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 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. |
| 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, 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 \(n\). |
| 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 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. |
