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. |
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]. |
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 [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 code | The quantum capacity of the AD channel [5] and the dephasing noise channel [6] are both known. The capacity of the displacement noise channel, the quantum analogue of AGWN, has been bounded using GKP codes [7,8]. Exact two-way assisted capacities have been obtained for the AD channels and quantum limited amplifiers in what is known as the PLOB bound [9]. These are examples of Gaussian channels, i.e., channels that map Gaussian states to Gaussian states [10–16]. |
Bravyi-Bacon-Shor (BBS) code | A class of BBS codes [17] saturate the subsystem bound \(kd = O(n)\) [18]. |
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 [19]. |
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 [20]. |
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 [21,22]. |
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 [23]. Concatenating Abelian LP codes with GKP codes can surpass the CSS Hamming bound [24]. |
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 [25]. |
EA quantum turbo code | Maximal-entanglement EA quantum turbo codes come close to achieving the EA hashing bound [26]; see [27; Footnote 2]. |
EA qubit code | There are EA versions of classical and quantum capacities [28]. EA hashing bounds on the minimum entanglement required to achieve the entanglement-assisted channel capacity are derived [29]. |
EA qubit stabilizer code | Asymptotically good EA qubit stabilizer codes exist [30]. |
Expander LP code | Expander lifted-product codes for non-Abelian groups include the first examples [31] 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 \(\sqrt{n}~\text{polylog}(n)\). |
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 [32]. |
Generalized bicycle (GB) code | GB codes can achieve an asymptotic rate of 1/4 [33]. 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. |
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 [7,8,34,35]. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel [7]. |
Guth-Lubotzky code | An explicit construction based on Coxeter groups yields a lower bound of \(13/72\) on the asymptotic rate [36]. |
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 [37,38]. 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 \(O(\sqrt{n})\) [39,40], 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 [41]. |
Hyperbolic Floquet code | Finite encoding rate whose value depends on the hyperbolic lattice. The asymptotic rate is 1/8 for a lattice of octagons [42]. |
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 \(K\) [43; 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 \(O(L)\). |
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 [44], 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. [45,46]; see [27; Footnote 2]. |
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 [47]. |
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. |
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 [48]. |
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 [49]. |
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 [37]. |
Qubit code | Exact two-way assisted capacities have been obtained for the erasure and dephasing channels [9]. |
SYK code | SYK codes can have a constant rate and distance scaling as \(n^c\) for some power \(c\) [50]. |
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 [51]. |
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 [52]. 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 [53]. |
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 \(d \in \Theta(n^{2/3})\) [54]. |
\([[144,12,12]]\) 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 \(2^r\). |
References
- [1]
- 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
- [2]
- 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
- [3]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
- [4]
- F. G. Jeronimo et al., “Explicit Abelian Lifts and Quantum LDPC Codes”, (2021) arXiv:2112.01647
- [5]
- M. M. Wolf, D. Pérez-García, and G. Giedke, “Quantum Capacities of Bosonic Channels”, Physical Review Letters 98, (2007) arXiv:quant-ph/0606132 DOI
- [6]
- 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
- [7]
- J. Harrington and J. Preskill, “Achievable rates for the Gaussian quantum channel”, Physical Review A 64, (2001) arXiv:quant-ph/0105058 DOI
- [8]
- 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
- [9]
- S. Pirandola et al., “Fundamental limits of repeaterless quantum communications”, Nature Communications 8, (2017) arXiv:1510.08863 DOI
- [10]
- B. Demoen, P. Vanheuverzwijn, and A. Verbeure, “Completely positive quasi-free maps of the CCR-algebra”, Reports on Mathematical Physics 15, 27 (1979) DOI
- [11]
- J. Eisert and M. M. Wolf, “Gaussian quantum channels”, (2005) arXiv:quant-ph/0505151
- [12]
- M. M. Wolf, “Not-So-Normal Mode Decomposition”, Physical Review Letters 100, (2008) arXiv:0707.0604 DOI
- [13]
- F. Caruso et al., “Multi-mode bosonic Gaussian channels”, New Journal of Physics 10, 083030 (2008) arXiv:0804.0511 DOI
- [14]
- A. S. Holevo, “The Choi–Jamiolkowski forms of quantum Gaussian channels”, Journal of Mathematical Physics 52, (2011) arXiv:1004.0196 DOI
- [15]
- F. Caruso et al., “Optimal unitary dilation for bosonic Gaussian channels”, Physical Review A 84, (2011) arXiv:1009.1108 DOI
- [16]
- J. S. Ivan, K. K. Sabapathy, and R. Simon, “Operator-sum representation for bosonic Gaussian channels”, Physical Review A 84, (2011) arXiv:1012.4266 DOI
- [17]
- T. J. Yoder, “Optimal quantum subsystem codes in two dimensions”, Physical Review A 99, (2019) arXiv:1901.06319 DOI
- [18]
- S. Bravyi, “Subsystem codes with spatially local generators”, Physical Review A 83, (2011) arXiv:1008.1029 DOI
- [19]
- W. Brown and O. Fawzi, “Short random circuits define good quantum error correcting codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1312.7646 DOI
- [20]
- S. Bravyi, B. Leemhuis, and B. M. Terhal, “Topological order in an exactly solvable 3D spin model”, Annals of Physics 326, 839 (2011) arXiv:1006.4871 DOI
- [21]
- 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
- [22]
- 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
- [23]
- K. Fukui et al., “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
- [24]
- N. Raveendran et al., “Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound”, Quantum 6, 767 (2022) arXiv:2111.07029 DOI
- [25]
- K. Guenda, S. Jitman, and T. A. Gulliver, “Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes”, (2016) arXiv:1606.00134
- [26]
- M. M. Wilde, M.-H. Hsieh, and Z. Babar, “Entanglement-Assisted Quantum Turbo Codes”, IEEE Transactions on Information Theory 60, 1203 (2014) arXiv:1010.1256 DOI
- [27]
- C.-Y. Lai, T. A. Brun, and M. M. Wilde, “Duality in Entanglement-Assisted Quantum Error Correction”, IEEE Transactions on Information Theory 59, 4020 (2013) arXiv:1302.4150 DOI
- [28]
- C. H. Bennett et al., “Entanglement-Assisted Classical Capacity of Noisy Quantum Channels”, Physical Review Letters 83, 3081 (1999) arXiv:quant-ph/9904023 DOI
- [29]
- G. Bowen, “Entanglement required in achieving entanglement-assisted channel capacities”, Physical Review A 66, (2002) arXiv:quant-ph/0205117 DOI
- [30]
- J. Qian and L. Zhang, “Entanglement-assisted quantum codes from arbitrary binary linear codes”, Designs, Codes and Cryptography 77, 193 (2014) DOI
- [31]
- P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
- [32]
- Y. Ouyang, “Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound”, IEEE Transactions on Information Theory 60, 3117 (2014) arXiv:1004.1127 DOI
- [33]
- R. Wang and L. P. Pryadko, “Distance bounds for generalized bicycle codes”, (2022) arXiv:2203.17216
- [34]
- 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
- [35]
- M. Rosati, A. Mari, and V. Giovannetti, “Narrow bounds for the quantum capacity of thermal attenuators”, Nature Communications 9, (2018) arXiv:1801.04731 DOI
- [36]
- N. P. Breuckmann and V. Londe, “Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance”, IEEE Transactions on Information Theory 68, 272 (2022) arXiv:2001.03568 DOI
- [37]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [38]
- N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
- [39]
- 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
- [40]
- E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
- [41]
- “Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002) DOI
- [42]
- A. Fahimniya et al., “Fault-tolerant hyperbolic Floquet quantum error correcting codes”, (2024) arXiv:2309.10033
- [43]
- T. Beth et al., Designs, Codes and Cryptography 29, 51 (2003) DOI
- [44]
- T.-C. Lin and M.-H. Hsieh, “Good quantum LDPC codes with linear time decoder from lossless expanders”, (2022) arXiv:2203.03581
- [45]
- I. Devetak, A. W. Harrow, and A. Winter, “A Family of Quantum Protocols”, Physical Review Letters 93, (2004) arXiv:quant-ph/0308044 DOI
- [46]
- I. Devetak, A. W. Harrow, and A. J. Winter, “A Resource Framework for Quantum Shannon Theory”, IEEE Transactions on Information Theory 54, 4587 (2008) arXiv:quant-ph/0512015 DOI
- [47]
- J. Bausch and F. Leditzky, “Quantum codes from neural networks”, New Journal of Physics 22, 023005 (2020) arXiv:1806.08781 DOI
- [48]
- 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
- [49]
- 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
- [50]
- G. Bentsen, P. Nguyen, and B. Swingle, “Approximate Quantum Codes From Long Wormholes”, (2023) arXiv:2310.07770
- [51]
- D. Bacon et al., “Sparse Quantum Codes From Quantum Circuits”, IEEE Transactions on Information Theory 63, 2464 (2017) arXiv:1411.3334 DOI
- [52]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
- [53]
- Z. Zhang, D. Aasen, and S. Vijay, “The X-Cube Floquet Code”, (2022) arXiv:2211.05784
- [54]
- A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI