Here is a list of codes related to cyclic quantum codes.
| Code | Description |
|---|---|
| Binary dihedral PI code | Multi-qubit PI code designed to realize gates from the binary dihedral group transversally. Can also be interpreted as a single-spin code. The codespace projection is a projection onto an irrep of the binary dihedral group \( \mathsf{BD}_{2N} = \langle\omega I, X, P\rangle \) of order \(8N\), where \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \). |
| Clifford spin code | A single-spin code designed to realize a discrete group of gates using \(SU(2)\) rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of a discrete subgroup of \(SU(2)\). |
| Combinatorial PI code | A member of a family of PI quantum codes whose correction properties are derived from solving a family of combinatorial identities. The code encodes one logical qubit in superpositions of Dicke states whose coefficients are square roots of ratios of binomial coefficients. |
| Cyclic quantum code | A block quantum code such that cyclic permutations of the subsystems leave the codespace invariant. In other words, the automorphism group of the code contains the cyclic group \(\mathbb{Z}_n\). |
| Five-rotor code | Extension of the five-qubit stabilizer code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable. |
| Four-qubit single-deletion code | Four-qubit PI code that is the smallest qubit code to correct one deletion error. |
| Frobenius code | A cyclic prime-qudit stabilizer code whose length \(n\) divides \(p^t + 1\) for some positive integer \(t\). |
| GNU PI code | PI code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution. |
| La-cross code | Code constructed using a hypergraph product of two copies of a cyclic LDPC code. The construction uses cyclic LDPC codes with generating polynomials \(1+x+x^k\) for some \(k\). Using a length-\(n\) seed code yields an \([[2n^2,2k^2]]\) family for periodic boundary conditions and an \([[(n-k)^2+n^2,k^2]]\) family for open boundary conditions. |
| Ouyang-Chao constant-excitation PI code | A constant-excitation PI Fock-state code whose construction is based on integer partitions. |
| PI qubit code | Block quantum code defined on two-dimensional subsystems such that any permutation of the subsystems leaves any codeword invariant. |
| Permutation-invariant (PI) code | Block quantum code such that any permutation of the subsystems leaves any codeword invariant. In other words, the automorphism group of the code contains the symmetric group \(S_n\). |
| Post-selected PI code | PI qubit code whose recovery succeeds at protecting against AD errors with a success probability less than one. |
| Quantum repetition code | Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). The code is called a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\). |
| Qudit GNU PI code | Extension of the GNU PI codes to those encoding logical qudits into physical qubits. Codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of polynomial coefficients, with the case of binomial coefficients reducing to the GNU PI codes. |
| Tetron code | A \([[2,1,2]]_{f}\) Majorana box qubit, encoding two fixed-fermion states into the four-dimensional ground-state space of two Kitaev chains, each of length two. The code encodes a logical qubit into four Majorana modes (i.e., two physical fermions), allowing it to be concatenated with various qubit codes such as surface codes and color codes. Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [1] and other qubit Hamiltonians on certain graphs [2,3]. It has been used throughout condensed matter physics under the name of the Majorana representation [4,5] or parton construction [6], allowing for a mean-field treatment of many models that are otherwise not amenable. Majorana stabilizer groups can be converted into ordinary qubit stabilizer groups via the parton mapping, while their corresponding states are converted via the Gutzwiller projection [7]. |
| Twisted XZZX toric code | A cyclic code that can be thought of as the XZZX toric code with shifted (a.k.a twisted) boundary conditions. Admits a set of stabilizer generators that are cyclic shifts of a particular weight-four \(XZZX\) Pauli string. For example, a seven-qubit \([[7,1,3]]\) variant has stabilizers generated by cyclic shifts of \(XZIZXII\) [8]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [9]. |
| Twisted \(1\)-group code | Block group-representation code realizing particular irreps of particular groups such that a distance of two is automatically guaranteed. Groups which admit irreps with this property are called twisted (unitary) \(1\)-groups and include the binary icosahedral group \(2I\), the \(\Sigma(360\phi)\) subgroup of \(SU(3)\), the family \(\{PSp(2b, 3), b \geq 1\}\), and the alternating groups \(A_{5,6}\). Groups whose irreps are images of the appropriate irreps of twisted \(1\)-groups also yield such properties, e.g., the binary tetrahedral group \(2T\) or qutrit Pauli group \(\Sigma(72\phi)\). |
| Very small logical qubit (VSLQ) code | The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), where the total Hilbert space is the tensor product of two transmon qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords). Since the code is intended to protect against losses, the qutrits can equivalently be thought of as oscillator Fock-state subspaces. |
| Wasilewski-Banaszek code | Three-oscillator constant-excitation Fock-state code encoding a single logical qubit. |
| \(((5,3,2))_3\) qutrit code | Smallest qutrit block code realizing the \(\Sigma(360\phi)=3.A_6\) subgroup of \(SU(3)\) transversally. The next smallest code is \(((7,3,2))_3\). |
| \(((5,6,2))\) qubit code | Six-qubit cyclic CWS code detecting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[5,2,2]]\) code, the best five-qubit stabilizer code with the same distance [10]. |
| \(((7,2,3))\) Pollatsek-Ruskai code | Seven-qubit PI code that realizes gates from the binary icosahedral group transversally. Can also be interpreted as a spin-\(7/2\) single-spin code. The codespace projection is a projection onto an irrep of the binary icosahedral group \(2I\). See Ref. [11] for other non-PI codes realizing \(2I\) gates transversally. |
| \(((9,12,3))\) qubit code | Nine-qubit cyclic CWS code correcting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[9,3,3]]\) code, the best nine-qubit stabilizer code with the same distance [12]. |
| \(((9,2,3))\) Ruskai code | Nine-qubit PI code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes. |
| \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code | PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) [13; Appx. D] (cf. [14]). |
| \([[13,1,5]]\) cyclic code | Thirteen-qubit twisted surface code for which there is a set of stabilizer generators consisting of cyclic permutations of the \(XZZX\)-type Pauli string \(XIZZIXIIIIIII\). The code can be thought of as a small twisted XZZX code [15; Exam. 11 and Fig. 3]. |
| \([[5,1,3]]\) Five-qubit perfect code | Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. |
| \([[5,1,3]]_q\) Galois-qudit code | True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(GF(q)\) over \(GF(p)\). |
| \([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code | An analog stabilizer version of the five-qubit perfect code, encoding one mode into five and correcting arbitrary errors on any one mode. |
| \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code | Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [16]; see also [17; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations. |
References
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