Here is a list of approximate quantum codes.
| Code | Description |
|---|---|
| 2T-qutrit code | Two-mode qutrit code constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group \(2T\), a.k.a. the 24-cell. |
| Amplitude-damping (AD) code | Block quantum code on either qubits or bosonic modes that is designed to detect and correct qubit or bosonic AD errors, respectively. |
| Amplitude-damping CWS code | Self-complementary CWS code that is designed to detect and correct AD errors. |
| Approximate quantum error-correcting code (AQECC) | Encodes quantum information so that it is possible to approximately recover that information from noise up to an error bound in recovery. |
| Approximate secret-sharing code | A family of \( [[n,k,d]]_q \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) qubits, 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 an \(t\)-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. |
| Binomial code | Bosonic rotation codes designed to approximately protect against errors consisting of powers of raising and lowering operators up to some maximum power. Binomial codes can be thought of as spin-coherent states embedded into an oscillator [1]. |
| Bosonic \(q\)-ary expansion | A one-to-one mapping between basis states on \(n\) prime-dimensional qudits (of dimension \(q=p\)) and the subspace of the first \(p^n\) single-mode Fock states. While this mapping offers a way to map qudits into a single mode, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [2]. |
| Bosonic rotation code | A single-mode Fock-state bosonic code whose codespace is preserved by a phase-space rotation by a multiple of \(2\pi/N\) for some \(N\). The rotation symmetry ensures that encoded states have support only on every \(N^{\textrm{th}}\) Fock state. For example, single-mode Fock-state codes for \(N=2\) encoding a qubit admit basis states that are, respectively, supported on Fock state sets \(\{|0\rangle,|4\rangle,|8\rangle,\cdots\}\) and \(\{|2\rangle,|6\rangle,|10\rangle,\cdots\}\). |
| Brown-Fawzi Clifford-circuit code | An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log^3 n)\). |
| Cat code | Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator which utilizes a constellation of \(q(S+1)\) coherent states distributed equidistantly around a circle in phase space of radius \(\alpha\). |
| Cat-repetition code | A concatenated qubit-into-\(n\)-mode code whose outer code is a quantum repetition code and whose inner code is the cat code in its cat basis. |
| Chebyshev code | Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. |
| Chuang-Leung-Yamamoto (CLY) code | Bosonic Fock-state code that encodes \(k\) qubits into \(n\) oscillators, with each oscillator restricted to having at most \(N\) excitations. Codewords are superpositions of oscillator Fock states which have exactly \(N\) total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions. |
| Circuit-to-Hamiltonian approximate code | Approximate qubit block code that forms the ground-state space of a frustration-free Hamiltonian with non-commuting terms. Its distance and logical-qubit number are both of order \(\Omega(n/\log^5 n)\) [3; Thm. 3.1]. The code is an approximate non-stabilizer QLWC code since the Hamiltonian consists of non-commuting weight-ten non-Pauli projectors, with each qubit acted on by order \(O(\text{polylog}(n)\) projectors. |
| Clifford group-representation QSC | Non-uniform QSC whose projection is onto a copy of an irreducible representation of the single-qubit Clifford group, taken as the binary octahedral subgroup of the group \(SU(2)\) of Gaussian rotations. Its codewords consist of non-uniform superpositions of 48 coherent states. |
| Clifford subgroup-orbit QSC | A \(((2^r,2,2-\sqrt{2},8))\) QSC for \(r \geq 2\) constructed using the real-Clifford subgroup-orbit code. Logical constellations are constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group [4] to \(2\) different vectors on the complex sphere. The code is known as the Witting code for \(r=2\) because its two logical constellations form vertices of Witting polytopes. |
| Coherent-state repetition code | A concatenated qubit-into-\(n\)-mode code (for odd \(n\)) whose outer code is a quantum repetition code and whose inner code is the two-component cat code in its coherent-state basis. |
| Concatenated cat code | A concatenated code whose outer code is a cat code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another cat outer code. Most examples encode physical qubits of an inner stabilizer code into the two-component cat code in its cat-state basis. |
| Conformal-field theory (CFT) code | Approximate code whose codewords lie in the low-energy subspace of a conformal field theory, e.g., the quantum Ising model at its critical point [5,6]. Its encoding is argued to perform source coding (i.e., compression) as well as channel coding (i.e., error correction) [5]. |
| Constant-excitation (CE) code | Code whose codewords lie in an excited-state eigenspace of a Hamiltonian governing the total energy or total number of excitations of the underlying quantum system. For qubit codes, such a Hamiltonian is often the total spin Hamiltonian, \(H=\sum_i Z_i\). For spin-\(S\) codes, this generalizes to \(H=\sum_i J_z^{(i)}\), where \(J_z\) is the spin-\(S\) \(Z\)-operator. For bosonic (and, similarly, for fermion) codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the total excitation or energy Hamiltonian, \(H=\sum_i \hat{n}_i\). |
| Covariant block quantum code | A block code on \(n\) subsystems that admits a group \(G\) of transversal gates. The group has to be finite for finite-dimensional codes due to the Eastin-Knill theorem. Continuous-\(G\) covariant codes, necessarily infinite-dimensional, are relevant to error correction of quantum reference frames [7] and error-corrected parameter estimation. |
| Crystalline-circuit qubit code | Code dynamically generated by unitary Clifford circuits defined on a lattice with some crystalline symmetry. A notable example is the circuit defined on a rotated square lattice with vertices corresponding to iSWAP gates and edges decorated by \(R_X[\pi/2]\), a single-qubit rotation by \(\pi/2\) around the \(X\)-axis. This circuit is invariant under space-time translations by a unit cell \((T, a)\) and all transformations of the square lattice point group \(D_4\). |
| Dual-rail quantum code | Two-mode bosonic code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used. |
| Dynamical code | Dynamically generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection. |
| Eigenstate thermalization hypothesis (ETH) code | An \(n\)-qubit approximate code whose codespace is formed by eigenstates of a translationally-invariant quantum many-body system which satisfies the Eigenstate Thermalization Hypothesis (ETH). ETH ensures that codewords cannot be locally distinguished in the thermodynamic limit. Relevant many-body systems include 1D non-interacting spin chains or frustration-free systems such as Motzkin chains and Heisenberg models. |
| Fock-state bosonic code | Qudit-into-oscillator code whose protection against AD noise (i.e., photon loss) stems from the use of disjoint sets of Fock states for the construction of each code basis state. The simplest example is the dual-rail code, which has codewords consisting of single Fock states \(|10\rangle\) and \(|01\rangle\). This code can detect a single loss error since a loss operator in either mode maps one of the codewords to a different Fock state \(|00\rangle\). More involved codewords consist of several well-separated Fock states such that multiple loss events can be detected and corrected. |
| Four-qubit single-deletion code | Four-qubit PI code that is the smallest qubit code to correct one deletion error. |
| GNU PI code | PI code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution. |
| Haar-random qubit code | Haar-random codewords are generated in a process involving averaging over unitary operations distributed according to the Haar measure. Haar-random codes are used to prove statements about the capacity of a quantum channel to transmit quantum information [8], but encoding and decoding in such \(n\)-qubit codes quickly becomes impractical as \(n\to\infty\). |
| Hessian QSC | Quantum spherical code encoding a logical qubit, with each codeword an equal superposition of vertices of a Hessian complex polyhedron. For the unit sphere, the codewords are \begin{align} |\overline{0}\rangle &= \frac{1}{\sqrt{27}}\left( \sum_{\mu,\nu=0}^{2} |0,\omega^{\mu},-\omega^{\nu}\rangle + |-\omega^{\nu},0,\omega^{\mu}\rangle + |\omega^{\mu},-\omega^{\nu},0\rangle \right) \tag*{(1)}\\ |\overline{1}\rangle &= \frac{1}{\sqrt{27}}\left( \sum_{\mu,\nu=0}^{2} |0,-\omega^{\mu},\omega^{\nu}\rangle + |\omega^{\nu},0,-\omega^{\mu}\rangle + |-\omega^{\mu},\omega^{\nu},0\rangle \right)~, \tag*{(2)}\end{align} where \(\omega = e^{\frac{2\pi i}{3}}\).  |
| Holographic code | Block quantum code whose features serve to model aspects of the AdS/CFT holographic duality and, more generally, quantum gravity. |
| Jump code | A CE code designed to detect and correct AD errors. An \(((n,K))\) jump code is denoted as \(((n,K,t))_w\) (which conflicts with modular-qudit notation), where \(t\) is the maximum number of qubits that can be corrected after each one has undergone a jump error \(|0\rangle\langle 1|\), and where each codeword is a uniform superposition of qubit basis states with Hamming weight \(w\). |
| Landau-level spin code | Approximate quantum code that encodes a qudit in the finite-dimensional Hilbert space of a single spin, i.e., a spherical Landau level. Codewords are approximately orthogonal Landau-level spin coherent states whose orientations are spaced maximally far apart along a great circle (equator) of the sphere. The larger the spin, the better the performance. |
| Local Haar-random circuit qubit code | An \(n\)-qubit code whose codewords are a pair of approximately locally indistinguishable states produced by starting with any two orthogonal \(n\)-qubit states and acting with a random unitary circuit of depth polynomial in \(n\). Two states are locally indistinguishable if they cannot be distinguished by local measurements. A single layer of the encoding circuit is composed of about \(n/2\) two-qubit nearest-neighbor gates run in parallel, with each gate drawn randomly from the Haar distribution on two-qubit unitaries. |
| Log-depth geometrically local Clifford-circuit code | A random \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log n)\) on a 1D Euclidean geometry. |
| Magnon code | An \(n\)-spin approximate code whose codespace of \(k=\Omega(\log n)\) qubits is efficiently described in terms of particular matrix product states or Bethe ansatz tensor networks. Magnon codewords are low-energy excited states of the frustration-free Heisenberg-XXX model Hamiltonian [9]. |
| Matrix-model code | Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry. |
| Monitored random-circuit code | Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittant random local projective Pauli measurements with random unitary time-evolution operators. An important sub-family consists of Clifford monitored random circuits, where unitaries are sampled from the Clifford group [10]. When the rate of projective measurements is independently controlled by a probability parameter \(p\), there can exist two stable phases, one described by volume-law entanglement entropy and the other by area-law entanglement entropy. The phases and their transition can be understood from the perspective of quantum error correction, information scrambling, and channel capacities [11,12]. |
| Neural network quantum code | An approximate qubit code obtained from a numerical optimization involving a reinforcement learning agent. |
| Number-phase code | Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states [13]. |
| Numerically optimized bosonic code | Bosonic Fock-state code obtained from a numerical minimization procedure, e.g., from enforcing error-correction criteria against some number of losses while minimizing average occupation number. Useful single-mode codes can be determined using basic numerical optimization [1,14], semidefinite-program recovery/encoding optimization [15,16], or reinforcement learning [17,18]. |
| Numerically optimized four-qubit AD code | One of several four-qubit codes that can (approximately) correct a single AD error with higher fidelity than the \([[4,1,2]]\) subcodes of the \([[4,2,2]]\) code. |
| One-hot quantum code | Encoding of a \(q\)-dimensional qudit into the single-excitation subspace of \(q\) modes. The \(j\)th logical state is the multi-mode Fock state with one photon in mode \(j\) and zero photons in the other modes. This code is useful for encoding and performing operations on qudits in multiple qubits [19–23]. |
| Ouyang-Chao constant-excitation PI code | A constant-excitation PI Fock-state code whose construction is based on integer partitions. |
| Pair-cat code | Two- or higher-mode extension of cat codes whose codewords are right eigenstates of powers of products of the modes' lowering operators. Many gadgets for cat codes have two-mode pair-cat analogues, with the advantage being that such gates can be done in parallel with a dissipative error-correction process. |
| Pauli group-representation QSC | Non-uniform QSC whose projection is onto a copy of an irreducible representation of the single-qubit Pauli group, the symmetry group of the \(\{2,2,4\}\) tesselation of the sphere. Each codeword is a quantum superposition of vertices of a tetrahedron with \(\pm 1\) coefficients. |
| Post-selected PI code | PI qubit code whose recovery succeeds at protecting against AD errors with a success probability less than one. |
| Quantum error-correcting code (QECC) | Encodes quantum information in a (logical) subspace of a (physical) Hilbert space such that it is possible to recover said information from errors that act as linear maps on the physical space. The state space of a QECC is contained in the space of complex \(L^2\)-normalizable functions of some configuration space, which usually corresponds to the alphabet of a classical code. |
| Quantum low-weight check (QLWC) code | Member of a family of \([[n,k,d]]\) stabilizer codes for which the number of sites participating in each stabilizer generator is bounded by a constant as \(n\to\infty\). |
| 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 \(|\phi_1\rangle = |-\rangle\). |
| Quantum spherical code (QSC) | Code whose codewords are superpositions of points on an \(n\)-dimensional real or complex sphere. Such codes can in principle be defined on any configuration space housing a sphere, but the focus of this entry is on QSCs constructed out of coherent-state constellations. |
| 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. |
| Qudit-into-oscillator code | Encodes \(K\)-dimensional Hilbert space into \(n\) bosonic modes. |
| Random quantum code | Quantum code whose construction is non-deterministic in some way, i.e., codes that utilize an elements of randomness somewhere in their construction. Members of this class range from fully non-deterministic codes (e.g., random-circuit codes), to codes whose multi-step construction is deterministic with the exception of a single step (e.g., expander lifter-product codes). |
| Random stabilizer code | An \(n\)-qubit, modular-qudit, or Galois-qudit stabilizer code whose construction is non-deterministic. Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits. |
| Random-circuit code | Code whose encoding is naturally constructed by randomly sampling from a large set of (not necessarily unitary) quantum circuits. |
| Renormalization group (RG) cat code | Code whose codespace is spanned by \(q\) field-theoretic coherent states which are flowing under the renormalization group (RG) flow of massive free fields. The code approximately protects against displacements that represent local (i.e., short-distance, ultraviolet, or UV) operators. Intuitively, this is because RG cat codewords represent non-local (i.e., long-distance) degrees of freedom, which should only be excitable by acting on a macroscopically large number of short-distance degrees of freedom. |
| SYK code | Approximate \(n\)-fermionic code whose codewords are low-energy states of the Sachdev-Ye-Kitaev (SYK) Hamiltonian [24,25] or other low-rank SYK models [26,27]. |
| Self-complementary qubit code | A qubit code which admits a basis of codewords of the form \(|c\rangle+|\overline{c}\rangle\), where \(c\) is a bitstring and \(\overline{c}\) is its negation a.k.a. complement. Their codewords generalize the two-qubit Bell states and three-qubit GHZ states and are often called (qubit) cat states or poor-man's GHZ states. Such codes were originally pointed out to perform well against AD noise [28]. |
| Singleton-bound approaching AQECC | Approximate quantum code 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 [29,30]. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability. |
| Smolin-Smith-Wehner (SSW) code | A family of \(((n=4k+2l+3,M_{k,l},2))\) self-complementary CWS codes, where \(M_{k,l} \approx 2^{n-2}(1-\sqrt{2/(\pi(n-1))})\). For \(n \geq 11\), these codes have a logical subspace whose dimension is larger than that of the largest stabilizer code for the same \(n\) and \(d\). A subset of these codes can be augmented to yield codes with one higher logical dimension [31]. |
| 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. |
| Squeezed cat code | Two-component cat code whose two coherent states have been squeezed in a direction perpendicular to the segment formed by the two coherent state values \(\pm\alpha\). |
| Squeezed fock-state code | Approximate bosonic code that encodes a qubit into the same Fock state, but one which is squeezed in opposite directions. |
| Topological code | A code whose codewords form the ground-state or low-energy subspace of a (typically geometrically local) code Hamiltonian realizing a topological phase. A topological phase may be bosonic or fermionic, i.e., constructed out of underlying subsystems whose operators commute or anti-commute with each other, respectively. Unless otherwise noted, the phases discussed are bosonic. |
| Two-component cat code | Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\). |
| Two-mode binomial code | Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients. |
| Valence-bond-solid (VBS) code | An \(n\)-qubit approximate \(q\)-dimensional spin code family whose codespace is described in terms of \(SU(q)\) valence-bond-solid (VBS) [32] matrix product states with various boundary conditions. The codes become exact when either \(n\) or \(q\) go to infinity. The original work on these codes studied the \(q=2\) case [33]. |
| 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. |
| W-state code | Approximate block quantum code whose encoding resembles the structure of the W state [34]. This code enables universal quantum computation with transversal gates. |
| Wasilewski-Banaszek code | Three-oscillator constant-excitation Fock-state code encoding a single logical qubit. |
| \(((8,1,3))\) Plenio-Vedral-Knight CE code | An eight-qubit single error-correcting code that is the first CE code. Each logical state is a superposition of computational basis states with four excitations. |
| \(((9,2,3))\) Ruskai code | Nine-qubit PI code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes. |
| \(U(d)\)-covariant approximate erasure code | Covariant code whose construction takes in an arbitrary erasure-correcting code to yield an approximate QECC that is also covariant with respect to the unitary group. |
| \([[2^D,D,2]]\) hypercube quantum code | Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. It can be generalized to a \([[4^D,D,4]]\) error-correcting family [35]. Various other concatenations give families with increasing distance (see cousins). |
| \([[2m,2m-2,2]]\) error-detecting code | Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [36; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [37]. |
| \([[4,2,2]]\) Four-qubit code | Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. |
| \([[6,4,2]]\) error-detecting code | Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHZ states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to equivalence [37; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [38]. |
| \([[8,3,2]]\) Smallest interesting color code | Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal \(CCZ\) gate. |
| \(\chi^{(2)}\) code | A \(3n\)-mode bosonic Fock-state code that requires only linear optics and the \(\chi^{(2)}\) optical nonlinear interaction for encoding, decoding, and logical gates. Codewords lie in Fock-state subspaces that are invariant under Hermitian combinations of the \(\chi^{(2)}\) nonlinearities \(abc^\dagger\) and \(i abc^\dagger\), where \(a\), \(b\), and \(c\) are lowering operators acting on one of the \(n\) triples of modes on which the codes are defined. Codewords are also \(+1\) eigenstates of stabilizer-like symmetry operators, and photon parities are error syndromes. |
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