Mapping cone code[1]
Description
A CSS code obtained from an input CSS code, called the embedded code, by adding physical qubits and parity checks in a way that guarantees a natural isomorphism between the logical operators of the input and output codes. The framework unifies various quantum code embedding procedures, including code concatenation, weight reduction, layer codes, geometrically local codes obtained by subdivision, and gauging-based fault-tolerant logical measurement [1].
Formally, a height-\(n\) cone is a chain complex \(C\) whose spaces decompose into direct sums, \(C_i = C_i^n \oplus C_i^{n-1} \oplus \cdots \oplus C_i^0\), with a boundary operator that is lower triangular with respect to the decomposition. The diagonal blocks \(\partial^s\) define chain complexes \(C^s\) called the levels of the cone, while the sub-diagonal blocks \(g_s\), called gluing maps, induce maps on homology that assemble the level homologies into the embedded complex \(H_n(\partial^n) \to H_{n-1}(\partial^{n-1}) \to \cdots \to H_0(\partial^0)\). If the levels have trivial homology away from the appropriate degrees, a condition called regularity, then the logical operators of the cone are naturally isomorphic to those of the embedded code [1]. The construction is an iterated application of the mapping cone of homological algebra.
Protection
A cleaning lemma states that, whenever a level boundary map and a gluing map satisfy an isoperimetric inequality, any logical representative of the cone can be cleaned out of the corresponding level, yielding a lower bound on the code distance of the cone in terms of the minimal weight of representatives of embedded-code logical operators [1]. This lemma unifies distance bounds for layer codes, subdivision-based geometrically local codes, weight-reduced codes, and codes obtained by gauging logical operators [1].Cousins
- Concatenated qubit code— The mapping cone framework can be regarded as a generalization of code concatenation for CSS codes [1].
- Distance-balanced code— Height-1 cones underlie Hastings’ coning procedure for weight reduction [2] as well as fault-tolerant logical measurement schemes [3–5], while general quantum embeddings require cones of height at least two [1].
- Good QLDPC code— Optimal embeddings of good QLDPC codes into \(D\)-dimensional Euclidean space via layer codes and subdivided square complexes fit within the mapping cone framework, which streamlines their logical-preservation and distance proofs [1].
- Homological code— Homological codes with various boundary conditions, which need not be realizable by a CW complex, can be obtained as mapping cones at the chain-complex level, without reference to the point-set topology of an underlying manifold [1].
- QLDPC code— Gauging-based fault-tolerant logical measurement schemes for QLDPC codes [3–5] are height-1 mapping cones: measuring a logical operator corresponds to adding it as a parity check and reducing its weight via an ancillary complex [1].
Member of code lists
Primary Hierarchy
References
- [1]
- A. C. Yuan, “Unified framework for quantum code embedding”, Physical Review A 113, (2026) arXiv:2507.05361 DOI
- [2]
- M. B. Hastings, “On Quantum Weight Reduction”, (2023) arXiv:2102.10030
- [3]
- B. Ide, M. G. Gowda, P. J. Nadkarni, and G. Dauphinais, “Fault-Tolerant Logical Measurements via Homological Measurement”, Physical Review X 15, (2025) arXiv:2410.02753 DOI
- [4]
- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, Nature Physics 22, 598 (2026) arXiv:2410.02213 DOI
- [5]
- A. W. Cross, Z. He, P. J. Rall, and T. J. Yoder, “Improved QLDPC Surgery: Logical Measurements and Bridging Codes”, (2025) arXiv:2407.18393
Page edit log
- Victor V. Albert (2026-07-29) — most recent
Cite as:
“Mapping cone code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/mapping_cone, arXiv:2606.11484