Here is a list of codes related to quantum locally testable codes.
| Code | Description |
|---|---|
| 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. |
| Dinur-Lin-Vidick (DLV) code | Member of a family of CSS codes defined by the degree-\(k\) cochains of a tensor-product sheaf on a \(t\)-dimensional cubical complex [1]. The complexes are \(t\)-order extensions of those underlying expander codes (\(t=1\)) and expander lifted-product codes (\(t=2\)). |
| 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. |
| Gauged sheaf QLTC | Qubit code whose dimension has a prime divisor that grows with \(n\), obtained by gauging three sheaf-code QLTC blocks with a cup-product trilinear form [2; Lemma 11]. The family has constant rate, linear distance, and constant operator soundness [2; Thm. 4]. |
| Good QLDPC code | Also called asymptotically good QLDPC codes. A family of QLDPC codes \([[n_i,k_i,d_i]]\) whose asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive. |
| Hemicubic code | Homological code constructed out of cubes in high dimensions. The hemicubic code family has a soundness lower bound of order \(\Omega(1/\log n)\), locality of stabilizer generators scaling as order \(O(\log n)\), and distance of order \(\Theta(\sqrt{n})\). |
| Hypersphere product code | Homological code based on products of hyperspheres. The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance of order \(\Theta(\sqrt{n})\). |
| Locally testable code (LTC) | Code for which one can efficiently check whether a given string is a codeword or is far from a codeword. Efficiency of the verification is quantified by the code’s query complexity \(u\), while effectiveness is quantified by the code’s soundness \(R\). |
| QLDPC code | Member of a family of stabilizer codes for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant as \(n\to\infty\). Sometimes, the two parameters are explicitly stated: each site of an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\). |
| Quantum check-product code | CSS code constructed from an extension of the check product (between two classical codes) to a product between a classical and a quantum code. |
| Quantum locally testable code (QLTC) | A local-projector Hamiltonian-based block quantum code which has a nonzero average-energy penalty for creating large errors. Informally, testing a randomly chosen local projector must reject states far from the codespace with probability bounded away from zero as \(n\) grows. |
| Qubit CSS code | An \([[n,k,d]]\) 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 \(\mathbb{Z}_2\) per the qubit CSS-to-homology correspondence below. |
| Reed-Solomon cubical sheaf code | DLV code on the degree-\(k\) cochains of a sheaf with projective Reed-Solomon local codes over an arithmetic quotient of a product of \(r\) Bruhat-Tits trees. Its parameters are chosen so that Ramanujan expansion and uniform product expansion of the local codes yield binary QLTCs with linear distance and constant soundness [3,4]. |
| 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 arbitrarily 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. |
| Transversal-\(CCZ\) Reed-Solomon cubical sheaf code | One of three degree-two Reed-Solomon cubical sheaf codes on a common six-dimensional complex, which together admit \(\Theta(N)\) addressable and parallel transversal logical \(CCZ\) gates for block length \(N\) [5]. Block \(a\in\{1,2,3\}\) uses lower-degree projective Reed-Solomon local codes in directions \(2a-1\) and \(2a\) than in the other four [5; Sec. 3.1]. |
List (
descendants_and_cousins): All descendants and cousins of Quantum locally testable code (QLTC).
References
- [1]
I. Dinur, T.-C. Lin, and T. Vidick, “Expansion of higher-dimensional cubical complexes with application to quantum locally testable codes”, (2025) arXiv:2402.07476
- [2]
F. Wei, Z. Han, Z. Li, and Z.-W. Liu, “NLTM Hamiltonians from gauged sheaf quantum locally testable codes”, (2026) arXiv:2609.40220
- [3]
W. Gay and F. G. Jeronimo, “Asymptotically Good Quantum Locally Testable Codes”, (2026) arXiv:2609.20780
- [4]
Y. Chen, M. M.-Y. Huang, Y. Liu, and E.-C. Tang, “Cubical Sheaf Complexes with Constant Expansion with Applications to Asymptotically Good qLTCs”, (2026) arXiv:2609.28028
- [5]
Y. Li, Z. Li, Z. Han, and Z.-W. Liu, “Parallelizable and addressable transversal non-Clifford gates on good quantum LDPC codes”, (2026) arXiv:2609.26691