrices \(\hat{H}_X\) and \(\hat{H}_Z\), with the CSS orthogonality condition imposed only among those rows [1]. The discarded latent rows need not commute with the active checks of the opposite type, so neither they nor their low-weight combinations automatically become logical operators. The distance can therefore exceed the stabilizer generator weight at rate at or above one half. |
| BB5 code | A BB code with weight-five stabilizer generators (contrasting with the weight-six checks of standard BB codes), designed and benchmarked for long chains of trapped ions [2]. |
| Ball code | A distance-two color code defined on a colorable \(D\)-ball, equivalently on a \(D\)-colex with boundary [3; Appx. A]. In the morphing construction of Ref. [3], 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). |
| Bivariate bicycle (BB) code | One of several Abelian 2BGA codes which admit time-optimal syndrome measurement circuits that can be implemented in a two-layer architecture, a generalization of the square-lattice architecture optimal for the surface codes. Codes can be classified by the weight of their checks, e.g., by BB\(w\) where \(w\) is the check weight. |
| Camara-Ollivier-Tillich code | A Hermitian qubit QLDPC code whose stabilizer generator matrix is constructed using two nested subgroups of \(\mathbb{F}_4^n\). |
| 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. [4] 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 [5] 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 [6]. |
| Color code | Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs. |
| Cornucopia code | GALA code with rate just above one half whose \(G\)-lift entries are affine permutations of a \(\mathbb{Z}_3\times\mathbb{Z}_q\) grid. The layout and the syndrome-extraction schedule are designed together, so that a full syndrome cycle takes twelve entangling layers regardless of the code size. |
| 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 [7,8] are not strictly QLDPC. |
| Fractal surface code | Kitaev surface code on a fractal geometry obtained by puncturing a higher-dimensional lattice at all length scales. A related construction, the fractal product code, is a homological product of two surface codes defined on Sierpinski carpet graphs [9]. The underlying classical codes form classical self-correcting memories [10–12]. |
| Freedman-Meyer-Luo code | Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [13]. 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 [14]. 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. |
| Group-action lift with active orthogonality (GALA) code | Actively orthogonal CSS code whose lifts are drawn from a direct product \(H_k \times C_m\), or a semidirect product \(H_k \ltimes C_m^k\), of a small non-Abelian group \(H_k\) and a large Abelian group \(C_m\) [15]. For a direct product, two lifts commute exactly when their \(H_k\) parts commute, so the non-Abelian top factor alone decides the active orthogonality pattern. The Abelian bottom factor is unconstrained by that pattern, and its shifts are code automorphisms. They also make the qubit permutation between syndrome-extraction rounds one rigid row move and one rigid column move, the parallel moves native to crossed acousto-optic deflectors [15]. |
| 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 a soundness lower bound of 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 [16,17]. |
| High-dimensional expander (HDX) code | CSS code obtained by applying the generalized distance-balancing/product construction of Ref. [18] 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 [19]. Certain double covers of hyperbolic tilings also yield admissible tilings [20]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [21]; see also a construction based on the more general quantum pin codes [22]. |
| 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 \(\Theta(\sqrt{n})\). |
| Kasai code | Actively orthogonal CSS code whose lifts are affine permutation matrices \(\mathrm{APM}(a,b,P)\), the permutation matrices of the maps \(x \mapsto ax+b \bmod P\) with \(\gcd(a,P)=1\) [1]. Two such matrices commute exactly when a quadratic congruence modulo \(P\) in their coefficients holds [1]. The commutation table of active orthogonality is therefore a system of congruences solved by the choice of coefficients, which yields regular, high-girth stabilizer generator matrices at encoding rate at or above one half. |
| Layer code | Member of a family of geometrically local 3D qubit QLDPC codes obtained by coupling layers of 2D surface codes according to the check-qubit incidence structure of an input QLDPC 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. |
| Lechner-Hauke-Zoller (LHZ) code | A \([[k(k+1)/2,k]]\) QLDPC CSS code whose physical qubits store logical bits and their pairwise parities. It was originally designed to convert the long-range interactions of a quantum annealer into local constraints [23,24]. Diagonal logical gates become physical single-qubit \(Z\)-rotations at arbitrary angle [25]. |
| 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 [26] yields a \(c^3\)-LTC [27]. Using two-sided expanders [28] yields an asymptotically good QLDPC code family [29]. |
| Mirror code | Stabilizer code defined by a finite group \(G\) and two subsets \(A,B\subseteq G\), with one qubit per group element [30]. Each stabilizer generator acts as \(Z\) on a translate of \(A\) and as \(X\) on a translate of \(B\), so the check weight is at most \(|A|+|B|\). Mirror codes contain all qubit Abelian 2BGA codes up to qubit permutations and Hadamard gates, and are not CSS in general. |
| Mitten code | Member of a family of LP codes \(\mathrm{LP}(A,B)\) with one-by-two base matrices \(A=[a_0~a_1]\) and \(B=[b_0~b_1]\) whose entries are weight-three elements of the group algebra \(\mathbb{F}_2[G]\) of a non-Abelian group \(G\) [31]. The \(G\)-lift of this shape yields five blocks of \(|G|\) data qubits and two blocks each of \(X\)- and \(Z\)-type checks, hence check weight nine and encoding rate at least \(20\%\). A non-Abelian \(G\) evades the distance ceiling of Abelian lifts of the same shape, allowing distances of about \(20\) with a few hundred data qubits. |
| Pair-partition CPM code | Qubit CSS code whose \(X\)- and \(Z\)-type stabilizer generator matrices are \(J\)-by-\(L\) arrays of \(P\)-by-\(P\) circulant permutation matrices (CPMs), with the CSS condition enforced by pairing up block columns [32]. Each block of \(H_XH_Z^T\) is a sum of \(L\) CPMs, one per block column. A pair partition, one per block, splits the block columns into \(L/2\) pairs, and the CPM exponents are constrained so that the two CPMs in each pair coincide and cancel over \(\mathbb{F}_2\). CSS orthogonality thereby becomes a homogeneous linear condition on the exponents [32]. |
| Perturbed bivariate bicycle (PBB) code | Qubit stabilizer code defined over the ring \(R=\mathbb{F}_2[x,y]/(x^{\ell}-1,y^{m}-1)\) by four polynomials \(A,B,C,D\in R\) [33]. The polynomials \(C\) and \(D\) add \(Z\)-type support to the \(X\)-type checks of the BB code defined by \(A\) and \(B\). Such codes are not CSS in general. |
| 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. [34] for explicit instances based on dihedral groups. This construction has been generalized to Schreier graphs [35]. |
| 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 \(\leq w\). |
| 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. |
| Tricycle code | A three-block CSS code whose commuting matrices represent three elements \(a,b,c\) of the group algebra \(\mathbb{F}_2[G]\) of a finite Abelian group \(G\). The code is the balanced product over \(\mathbb{F}_2[G]\) of the three classical group-algebra codes defined by \(a\), \(b\), and \(c\), extending the two-block 2BGA construction to three homological dimensions [36,37]. The third dimension provides \(Z\)-type metachecks, which make the code single-shot decodable in the \(Z\) basis, and a cup product, which yields constant-depth transversal \(CCZ\) circuits for magic-state generation [37]. |
| Two-branch coset CSS code | Qubit CSS code whose base stabilizer generator matrices place the support of each check on translated cosets of a multiplicative subgroup \(M\) of a finite field, once in each of two copies of the qubit set called branches [38]. The branch coefficients are chosen so that any overlap between an \(X\)-type and a \(Z\)-type check occurs once in each branch and is therefore even. Any overlap between two checks of the same type occurs in at most one branch, which excludes four-cycles. A cyclic lift of the base pair by circulant permutation matrices can then randomize the edge connections while preserving these constraints [38]. |
| \([[1152,580,\leq 12]]\) co-designed Kasai code | Kasai code with \(J=3\), \(L=12\), and lift size \(P=96\) whose affine permutations are chosen so that every transition permutation of syndrome extraction commutes with a fixed reference affine permutation having three orbits of length \(32\) [39]. In the qubit ordering along those orbits, each transition is a cyclic shift within the orbits plus a permutation between them, which yields a three-by-thirty-two layout with a syndrome-extraction schedule of constant qubit-movement cost [39]. |
| \([[132,30,12]]\) self-dual GALA code | A hyperbolic self-dual GALA code whose Hadamard-type and phase-type fold-transversal Clifford operations are implemented physically by depth-one layers of single-qubit gates. |
| \([[27,3,3]]\) bivariate tricycle code | Bivariate tricycle code that can code switch into a BB code and that admits a depth-two cup-product \(CCZ\) gate [40]. It is defined over \(\mathbb{F}_2[x,y]/(x^3+1,y^3+1)\) by the binomial elements \(a=x^2y+x^2y^2\), \(b=1+xy^2\), and \(c=x+x^2y\) [40; Table III]. |
| \([[2^D,D,2]]\) 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 [41]. 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 [42,43]. Each code is a color code defined on a simplex in \(r-1\) dimensions [44,45], 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 [46; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [47]. |
| \([[30,8,3]]\) 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. |
| \([[36,6,6]]\) mirror code | Abelian mirror code on \(G=\mathbb{Z}_6\times\mathbb{Z}_6\) with check weight six that is not equivalent to a CSS code via Hadamard gates [30; Fig. 1]. Its symplectic double is the half-gross code. |
| \([[8,2,2]]\) hyperbolic color code | An \([[8,2,2]]\) hyperbolic color code defined on the projective plane. It is a self-dual CSS code. |
| \([[81,3,5]]\) bivariate tricycle code | Bivariate tricycle code that can code switch into a BB code and that admits a depth-two cup-product \(CCZ\) gate [40]. It is defined over \(\mathbb{F}_2[x,y]/(x^3+1,y^9+1)\) by the binomial elements \(a=xy^3+x^2y\), \(b=1+xy^8\), and \(c=x^2y^4+x^2y^6\) [40; Table III]. |
List: Codes that are descendants of Qubit QLDPC code and not descendants of Lattice stabilizer code.
References
- [1]
- K. Kasai, “Breaking the Orthogonality Barrier in Quantum LDPC Codes”, Quantum 10, 2205 (2026) arXiv:2601.08824 DOI
- [2]
- M. Ye and N. Delfosse, “Quantum error correction for long chains of trapped ions”, Quantum 9, 1920 (2025) arXiv:2503.22071 DOI
- [3]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [4]
- E. T. Campbell, “A theory of single-shot error correction for adversarial noise”, Quantum Science and Technology 4, 025006 (2019) arXiv:1805.09271 DOI
- [5]
- D. Ostrev, D. Orsucci, F. Lázaro, and B. Matuz, “Classical product code constructions for quantum Calderbank-Shor-Steane codes”, Quantum 8, 1420 (2024) arXiv:2209.13474 DOI
- [6]
- D. Ostrev, “Quantum LDPC Codes From Intersecting Subsets”, IEEE Transactions on Information Theory 70, 5692 (2024) arXiv:2306.06056 DOI
- [7]
- J. Farinholt, “Quantum LDPC Codes Constructed from Point-Line Subsets of the Finite Projective Plane”, (2012) arXiv:1207.0732
- [8]
- B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081
- [9]
- C. G. Brell, “A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)”, New Journal of Physics 18, 013050 (2016) arXiv:1411.7046 DOI
- [10]
- A. Vezzani, “Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result”, Journal of Physics A: Mathematical and General 36, 1593 (2003) arXiv:cond-mat/0212497 DOI
- [11]
- R. Campari and D. Cassi, “Generalization of the Peierls-Griffiths theorem for the Ising model on graphs”, Physical Review E 81, (2010) arXiv:1002.1227 DOI
- [12]
- M. Shinoda, “Existence of phase transition of percolation on Sierpiński carpet lattices”, Journal of Applied Probability 39, 1 (2002) DOI
- [13]
- M. H. Freedman, “Z\({}_{\text{2}}\)–Systolic-Freedom”, Geometry & Topology Monographs 113 (1999) DOI
- [14]
- E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
- [15]
- W. Yang, C. Duckering, and A. Dua, “Designer Codes from GALA: Compact, Self-Dual, and Rate-1/2 QEC on Reconfigurable Atom Arrays”, (2026) arXiv:2608.07431
- [16]
- N. Delfosse, M. E. Beverland, and M. A. Tremblay, “Bounds on stabilizer measurement circuits and obstructions to local implementations of quantum LDPC codes”, (2021) arXiv:2109.14599
- [17]
- N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
- [18]
- S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
- [19]
- E. B. da Silva and W. S. Soares, “Hyperbolic quantum color codes”, (2018) arXiv:1804.06382
- [20]
- N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory 917 (2013) arXiv:1301.6588 DOI
- [21]
- H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
- [22]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [23]
- W. Lechner, P. Hauke, and P. Zoller, “A quantum annealing architecture with all-to-all connectivity from local interactions”, Science Advances 1, (2015) DOI
- [24]
- F. Pastawski and J. Preskill, “Error correction for encoded quantum annealing”, Physical Review A 93, (2016) arXiv:1511.00004 DOI
- [25]
- M. Fellner, A. Messinger, K. Ender, and W. Lechner, “Universal Parity Quantum Computing”, Physical Review Letters 129, (2022) arXiv:2205.09505 DOI
- [26]
- M. Capalbo, O. Reingold, S. Vadhan, and A. Wigderson, “Randomness conductors and constant-degree lossless expanders”, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing 659 (2002) DOI
- [27]
- T.-C. Lin and M.-H. Hsieh, “\(c^3\)-Locally Testable Codes from Lossless Expanders”, (2022) arXiv:2201.11369
- [28]
- J.-T. Hsieh, A. Lubotzky, S. Mohanty, A. Reiner, and R. Y. Zhang, “Explicit Lossless Vertex Expanders”, (2025) arXiv:2504.15087
- [29]
- T.-C. Lin and M.-H. Hsieh, “Good quantum LDPC codes with linear time decoder from lossless expanders”, (2022) arXiv:2203.03581
- [30]
- A. B. Khesin and J. Z. Lu, “Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime”, (2026) arXiv:2603.05496
- [31]
- A. Bhardwaj, M. Ma, N. Meister, R. King, D. Bluvstein, J. Preskill, M. Cain, Q. Xu, and H.-Y. Huang, “High-rate qLDPC processors”, (2026) arXiv:2607.28795
- [32]
- K. Okada and K. Kasai, “Pair-Partition Constructions for CPM-Based Quantum LDPC Codes”, (2026) arXiv:2607.14091
- [33]
- J. Cruz-Benito, A. W. Cross, D. Kremer, and I. Faro, “Evolutionary Discovery of Bivariate Bicycle Codes with LLM-Guided Search”, (2026) arXiv:2606.02418
- [34]
- R. K. Radebold, S. D. Bartlett, and A. C. Doherty, “Explicit Instances of Quantum Tanner Codes”, (2025) arXiv:2508.05095
- [35]
- O. Å. Mostad, E. Rosnes, and H.-Y. Lin, “Generalizing Quantum Tanner Codes”, (2024) arXiv:2405.07980
- [36]
- A. Jacob, C. McLauchlan, and D. E. Browne, “Single-Shot Decoding and Fault-tolerant Gates with Trivariate Tricycle Codes”, (2026) arXiv:2508.08191
- [37]
- V. Menon, J. P. Bonilla Ataides, R. Mehta, A. Gu, D. B. Tan, and M. D. Lukin, “Magic Tricycles: Efficient Magic-State Generation with Finite Block-Length Quantum LDPC Codes”, Physical Review X 16, (2026) arXiv:2508.10714 DOI
- [38]
- K. Okada and K. Kasai, “A Two-Branch Finite-Field Construction for Regular CSS LDPC Bases”, (2026) arXiv:2605.23894
- [39]
- C. Zhao, C. Duckering, A. Gu, N. Maskara, and H. Zhou, “Towards Ultra-High-Rate Quantum Error Correction with Reconfigurable Atom Arrays”, (2026) arXiv:2604.16209
- [40]
- C. Li, J. Preskill, and Q. Xu, “Transversal dimension jump for product qLDPC codes”, (2026) arXiv:2510.07269
- [41]
- D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
- [42]
- B. Zeng, H. Chung, A. W. Cross, and I. L. Chuang, “Local unitary versus local Clifford equivalence of stabilizer and graph states”, Physical Review A 75, (2007) arXiv:quant-ph/0611214 DOI
- [43]
- S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
- [44]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [45]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
- [46]
- N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) 791 (2018) arXiv:1803.06987 DOI
- [47]
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006