Here is a list of non-qubit quantum CSS codes. For qubit CSS codes, see Qubit CSS codes.
| Code | Description |
|---|---|
| Abelian LP code | A lifted-product code whose lift group \(G\) is Abelian. The case of \(G\) being a cyclic group is a GB code (a.k.a. a quasi-cyclic LP code) [1; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with parameters \([[n,k=\Theta(\log n),d=\Theta(n/\log n)]]\) [1]. |
| Analog repetition code | An \([[n,1]]_{\mathbb{R}}\) analog stabilizer version of the quantum repetition code, encoding the position states of one mode into an odd number \(n\) of modes. |
| Analog surface code | An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory. |
| Approximate secret-sharing code | A family of \( [[n,k,d]]_q \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) Galois qudits, i.e., with approximate distance approaching the no-cloning bound \(n/2\). Constructed using a non-degenerate CSS code, such as a polynomial quantum code, and a classical authentication scheme. The code can be viewed as a \(t\)-error-tolerant secret sharing scheme. Since the code yields a small logical subspace using large registers that contain both classical and quantum information, it is not useful for practical error correction problems, but instead demonstrates the power of approximate quantum error correction. |
| Balanced product (BP) code | Family of CSS quantum codes obtained from two classical-code chain complexes that share a common group symmetry. The balanced product can be understood as taking the usual tensor or hypergraph product and then quotienting by the shared symmetry action. This can reduce the overall number of physical qubits \(n\) while, in favorable cases, preserving the number of encoded qubits and the code distance, thereby improving the encoding rate \(k/n\) and normalized distance \(d/n\) compared to the underlying tensor or hypergraph product. |
| Binary quantum Goppa code | A quantum AG code obtained from algebraic-geometric Goppa codes via the Galois-qudit CSS construction. |
| Bosonic CSS code | Bosonic stabilizer code admitting a set of stabilizer generators that are either position or momentum displacements. |
| Calderbank-Shor-Steane (CSS) stabilizer code | A stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type operators. The two sets of stabilizer generators can often be related to parts of a chain complex over the appropriate ring or field. |
| Compactified \(\mathbb{R}\) gauge theory code | An integer-homology bosonic CSS code realizing 2D \(U(1)\) gauge theory on bosonic modes. The code can be obtained from the analog surface code by condensing certain anyons [2]. This results in a pinning of each mode to the space of periodic functions, which is the Hilbert space of a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code. |
| Distance-balanced code | Galois-qudit CSS code obtained from a CSS code by increasing the smaller of the \(X\)- and \(Z\)-distances using a homological-product-based balancing step or one of its generalizations. The initial code is said to be unbalanced, i.e., tailored to noise biased toward either bit- or phase-flip errors, and the procedure can result in a code that treats both types of errors on a more equal footing. |
| Expander LP code | Family of \(G\)-lifted product codes constructed using two classical expander codes, equivalently two regular Tanner codes defined on the same expander graph [3]. For certain parameters, this construction yields the first asymptotically good QLDPC codes. Classical codes resulting from the same lifted-product complexes are one of the first two families of \(c^3\)-LTCs [4]. |
| Folded quantum RS (FQRS) code | CSS code on \(q^m\)-dimensional Galois-qudits that is constructed from folded RS (FRS) codes (i.e., an RS code whose coordinates have been grouped together) via the Galois-qudit CSS construction. This code is used to construct Singleton-bound approaching approximate quantum codes. |
| Four-rotor code | \([[4,2,2]]_{\mathbb Z}\) CSS rotor code that is an extension of the four-qubit code to the integer alphabet, i.e., the angular momentum states of a rotor. |
| Galois-qudit CSS code | An \([[n,k,d]]_q \) Galois-qudit true stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Galois-qudit Pauli strings. Codes can be defined from chain complexes over \(\mathbb{F}_q\) via an extension of qubit CSS-to-homology correspondence to Galois qudits. |
| Galois-qudit HGP code | A member of a family of Galois-qudit CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear \(q\)-ary codes. |
| Galois-qudit color code | Extension of the color code to 2D lattices of Galois qudits. |
| Galois-qudit expander code | Galois-qudit CSS code obtained from tensor products of chain complexes associated with an explicit family of expander codes with Reed-Solomon local checks. |
| Galois-qudit surface code | Extension of the surface code to 2D lattices of Galois qudits. |
| Generalized bicycle (GB) code | A quasi-cyclic Galois-qudit CSS code constructed using a generalized version of the bicycle ansatz [5] from a pair of equivalent index-two quasi-cyclic linear codes. Equivalently, the codes can be constructed via the lifted-product construction for \(G\) being a cyclic group [1; Sec. III.E]. |
| Generalized homological-product CSS code | CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. |
| Hayden-Nezami-Salton-Sanders bosonic code | An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structures associated with an \(n-1\) simplex. Relevant to the study of spacetime replication of quantum information [6]. |
| Homological rotor code | A CSS rotor code stabilized by a group of rotor \(X\)-type and \(Z\)-type generalized Pauli operators. Codes are formulated using an extension of the qubit CSS-to-homology correspondence to rotors. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension, i.e., encoding logical qudits instead of only logical rotors. |
| Integer-homology bosonic CSS code | A bosonic stabilizer code whose physical modes have been restricted, via a single GKP stabilizer, from the space of functions on the real line to the space of periodic functions. This restriction effectively realizes a rotor on each physical mode, allowing one to construct homological rotor codes out of displacement stabilizer groups. The stabilizer group is continuous, but contains discrete components in the form of the single-mode GKP stabilizers. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension. |
| Kitaev current-mirror qubit code | Member of the family of \([[2n,(0,2),(2,n)]]_{\mathbb{Z}}\) homological rotor codes storing a logical qubit on a thin Möbius strip. The ideal code can be obtained from a Josephson-junction [7] system [8]. |
| Lifted-product (LP) code | Galois-qudit code that utilizes the notion of a lifted product in its construction. Lifted products of certain classical Tanner codes are the first (asymptotically) good QLDPC codes. |
| Modular-qudit 3D surface code | A generalization of the 3D surface code to modular qudits. Qudits are placed on edges, \(Z\)-type stabilizer generators are placed on square plaquettes oriented in all three directions, and \(X\)-type stabilizers are placed on the six edges neighboring every vertex [9]. |
| Modular-qudit CSS code | An \(((n,K,d))_q\) modular-qudit stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over the ring \(\mathbb{Z}_q\) via an extension of qubit CSS-to-homology correspondence to modular qudits [10]. The homology group of the logical operators has a torsion component because the chain complexes are defined over a ring, which yields codes whose logical dimension is not a power of \(q\). |
| Modular-qudit GKP code | Modular-qudit analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value. |
| Modular-qudit lattice color code | Extension of the color code to lattices of modular qudits. Codes are defined analogously to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizers commute. This can be done by puncturing a hyperspherical lattice [11] or constructing a star-bipartition; see [12; Sec. III]. Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and/or any twist defects present. |
| Modular-qudit shift-resistant code | Monolithic code encoding a qubit into a single modular qudit and protecting against either \(Z\)-type or \(X\)-type modular-qudit Pauli shifts. |
| Modular-qudit surface code | Extension of the surface code to prime-dimensional [13,14] and more general modular qudits. Stabilizer generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. |
| Prime-qudit RM code | Modular-qudit stabilizer code constructed from GRM codes or their duals via the modular-qudit CSS construction. |
| Prime-qudit RS code | Prime-qudit CSS code constructed using two RS codes. |
| Prime-qudit triorthogonal code | An \(m \times n\) matrix over \(\mathbb{F}_p=\mathbb{Z}_p\) is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\) modulo \(p\), where addition and multiplication are done on \(\mathbb{F}_p\). The triorthogonal prime-qudit CSS code associated with the matrix is constructed by mapping nonzero entries in self-orthogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [15,16]. |
| Quantum Tamo-Barg (QTB) code | A member of a family of Galois-qudit CSS codes whose underlying classical codes consist of Tamo-Barg codes together with specific low-weight codewords. Folded versions of QTB codes, or FQTB codes, defined on qudits whose dimension depends on \(n\), yield explicit examples of QLRCs of arbitrary locality \(r\) [17; Cor. 64]. |
| Quantum quadratic-residue (QR) code | Galois-qudit \([[n,1]]_q\) pure self-dual Galois-qudit CSS code constructed from a dual-containing QR code via the Galois-qudit CSS construction. For \(q\) not divisible by \(n\), its distance satisfies \(d^2-d+1 \geq n\) when \(n \equiv 3\) modulo 4 [18; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [18; Thm. 41]. |
| Qudit X-cube model code | Generalization of the X-cube model code to modular qudits. |
| Rotor GKP code | GKP code protecting against small angular position and momentum shifts of a planar rotor. |
| Singleton-bound approaching AQECC | A member of an approximate quantum code family of rate \(R\) that can tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\). The formulation of such codes relies on a notion of quantum list decoding [19,20]. |
| Skew-cyclic CSS code | A member of a family of Galois-qudit CSS codes constructed from skew-cyclic classical codes over rings [21; Thm. 5.8]. See related study [22] that uses cyclic codes over rings. |
| Square-lattice GKP code | Single-mode GKP qudit-into-oscillator CSS code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension. |
| Triorthogonal code | Qubit CSS code whose \(X\)-type logicals and stabilizer generators form a triorthogonal matrix (defined below) in the symplectic representation. |
| Two-block CSS code | Galois-qudit CSS code whose stabilizer generator matrices \(H_X=(A_1,B_1)\) and \(H_Z=(B^T_2,-A^T_2)\), are constructed from four matrices satisfying \(A_1 B_2 - B_1 A_2 = 0\). In the case the two pairs are equal, we have \(H_X=(A,B)\) and \(H_Z=(B^T,-A^T)\), constructed from a pair of square commuting matrices \(A\) and \(B\). |
| Two-block group-algebra (2BGA) codes | 2BGA codes are the one-by-one, or smallest, LP codes: \(LP(a,b)\) is defined by a pair of group algebra elements \(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group. If \(|G|=\ell\), then the code has length \(n=2\ell\). |
| Type-II fractal spin-liquid code | A type-II fracton prime-qudit CSS code defined on a cubic lattice [23; Eqs. (D9-D10)]. |
| Zero-pi qubit code | A \([[2,(0,2),(2,1)]]_{\mathbb{Z}}\) homological rotor code on the smallest tiling of the projective plane \(\mathbb{R}P^2\). The ideal code can be obtained from a four-rotor Josephson-junction [7] system after a choice of grounding [8]. |
| \([[11,1,5]]_3\) qutrit Golay code | An \([[11,1,5]]_3\) code constructed from the ternary Golay code via the CSS construction. The code’s stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix of the ternary Golay code. |
| \([[13,1,5]]\) quantum QR code | Thirteen-qubit cyclic Hermitian qubit code derived from a quaternary quadratic-residue code using the Hermitian construction [25][24; pg. 11]. The code admits a stabilizer tableau whose rows are cyclic permutations of the Pauli string \(XXZZIZIIIZIZZ\). |
| \([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code | A family of CSS codes extending quantum Hamming codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [26]. |
| \([[3,1,2]]_3\) Three-qutrit code | A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. It has stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound. |
| \([[3,1,2]]_4\) three-Galois-quartrit code | Three-Galois-qudit CSS code over \(\mathbb{F}_4=\{0,1,\omega,\omega^2\}\) that encodes one logical Galois qudit and detects a single-qudit error. |
| \([[3,1,2]]_{\mathbb{Z}}\) Three-rotor code | \([[3,1,2]]_{\mathbb Z}\) rotor code that is an extension of the \([[3,1,2]]_3\) qutrit CSS code to the integer alphabet, i.e., the angular momentum states of a rotor. |
| \([[5,1,3]]_4\) Galois-qudit CSS code | Five-Galois-qudit CSS code over \(\mathbb{F}_4=\{0,1,\omega,\omega^2\}\) that encodes one logical Galois qudit and corrects a single-qudit error. |
| \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code | An analog stabilizer version of Shor’s nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode. |
| \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code to \(q\)-level systems. |
| \([[9m-k,k,2]]_3\) triorthogonal code | Member of the \([[9m-k,k,2]]_3\) family of triorthogonal qutrit codes (for \(k\leq 3m-2\)) that admit a transversal diagonal gate in the third level of the qudit Clifford hierarchy and that are relevant for magic-state distillation. |
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