Here is a list of all quantum codes that specify what rate they have.

Name | Rate |
---|---|

2D color code | For general 2D manifolds, \(kd^2 \leq c(\log k)^2 n\) for some constant \(c\) in what can be thought of as an extension of the BPT bound to codes on hyperbolic geometries [1], meaning that color codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\). |

2D hyperbolic surface code | 2D hyperbolic surface codes have an asymptotically constant encoding rate \( k/n \) with a distance scaling logarithmically with \( n\) when the surface is closed. The encoding rate depends on the tiling \( {r,s} \) and is given by \( k/n = (1-2/r - 2/s) + 2/n \), which approaches a constant value as the number of physical qubits grows. The weight of the stabilizers is \( r \) for \( Z \)-checks and \( s \) for \( X \)-checks. For open boundary conditions, the code reduces to constant distnace. |

Bacon-Shor code | A non-LDPC family of Bacon-Shor codes achieves a distance of \(\Omega(n^{1-\epsilon})\) with sparse gauge operators. |

Balanced product (BP) code | A notable family of balanced product codes encode \(k \in \Theta(n^{4/5})\) logical qubits with distance \(d \in \Omega(n^{3/5})\) for any number of physical qubits \(n\). Additionally, it is known that the code constructed from the balanced product of two good classical LDPC codes over groups of order \(\Theta(n)\) has a constant encoding rate [2]. |

Bivariate bicycle (BB) code | When ancilla qubit overhead is included, the encoding rate surpasses that of the surface code. A general \([[n,k,d]]\) bivariate bicycle code requires \(n\) ancilla qubits for encoding, meaning that its ancilla-added encoding rate is \(k/2n\). |

Bosonic code | The quantum capacity of the pure-loss channel [3] and the dephasing noise channel [4] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [5,6]. Exact two-way assisted capacities have been obtained for the pure-loss channels and quantum limited amplifiers in what is known as the PLOB bound [7]. These are examples of Gaussian channels, i.e., channels that map Gaussian states to Gaussian states [8–14]. |

Bravyi-Bacon-Shor (BBS) code | A class of BBS codes [15] saturate the subsystem bound \(kd = O(n)\) [16]. |

Chamon model code | The number of logical qubits is \(k = \text{gcd}(a,b,c)\) for a code constructed as a hypergraph product of three repetition codes of length \(a\), \(b\), and \(c\), respectively [17]. |

Chuang-Leung-Yamamoto (CLY) code | Code rate is \(\frac{k}{n \log_2(N+1)}\). To correct the loss of up to \(t\) excitations with \(K+1\) codewords, a code exists with scaling \(N \sim t^3 K/2\). |

Coherent-state constellation code | Coherent-state constellation codes consisting of points from a Gaussian quadrature rule can be concatenated with quantum polar codes to achieve the Gaussian coherent information of the thermal noise channel [18,19]. |

Dinur-Hsieh-Lin-Vidick (DHLV) code | Asymptotically good QLDPC codes. |

Expander LP code | Expander lifted-product codes include the first examples [20] of (asymptotically) good QLDPC codes, i.e., codes with asymptotically constant rate and linear distance. Another notable family encodes \(k \in \Theta(n^\alpha \log n)\) logical qubits with distance \(d \in \Omega(n^{1 - \alpha} / \log n)\) for any number of physical qubits \(n\) and any real parameter \(0 \leq \alpha < 1\) [21]. |

Fiber-bundle code | Rate \(k/n = \Omega(n^{-2/5}/\text{polylog}(n))\), distance \(d=\Omega(n^{3/5}/\text{polylog}(n))\). This is the first QLDPC code to achieve a distance scaling better than \(\sqrt{n}~\text{polylog}(n)\). |

Freedman-Meyer-Luo code | Codes held a 20-year record the best lower bound on asymptotic scaling of the minimum code distance, \(d=\Omega(\sqrt{n \sqrt{\log n}})\), broken by Ramanujan tensor-product codes. |

Generalized bicycle (GB) code | GB codes can achieve an asymptotic rate of 1/4 [22]. For an odd prime \(\ell\), let a prime \(p\) be a quadratic residue modulo \(\ell\), i.e. \(p=m^{2}\text{mod}\ell\) for some integer \(m\). Then, \(x^{\ell}-1\) has only three irreducible factors in \(\mathbb{F}_q(x)\), and there is a quadratic-residue cyclic code \([\ell,(\ell+1)/2, d]_p\) with \(d\geq\sqrt{\ell}\) and an irreducible generator polynomial. Using the GV distance \(d_{GV}\), a prime-field GB code with parameters \([[ 2\ell,(\ell-1)/2,d\geq \ell^{1/2}]]_p\) exists. |

Golden code | Nonvanishing rate and asymptotic distance lower bounded by \(n^0.1\). |

Gottesman-Kitaev-Preskill (GKP) code | Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of displacement-noise and thermal-noise Gaussian loss channels [5,6,23,24]. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel [5]. |

Guth-Lubotzky code | An explicit construction based on Coxeter groups yields a lower bound of \(13/72\) on the asymptotic rate [25]. |

Haar-random qubit code | The rate of the code is equal to the coherent information of the channel (i.e. the quantum channel capacity). |

Heavy-hexagon code | \(1/n\) for a distance-\(d\) heavy-hexagon code on \(n = (5d^2-2d-1)/2\) qubits. |

Hierarchical code | Rate vanishes as \(\Omega(1/\log(n)^2)\). |

High-dimensional expander (HDX) code | For 2D Ramanujan complexes, the rate is \(\Omega(\sqrt{ \frac{1}{n \log n} })\), with minimum distance \(d = \Omega(\sqrt{n \log n}) \). For 3D, the rate is \( \Omega(\frac{1}{\sqrt{n}\log n}) \) with minimum distance \(d \geq \sqrt{n} \log n \). |

Homological code | Rate depends on the underlying cellulation and manifold [26,27]. For general 2D manifolds, \(kd^2\leq c(\log k)^2 n\) for some constant \(c\) [1], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as \(O(\sqrt{n})\) [28,29], and (2) surface codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\). Higher-dimensional manifolds yield distances scaling more favorably. Loewner's theorem provides an upper bound for any bounded-geometry surface code [30]. |

Hyperbolic Floquet code | Finite encoding rate whose value depends on the hyperbolic lattice. The asymptotic rate is 1/8 for a lattice of octagons [31]. |

Hyperbolic color code | In the double-cover construction [1], an \(\{\ell,m\}\) input tiling yields a code family with an asymptotic rate of \(1 - 1/\ell - 1/m\). |

Kitaev surface code | Both the planar and toric codes saturate the BPT bound, which states that \(k d^2 = O(L^2)\) for codes on a 2D lattice of length \(O(L)\). |

Layer code | Code parameters on a cube, \([[n,\Theta(n^{1/3}),\Theta(n^{1/3})]]\), achieve the 3D BPT bound when asymptotically good QLDPC codes are used in the construction. |

Lifted-product (LP) code | There is no known simple way to compute the logical dimension \(k\) in the general case [21]. |

Lossless expander balanced-product code | Asymptotically good QLDPC codes [32], assuming the existence of two-sided lossless expanders. |

Movassagh-Ouyang Hamiltonian code | The rate depends on the classical code, but distance can scale linearly with \(n\). |

Neural network code | Neural network codes can be obtained by optimizing the coherent information [33]. |

Pastawski-Yoshida-Harlow-Preskill (HaPPY) code | The pentagon HaPPY code has an asymptotic rate \(\frac{1}{\sqrt{5}} \approx 0.447\). The pentagon/hexagon HaPPY code, with alternating layers of pentagons and hexagons in the tiling, has a rate of \(0.299\) if the last layer is a pentagon layer and a rate of \(0.088\) if the last layer is a hexagon layer. |

Projective-plane surface code | The rate is \(1/n\), where \(n\) is the number of edges of the particular cellulation. |

Quantum Reed-Muller code | \(\frac{k}{n}\), where \(k = 2^r - {r \choose t} + 2 \sum_{i=0}^{t-1} {r \choose i}\). Additionally, CSS codes formed from binary Reed-Muller codes achieve channel capacity on erasure channels [34]. |

Quantum Tanner code | Asymptotically good QLDPC codes. When \(C_A\) and \(C_B\) are chosen to have rates not equal to a half, the number of encoded qubits scales as \(k=\Theta(n)\). |

Quantum expander code | \([[n,k=\Theta(n),d=O(\sqrt{n})]]\) code with asymptotically constant rate. |

Quantum polar code | The rate approaches the symmetric coherent information of arbitrary quantum channels [35]. |

Qubit CSS code | For a depolarizing channel with probability \(p\), CSS codes allowing for arbitrarily accurate recovery exist with asymptotic rate \(1-2h(p)\), where \(h\) is the binary entropy function [26]. |

Qubit code | Exact two-way assisted capacities have been obtained for the erasure and dephasing channels [7]. |

SYK code | SYK codes can have a constant rate and distance scaling as \(n^c\) for some power \(c\) [36]. |

Singleton-bound approaching AQECC | Given rate \(R\), tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\). |

Stellated color code | Stellated color codes have negative curvature around the central defect, and thus circumvent the BPT bound for codes on Euclidean lattices. |

Subsystem spacetime circuit code | The spacetime circuit code construction is used to show the existance of spatially local subsystem codes that nearly saturate the subsystem BT bound [37]. |

Triangular surface code | For specific triangle codes, the rates are \(7/13\) or \(7/15\) both with distance \(3\) and weight-four check operators. In general, for \(d\) distance, there are \(3d^2 + O(d)\), \(9d^2/4 + O(d)\), or \(6d^2/4 + O(d)\) physical qubits per logical qubit, depending on the type of initialization and measurement procedures. |

Twist-defect surface code | Twist-defect surface codes have negative curvature around their defects, and thus circumvent the BPT bound for codes on Euclidean lattices. |

Two-block group-algebra (2BGA) codes | The 2BGA construction gives some of the best short codes with small stabilizer weights. A number of 2BGA codes \([[n,k,d]]_q\) with row weights \(W\le 8\), block lengths \(n\le 100\), and parameters such that \(kd\ge n\) have been constructed by exhaustive enumeration [38]. Examples include GB codes with parameters \([[70,8,10]]_2\), \([[72,10,9]]_2\), Abelian 2BGA for groups \(\mathbb{Z}_{mh}=\mathbb{Z}_m\times \mathbb{Z}_2\) (index-4 QC codes) with parameters \([[48,8,6]]_2\) and \([[56,8,7]]_2\), and non-Abelian codes with parameters \([[64,8,8]]_2\), \([[82,10,9]]_2\), \([[96,10,12]]_2\), and \([[96,12,10]]_2\) (all of these have stabilizer generators of weight \(W=8\).) |

X-cube Floquet code | Logical dimension grows with system size [39]. |

XYZ product code | Not much has been proven about the relationship between XYZ product codes and other codes. The logical dimension depends on properties of the input classical codes, specifically similarity invariants from abstract algebra. It is conjectured that specific instances of XYZ product codes have a constant encoding rate and a minimum distance of \(d \in \Theta(n^{2/3})\) [40]. |

\([[144,12,12]]\) gross code | An ancilla-added rate of \(1/24\). In contrast, the distance-13 surface code has ancilla-added rate \(1/338\). |

\([[k+4,k,2]]\) H code | The H codes are dense, i.e., the rate \(\frac{k}{k+4}\rightarrow 1\) as \(k \rightarrow \infty\). The distance is 2. However an \(r\)-level concatenation of H codes gives a distance of \(2^r\). |

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