Here is a list of Hamiltonian-based quantum codes that are non-stabilizer and non-constant-excitation (CE). All stabilizer codes [see Stabilizer codes (non-CSS, non-qubit), Qubit stabilizer codes (non-CSS), Quantum CSS codes (non-qubit), and Qubit CSS codes] are automatically Hamiltonian-based since their codewords are ground states of the stabilizer generator Hamiltonian. All CE codes [see Constant-excitation quantum codes] are associated with a Hamiltonian that counts the number of excitations, so they are also Hamiltonian-based.

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Code Description
3D Kitaev honeycomb code 3D subsystem stabilizer code whose Hamiltonian is a 3D generalization of the Kitaev honeycomb model. One of the phases realized by the 3D Kitaev honeycomb Hamiltonian is that of the 3D fermionic surface code [1].
Abelian TQD code TQD code whose codewords realize a 2D Abelian twisted-quantum-double topological order. For Abelian TQDs, the corresponding anyon theory is defined by an Abelian group and a group cocycle built from Type-I, Type-II, or Type-III 3-cocycles [2–4]. Abelian TQDs with Type-I and -II cocycles account for all 2D Abelian topological orders that admit gapped boundaries [5]. Abelian TQDs with Type-III cocycles may admit non-Abelian topological orders.
Abelian topological code Code whose codewords realize topological order associated with an Abelian anyon theory. In 2D, this is equivalent to a unitary braided fusion category which is also an Abelian group under fusion [6]. Unless otherwise noted, the phases discussed are bosonic.
Brickwork \(XS\) stabilizer code An \(XS\) stabilizer code that realizes the topological order of the Type-III \(G=\mathbb{Z}^3_2\) TQD model [7,8], which is the same topological order as the \(G=D_4\) quantum double [9]. Its qubits are placed on a 2D square lattice, and the stabilizers are defined using two overlapping rectangular tilings.
Cage-net code A commuting-projector code family obtained by coupling layers of two-dimensional topological orders and condensing extended one-dimensional flux strings [10]. A modern lattice realization starts from isotropic stacks of \(G\)-graded string-net models [11]. The family includes stabilizer examples such as the \(\mathbb{Z}_2\) string-membrane-net realization of the X-cube model [12] as well as non-stabilizer examples with non-Abelian restricted-mobility excitations [10]. String-membrane-net and cage-net constructions can realize the same fracton phases; for isotropic stacks, this equivalence can be understood via generalized local unitaries [11]. The cage-net construction can be used to realize various fracton phases, stabilizer and otherwise.
Chen-Hsin invertible-order code A geometrically local commuting-projector code that realizes beyond-group-cohomology invertible topological phases in arbitrary dimensions. Its code Hamiltonian terms include Pauli-\(Z\) operators and products of Pauli-\(X\) operators and \(CZ\) gates [13; Eq. (3.25)]. Instances of the code in 4D realize the 3D \(\mathbb{Z}_2\) gauge theory with fermionic charge and either bosonic (FcBl) or fermionic (FcFl) loop excitations at their boundaries [14,15]; see Ref. [16] for a different lattice-model formulation of the FcBl boundary code.
Chiral semion subsystem code Modular-qudit subsystem stabilizer code with qudit dimension \(q=4\) that is characterized by the chiral semion topological phase. Admits a set of geometrically local stabilizer generators on a torus.
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 \(\Omega(n/\log^5 n)\) [17; Thm. 3.1]. The code is an approximate non-stabilizer QLWC code since the Hamiltonian consists of non-commuting 9-local non-Pauli projectors, with each qubit acted on by order \(O( \text{polylog}(n) )\) projectors.
Commuting-projector Hamiltonian code Hamiltonian-based code whose Hamiltonian terms can be expressed as orthogonal projectors (i.e., Hermitian operators with eigenvalues 0 or 1) that commute with each other.
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 [18,19]. Its encoding is argued to perform source coding (i.e., compression) as well as channel coding (i.e., error correction) [18].
Cubic theory code A geometrically local commuting-projector code family defined on triangulations in arbitrary spatial dimensions. Its Hamiltonian contains Pauli-\(Z\) flux terms and non-Pauli Gauss-law terms built from products of Pauli-\(X\) operators and \(CZ\) gates. These commuting non-Pauli stabilizers realize higher-form \(\mathbb{Z}_2^3\) gauge theories with Abelian electric excitations and non-Abelian magnetic excitations.
Dihedral \(G=D_m\) quantum-double code Quantum-double code whose codewords realize topological order associated with the dihedral group \(D_m\) of order \(2m\). For \(m \geq 3\), these codes are non-Abelian, with the simplest case given by \(D_3=S_3\), the permutation group on three objects. On an oriented lattice, each edge hosts a \(2m\)-dimensional group qudit, and the codespace is the ground-state subspace of the corresponding quantum double Hamiltonian [20].
Dijkgraaf-Witten gauge theory code A code whose codewords realize \(D\)-dimensional lattice Dijkgraaf-Witten gauge theory [21,22] for a finite group \(G\) and a \(D+1\)-cocycle \(\omega\) in the cohomology class \(H^{D+1}( G, U(1) )\). When the cocycle is non-trivial, the gauge theory is called a twisted gauge theory. There exist lattice-model formulations in arbitrary spatial dimension [23]. Boundaries and excitations have been studied for arbitrary dimension [24].
Double-semion string-net code An \(XS\) stabilizer code that realizes the 2D double semion topological phase. The model can be extended to other spatial dimensions [25].
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.
Fibonacci string-net code Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings.
Frustration-free Hamiltonian code Hamiltonian-based code whose Hamiltonian is frustration free, i.e., whose ground states minimize the energy of each term.
Generalized 2D color code Member of a family of non-Abelian 2D topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)). Hamiltonian terms are built from group-based right- and left-multiplication \(X\)-type operators together with \(Z\)-type operators.
Groupoid toric code Extension of the Kitaev surface code from Abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism [26]. Some models admit fracton-like features such as extensive ground-state degeneracy and excitations with restricted mobility. The robustness of these features has not yet been established.
Hamiltonian-based code Code whose codespace corresponds to a set of energy eigenstates of a quantum-mechanical Hamiltonian i.e., a Hermitian operator whose expectation value measures the energy of its underlying physical system. The codespace is typically a set of low-energy eigenstates or ground states, but can include subspaces of arbitrarily high energy. Hamiltonians whose eigenstates are the canonical basis elements are called classical; otherwise, a Hamiltonian is called quantum.
Hexagonal \(CZ\) code A hexagonal-lattice realization of the \(2+1\)D \(l=m=n=1\) cubic theory / Type-III \(\mathbb{Z}_2^3\) twisted quantum double phase. Its stabilizers are products of Pauli-\(Z\) operators and \(CZ\) gates [7; Fig. 6][27; Fig. 3]. The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the Type-III \(G=\mathbb{Z}^3_2\) Abelian TQD model [7,8], which is the same topological order as the \(G=D_4\) non-Abelian quantum double [9]. The stabilizers include \(CZ\) operators acting on hexagonal loops, but a reduced version exists where only two \(CZ\) gates act on each loop [27].
Hopf-algebra quantum-double code Code whose codewords realize 2D gapped topological order defined on qudits valued in a Hopf algebra \(H\). The code Hamiltonian is a generalization [28,29] of the quantum double model from group algebras to Hopf algebras, as anticipated by Kitaev [20]. Boundaries of these models have been examined [30,31].
Kitaev honeycomb code Subsystem qubit stabilizer code underlying the Kitaev honeycomb model [32,33]. Its gauge generators are the two-qubit \(XX\), \(YY\), and \(ZZ\) link operators on the three edge types of the honeycomb lattice [33; Sec. 3.2]. Its stabilizer group is generated by loop operators, and syndrome extraction can be reduced to ordered measurements of the two-qubit link operators [33; Sec. 3.2]. This is the \(q=2\) instance of the \(\mathbb{Z}_q^{(1)}\) subsystem code and does not encode any logical qubits [33][34; Sec. 7.3].
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 [35].
Matrix-model code Multimode Fock-state bosonic approximate code derived from a matrix model, i.e., a bosonic theory with a large non-Abelian 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.
Movassagh-Ouyang Hamiltonian code This is a family of codes derived via an algorithm that takes as input any binary classical code and outputs a quantum code (note that this framework can be extended to \(q\)-ary codes). The algorithm is probabilistic but succeeds almost surely if the classical code is random. An explicit code construction does exist for linear distance codes encoding one logical qubit using Radon’s theorem [36,37]. For finite rate codes, there is no rigorous proof that the construction algorithm succeeds, and approximate constructions are described instead.
Multi-fusion string-net code Family of codes resulting from the string-net construction but whose input is a unitary multi-fusion category (as opposed to a unitary fusion category).
Non-Abelian Kitaev honeycomb code Code whose logical subspace in the gapped non-Abelian phase of the Kitaev honeycomb model with a magnetic field is labeled by different fusion outcomes of Ising anyons [32].
Quantum locally testable code (QLTC) A local commuting-projector Hamiltonian-based block quantum code which has a nonzero average-energy penalty for creating large errors. Informally, states that are far away from the codespace of a QLTC have to be excited states of a number of the code’s local projectors that scales linearly with \(n\).
Quantum-double code Group-based code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\). The code’s generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tessellation). The original Hamiltonian can be re-expressed via group-based right- and left-multiplication \(X\)-type as well as \(Z\)-type error operators [38; Sec. 3.3].
Quantum-triple code Group-based code whose codewords realize 3D topological order defined by a finite group \(G\).
SYK code Approximate \(n\)-fermionic code whose codewords are low-energy states of the Sachdev-Ye-Kitaev (SYK) Hamiltonian [39,40] or other low-rank SYK models [41,42].
Self-correcting quantum code A block quantum code that forms the ground-state subspace of an \(n\)-body geometrically local Hamiltonian whose logical information is recoverable for arbitrary long times in the \(n\to\infty\) limit after interaction with a sufficiently cold thermal environment. Typically, one also requires a decoder whose decoding time scales polynomially with \(n\) and a finite energy density.
String-net code A non-stabilizer commuting-projector code whose codewords realize a 2D topological order rendered by a Turaev-Viro topological field theory. The corresponding anyon theory is defined by a (multiplicity-free) unitary fusion category \( \mathcal{C} \). The code is defined on a cell decomposition dual to a triangulation of a two-dimensional surface, with a qudit of dimension \( |\mathcal{C}| \) located at each edge of the decomposition. For modular input categories, the emergent anyon theory is the doubled category \( \mathcal{C}\otimes\mathcal{C}^{*} \) [43]. These models realize local topological order (LTO) [44].
Symmetry-protected self-correcting quantum code A code which admits a restricted notion of thermal stability against symmetric perturbations, i.e., perturbations that commute with a set of operators forming a group \(G\) called the symmetry group.
Symmetry-protected topological (SPT) code A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing symmetry-protected topological (SPT) order.
Three-fermion (3F) subsystem code 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [45–47]. One version uses two qubits at each site [34], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [46,48]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit.
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.
Twisted quantum double (TQD) code Code whose codewords realize a 2D topological order rendered by a Chern-Simons topological field theory. The corresponding anyon theory is defined by a finite group \(G\) and a 3-cocycle \(\omega\in H^3( G, U(1) )\) [2,3,49]. Canonical TQD models [2] are defined on group-valued qudits.
Twisted quantum triple (TQT) code Group-based code realizing a 3D topological order rendered by a Dijkgraaf-Witten gauge theory. The corresponding anyon theory is defined by a finite group \(G\) and a Type-IV four-cocycle \(\omega\). Canonical TQT models [23,50] and other formulations whose ground states are in the same phase are all defined on group-valued qudits.
Two-gauge theory code A code whose codewords realize lattice two-gauge theory [51–59] for a finite two-group (a.k.a. a crossed module) in arbitrary spatial dimension. There exist several lattice-model formulations in arbitrary spatial dimension [60,61] as well as explicitly in 3D [62–65] and 4D [65], with the 3D case realizing the Yetter model [66–69].
Valence-bond-solid (VBS) code A member of an approximate \(q\)-dimensional spin-code family whose codespace is described in terms of \(SU(q)\) valence-bond-solid (VBS) [70] 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 [71].
Walker-Wang model code A non-stabilizer commuting-projector 3D topological code defined by a unitary braided fusion category \( \mathcal{C} \) (also known as a unitary premodular category). The code is defined on a cubic lattice that is resolved to be trivalent, with a qudit of dimension \( |\mathcal{C}| \) located at each edge. The codespace is the ground-state subspace of the Walker-Wang model Hamiltonian [72] and realizes the Crane-Yetter model [73–75]. A single-state version of the code provides a resource state for MBQC [76].
\(((4,2,2))\) Four-qubit single-deletion code Four-qubit PI code that is the smallest qubit code to correct one deletion error.
\(((n,2,2))\) Bravyi-Lee-Li-Yoshida PI code PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) [77; Appx. D] (cf. [78]).
\(G\)-enriched Walker-Wang model code A 3D topological code defined by a unitary \(G\)-crossed braided fusion category \( \mathcal{C} \) [79,80], where \(G\) is a finite group. The model realizes TQFTs that include two-gauge theories, those behind Walker-Wang models, as well as the Kashaev TQFT [81,82]. It has been generalized to include domain walls [83].
\([[4,2,2]]_{G}\) four group-qudit code \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits.
\(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code Modular-qudit 2D subsystem stabilizer code whose low-energy excitations realize a non-modular anyon theory with \(\mathbb{Z}_3\times\mathbb{Z}_9\) fusion rules. Encodes two qutrits when put on a torus.
\(\mathbb{Z}_q^{(1)}\) subsystem code Modular-qudit subsystem code, based on the Kitaev honeycomb model [32] and its generalization [84], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [85], which is modular for odd prime \(q\) and non-modular otherwise. Encodes a single \(q\)-dimensional qudit when put on a torus for odd \(q\), and a \(q/2\)-dimensional qudit for even \(q\). This code can be constructed using geometrically local gauge generators, but does not admit geometrically local stabilizer generators. For \(q=2\), the code reduces to the subsystem code underlying the Kitaev honeycomb model code as well as the honeycomb Floquet code.

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