Here is a list of codes related to quantum locally testable codes.
| Code | Description |
|---|---|
| Dinur-Lin-Vidick (DLV) code | Member of a family of codes constructed using cubical chain complexes, which are \(t\)-order extensions of the complexes underlying expander codes (\(t=1\)) and expander lifted-product codes (\(t=2\)). |
| Distance-balanced code | Galois-qudit CSS code constructed from a CSS code and a classical code using a distance-balancing procedure based on a generalized homological product. 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 is treats both types of errors on a more equal footing. The original distance-balancing procedure [1], later generalized [2; Thm. 4.2], can yield QLDPC codes [1; Thm. 1]. |
| Hemicubic code | Homological code constructed out of cubes in high dimensions. The hemicubic code family has asymptotically diminishing soundness that scales as 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 check product (between two classical codes) to a product between a classical and a quantum code. |
| 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, QLTC error states that are far away from the codespace have to be excited states by a number of the code’s local projectors that scales linearly with \(n\). |
| 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. |
| 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. The original criteria for a self-correcting quantum memory, informally known as the Caltech rules [3,4], also required finite-spin Hamiltonians. A concatenated quantum code with self-simulating control elements based on work by Gacs [5–9] yields a self-correcting quantum memory in 2D [10]. |
References
- [1]
- M. B. Hastings, “Weight Reduction for Quantum Codes”, (2016) arXiv:1611.03790
- [2]
- S. Evra, T. Kaufman, and G. Zémor, “Decodable quantum LDPC codes beyond the \(\sqrt{n}\) distance barrier using high dimensional expanders”, (2020) arXiv:2004.07935
- [3]
- C. G. Brell, “A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)”, New Journal of Physics 18, 013050 (2016) arXiv:1411.7046 DOI
- [4]
- O. Landon-Cardinal, B. Yoshida, D. Poulin, and J. Preskill, “Perturbative instability of quantum memory based on effective long-range interactions”, Physical Review A 91, (2015) arXiv:1501.04112 DOI
- [5]
- P. Gács, “Reliable computation with cellular automata”, Proceedings of the fifteenth annual ACM symposium on Theory of computing - STOC ’83 32 (1983) DOI
- [6]
- P. Gács, “Reliable computation with cellular automata”, Journal of Computer and System Sciences 32, 15 (1986) DOI
- [7]
- P. Gács, “Reliable Cellular Automata with Self-Organization”, Journal of Statistical Physics 103, 45 (2001) DOI
- [8]
- Gács, Peter. “Self-Correcting Two-Dimensional Arrays.” Adv. Comput. Res. 5 (1989): 223-326.
- [9]
- P. Gács, “Reliable Cellular Automata with Self-Organization”, Journal of Statistical Physics 103, 45 (2001) arXiv:math/0003117 DOI
- [10]
- G. Dünnweber, G. Styliaris, and R. Trivedi, “Quantum Memory and Autonomous Computation in Two Dimensions”, (2026) arXiv:2601.20818