Here is a list of all quantum codes that specify what rate they have.
Name Rate
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 [1].
Bivariate bicycle 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 \(k/2n\). For example, the \([[144,12,12]]\) code has ancilla-added rate \(1/24\). In contrast, the distance-13 surface code has ancilla-added rate \(1/338\).
Bosonic code The quantum capacity of the pure-loss channel [2] and the dephasing noise channel [3] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [4,5].
Bravyi-Bacon-Shor (BBS) code A class of BBS codes [6] saturate the subsystem bound \(kd = O(n)\) [7].
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 [8,9].
Dinur-Hsieh-Lin-Vidick (DHLV) code Asymptotically good QLDPC codes.
Expander LP code Expander lifted-product codes include the first examples [10] 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 \(0 \leq \alpha < 1\) [11].
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 [12]. 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.
Generalized surface code Rate depends on the underlying cellulation and manifold [13,14]. For general 2D manifolds, \(kd^2\leq c(\log k)^2 n\) for some constant \(c\) [15], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as \(O(\sqrt{n})\) [16,17], 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 [18].
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 [4,5,19,20]. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel [4].
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 \(d \geq \sqrt{n} \log n \).
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 \(O(L)\).
Lifted-product (LP) code There is no known simple way to compute the logical dimension \(k\) in the general case [11].
Lossless expander balanced-product code Asymptotically good QLDPC codes [21], 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 \(n\).
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 [22].
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 [23].
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 [13].
SYK code SYK codes can have a constant rate and distance scaling as \(n^c\) for some power \(c\) [24].
Singleton-bound approaching AQECC Given rate \(R\), tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\).
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 [25]. 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 \(W=8\).)
Two-dimensional color code For general 2D manifolds, \(kd^2 \leq c(\log k)^2 n\) for some constant \(c\) [15], meaning that color codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\).
Two-dimensional hyperbolic surface code Two-dimensional 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.
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 \(d \in \Theta(n^{2/3})\) [26].
\([[k+4,k,2]]\) 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 \(2^r\).

References

[1]
N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
[2]
M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0606132 DOI
[3]
L. Lami and M. M. Wilde, “Exact solution for the quantum and private capacities of bosonic dephasing channels”, Nature Photonics 17, 525 (2023) arXiv:2205.05736 DOI
[4]
J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
[5]
K. Noh, V. V. Albert, and L. Jiang, “Quantum Capacity Bounds of Gaussian Thermal Loss Channels and Achievable Rates With Gottesman-Kitaev-Preskill Codes”, IEEE Transactions on Information Theory 65, 2563 (2019) arXiv:1801.07271 DOI
[6]
T. J. Yoder, “Optimal quantum subsystem codes in two dimensions”, Physical Review A 99, (2019) arXiv:1901.06319 DOI
[7]
S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
[8]
F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent-state constellations and polar codes for thermal Gaussian channels”, Physical Review A 95, (2017) arXiv:1603.05970 DOI
[9]
F. Lacerda, J. M. Renes, and V. B. Scholz, “Coherent state constellations for Bosonic Gaussian channels”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) DOI
[10]
P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
[11]
P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022) arXiv:2012.04068 DOI
[12]
R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes”, (2022) arXiv:2203.17216
[13]
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[14]
N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
[15]
N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1301.6588 DOI
[16]
S. Bravyi, D. Poulin, and B. Terhal, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems”, Physical Review Letters 104, (2010) arXiv:0909.5200 DOI
[17]
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
[18]
“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002) DOI
[19]
K. Sharma et al., “Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels”, New Journal of Physics 20, 063025 (2018) arXiv:1708.07257 DOI
[20]
M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018) arXiv:1801.04731 DOI
[21]
T.-C. Lin and M.-H. Hsieh, “Good quantum LDPC codes with linear time decoder from lossless expanders”, (2022) arXiv:2203.03581
[22]
S. Kumar, R. Calderbank, and H. D. Pfister, “Reed-muller codes achieve capacity on the quantum erasure channel”, 2016 IEEE International Symposium on Information Theory (ISIT) (2016) DOI
[23]
M. M. Wilde and J. M. Renes, “Quantum polar codes for arbitrary channels”, 2012 IEEE International Symposium on Information Theory Proceedings (2012) arXiv:1201.2906 DOI
[24]
G. Bentsen, P. Nguyen, and B. Swingle, “Approximate Quantum Codes From Long Wormholes”, (2023) arXiv:2310.07770
[25]
H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
[26]
A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI