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. |

Abelian LP code | Expander LP codes for Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) [2]; this construction can be derandomized by being reformulated as a balanced product code [3]. Other explicit versions of codes with such parameters have been developed [4]. |

Amplitude-damping (AD) code | The quantum capacity of the AD channel is \(\max\{0, \log \frac{1-\gamma}{\gamma}\} \) [5]. Quantum capacities of the qubit AD channel are also determined [6,7], including for the case fo memory [8,9]. Capacities of qudit extensions have also been studied [10]. |

Approximate quantum error-correcting code (AQECC) | An extension of the BPT bound to approximate codes is done in Ref. [11]. |

BPSK c-q code | The single-degeneracy code yields an improved PIE by \(2.8\%\) over BPSK [12] (see [13]). |

Bacon-Shor code | A non-LDPC family of Bacon-Shor codes achieves a distance of order \(\Omega(n^{1-\epsilon})\) 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 whose size is of order \(\Theta(n)\) has a constant encoding rate [3]. |

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 c-q code | The Holevo capacity has been calculated for various bosonic quantum channels [14–16] such as the lossy bosonic channel [17] or quantum AGWN [18]. The energy-constrained capacity of the noiseless bosonic c-q channel is finite due to quantum effects [19,20], while the Shannon capacity can be infinite. Gordon was the first to calculate such capacities (in a published work) for a specific case [21–23], and a related discussion is given by Forney [24]. The most information-efficient format of a transmitted message is inditsinguishable from black-body radiation [25]. |

Bosonic code | The quantum capacity of the AD channel [5] and the dephasing noise channel [26] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [27,28]. Exact two-way assisted capacities have been obtained for the AD channels and quantum limited amplifiers in what is known as the PLOB bound [29]. These are examples of Gaussian channels, i.e., channels that map Gaussian states to Gaussian states [30–36]. Non-Gaussian channel capacities can be bounded for single [37] and multiple [38; Lemma 14] modes. |

Bravyi-Bacon-Shor (BBS) code | A class of BBS codes [39] saturate the subsystem bound \(kd = O(n)\) [40]. |

Brown-Fawzi random Clifford-circuit code | The achievable distance of these codes is asymptotically the same as a code whose encoder is a random (not necessarily log-depth) general Clifford unitary [41]. |

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 [42]. |

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 \(N \sim t^3 K/2\). |

Classical-quantum (c-q) code | The Holevo channel capacity, \begin{align} C=\lim_{n\to\infty}\frac{1}{n}\chi\left({\cal N}^{\otimes n}\right)~, \tag*{(1)}\end{align} where \(\chi\) is the Holevo information, is the highest rate of classical information transmission through a quantum channel with arbitrarily small error rate [43–45]. This capacity is equal to the single-letter Holevo information of a single copy of the channel, \(\chi(\cal{N})\), for all known deterministically constructed channels. However, it is known to be superadditive, i.e., not equal to the single-copy case, for particular random channels [46]. Corrections to the Holevo capacity and tradeoff between decoding error, code rate and code length are determined in quantum generalizations of small [47], moderate [48,49], and large [50] deviation analysis. Bounds also exist on the one-shot capacity, i.e., the achievability of classical codes given only one use of the quantum channel [51–56]; see [56; Table 2] for a summary. Ideal decoding error scales is suppressed exponentially with the number of subsystems \(n\) (for c-q block codes), and the exponent has been studied in Ref. [57]. Unambiguous state discrimination (USD) can be used to achieve Holevo capacity on a general pure-state c-q channel [58]. |

Coherent-state c-q code | Random Gaussian-distributed coherent-state c-q codes achieve the capacity of the lossy bosonic channel [17]. |

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 [59,60]. |

Concatenated GKP code | Recursively concatenating the \([[6,2,2]]\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [61]. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound [62]. |

Concatenated c-q code | Concatenated codes can achieve Holevo capacity [63]. |

Constant-excitation (CE) code | Fock-state CE codes can be used in a protocol that achieves the two-way quantum capacity of the AD Gaussian channel [64]. |

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

EA quantum LCD code | Asymptotically good maximal-entanglement EA Galois-qudit stabilizer codes can be constructed from LCD codes [65]. |

EA quantum turbo code | Maximal-entanglement EA quantum turbo codes come close to achieving the EA hashing bound [66]; see [67; Footnote 2]. |

EA qubit code | There are EA versions of classical and quantum capacities [68], and the ratio of the entanglement-assisted and unassisted classical capacities of a channel is bounded by a function of the input channel's dimension [69]. EA hashing bounds on the minimum entanglement required to achieve the entanglement-assisted channel capacity are derived [70]. |

EA qubit stabilizer code | Asymptotically good EA qubit stabilizer codes exist [71]. |

Entanglement-assisted (EA) hybrid quantum code | Tradeoff between classical communication, quantum communication, and entanglement distribution has been examined [72–74]; see also Ref. [75]. |

Expander LP code | Expander lifted-product codes for non-Abelian groups include the first examples [76] of (asymptotically) good QLDPC codes, i.e., codes with asymptotically constant rate and linear distance. Expander LP codes for Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) [2]; this construction can be derandomized by being reformulated as a balanced product code [3].Other explicit versions of codes with such parameters have been developed [4]. |

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 \(\sqrt{n}~\text{polylog}(n)\). |

Finite-dimensional quantum error-correcting code | One can achieve a transmission rate \(r\) over a quantum channel \(\mathcal{E}\) iff, for sufficiently large \(n\), \(m=\lfloor r n \rfloor\), and for all \(\epsilon>0\), \begin{align} ||\mathcal{D}\mathcal{E}\mathcal{U}-I^{\otimes m}||_1\leq \epsilon \tag*{(2)}\end{align} for some encoding channel \(\mathcal{U}\) and some recovery channel \(\mathcal{D}\). The quantum capacity \(Q\) of \(\mathcal{E}\) is defined as the supremum over \(n\) of achievable transmission rates [77]. |

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. |

Galois-qudit GRS code | Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum GV bound [78]. |

Generalized bicycle (GB) code | GB codes can achieve an asymptotic rate of 1/4 [79]. 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 \(n^0.1\). However, the smallest number of physical qubits in this family is 234,000. |

Good QLDPC code | The codes'' rate and distance are both separated from zero as block length goes to infinity. Rains shadow enumerators can be used to show that the distance of an asymptotically good QLDPC code should be bounded as \(d\leq n/3\) [80]; see Ref. [81]. AEL distance amplification [82,83] can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound [84; Corr. 5.3]. |

Gottesman-Kitaev-Preskill (GKP) code | Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of AD, displacement-noise, and thermal-noise Gaussian loss channels [27,28,85,86]. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel [27]. |

Guth-Lubotzky code | An explicit construction based on Coxeter groups yields a lower bound of \(13/72\) on the asymptotic rate [87]. |

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

Hadamard BPSK c-q code | Using a joint-detection receiver, the code achieves the Holevo capacity of the lossy bosonic channel [13]. |

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 a function of order \(\Omega(1/\log(n)^2)\). |

High-dimensional expander (HDX) code | For 2D Ramanujan complexes, the rate is of order \(\Omega(\sqrt{ \frac{1}{n \log n} })\), 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 [88,89]. 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 \(O(\sqrt{n})\) [90,91], 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 [92]. |

Hybrid QECC | The capacity of a hybrid quantum memory is determined by a convex region in the classical-quantum entropy plane [93]. The quantum capacity for simultaneous transmission of classical and quantum information has been derived [94]. The existence of a hybrid code protecting against a channel depends on certain matricial ranges [95]. |

Hyperbolic Floquet code | Finite encoding rate whose value depends on the hyperbolic lattice. The asymptotic rate is 1/8 for a lattice of octagons [96]. |

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

Jump code | An infinite family of jump codes asymptotically attains an upper bound on \(K\) [97; Thm. 27]. |

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 \(O(L)\). |

Lattice stabilizer code | BPT bound: Lattice qubit codes are limited by the Bravyi-Poulin-Terhal (BPT) bound [90] (see also [11,40,98]), which states that \(d \leq O(n^{1-1/D})\) and \(k d^{2/D-1} = O(n)\) for \(D\)-dimensional lattice geometries. The Bravyi-Terhal (BT) bound states that \(d = O(L^{D-1})\) [98]. Codes on a \(D\)-dimensional homogeneous Riemannian manifold with diameter \(L\) satisfy \(k = O(L^{D-2})\) [99]. Some non-locality is necessary to circumvent these bounds [100]. |

Lattice subsystem code | Subsystem BT bound: The subsystem BT bound is an upper bound of \(d = O(L^{D-1})\) on the distance [98] of lattice subsystem stabilizer codes arranged in a \(D\)-dimensional lattice of length \(L\) with \(n=L^D\). In particular, \(D=2\)-dimensional subsystem codes satisfy \(kd = O(n)\) [40]. More generally, there is a tradeoff theorem [101] stating that, for any logical operator, there is an equivalent logical operator with weight \(\tilde{d}\) such that \(\tilde{d}d^{1/(D-1)}=O(L^{D})\). ' |

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 [2]. |

Lossless expander balanced-product code | Asymptotically good QLDPC codes [102], assuming the existence of two-sided lossless expanders. |

Maximal-entanglement EA Galois-qudit stabilizer code | Maximal entanglement is required to achieve the EA hashing bound for the depolarizing channel using the father protocol from Refs. [103,104]; see [67; Footnote 2]. |

Monitored random-circuit code | Rate can be finite [105], depending on the family of random codes generated by the circuit. |

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

NTRU-GKP code | Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability. |

Neural network code | Neural network codes can be obtained by optimizing the coherent information [106]. |

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 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. |

Perfect quantum code | \(k/n\to 1\) asymptotically with \(n\). |

Polar c-q code | Codes achieve the symmetric Holevo information for sending classical information over any quantum channel. |

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

Quantum AG code | Quantum AG codes [107] can be asymptotically good. There exist three such families [108–110] that admit a diagonal transversal gate at the third level of the Clifford hierarchy. |

Quantum Goppa code | Quantum Goppa codes [111] can be asymptotically good. |

Quantum LDPC (QLDPC) code | Asymptotic scaling of \(k\) and \(d\) with \(n\) depends heavily on the code construction. Bounds generalizing the BPT bound to QLDPC codes depend on the separation profile of the code's underlying connectivity graph [112,113]. A constant relative minimum distance can be achieved only for graphs that contain expanders [112]. Conversely, a code with parameters \(k\) and \(d\) requires a graph with order \(\Omega(d)\) edges of length of order \(\Omega(d/n^{1/D})\) [114]. Random QLDPC codes found by solving certain constraint satisfaction problems (CSPs) practically achieve the capacity of the erasure channel [115]. Qubit QLDPC codes cannot attain the capacity of the erasure channel [116], but this capacity can be attained by code families with weight \(w = O(\text{polylog}n)\) [117]. |

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 [118]. |

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 error-correcting code (QECC) | The quantum channel capacity, i.e., the regularized coherent information, is the highest rate of quantum information transmission through a quantum channel with arbitrarily small error rate [119–121]. See [122; Ch. 24] for definitions and a history. |

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 [123]. |

Quasi-hyperbolic color code | A construction based on the Torelli mapping yields a code with constant rate with similar gates [124]. |

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 [88]. |

Qubit code | Exact two-way assisted capacities have been obtained for the erasure and dephasing channels [29]. |

Qubit stabilizer code | The hashing bound states that there is a qubit stabilizer code achieving a rate \(R = 1 - H(\mathbf{p})\) for a Pauli noise channel with Pauli error probabilities \(\mathbf{p}=(p_I,p_X,p_Y,p_Z)\), where \(H(\mathbf{p})\) is the entropy of the argument [125; Thm. 23.6.2]. Finite block length bounds and a refinement of the hashing bound have been developed [126]. |

Random stabilizer code | Random qubit stabilizer codes achieve the quantum GV bound [127,128]; see notes [129]. In fact, sampling random CSS codes is sufficient [130]. |

SYK code | SYK codes can have a constant rate and distance scaling as \(n^c\) for some power \(c\) [131]. |

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

Six-qubit-tensor holographic code | Zero-rate version of the code surpasses the hashing bound certain Pauli noise [132]. |

Sparse subsystem code | There exists a family of sparse subsystem codes with \(d = n^{1-\epsilon}\), where \(\epsilon = O(1/\sqrt{\log n})\) [133]. |

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 [133]. |

Surface-code-fragment (SCF) holographic code | Zero-rate version of the code surpasses the hashing bound under certain Pauli noise [132]. |

Topological code | 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 [134,135], \begin{align} K=\sum_{a\in A}\left(d_{a}/D\right)^{\chi(\Sigma^{2})}~, \tag*{(3)}\end{align} and a generalization of the formula to the non-orientable case can be found in Ref. [136]. |

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 [137]. 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 [138]. |

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 \(d \in \Theta(n^{2/3})\) [139]. |

\([[144,12,12]]\) 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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