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 bounds the \(Z\)-distance of a height-2 cone from below [1; Sec. I.B]. It applies when the top-level boundary operator and the gluing map \(g_2\) satisfy an isoperimetric inequality with coefficient \(\alpha\). The \(Z\)-distance is then at least \(\alpha\) times the minimum weight of a middle-level representative of a nontrivial logical operator of the embedded code. The lemma yields distance bounds for subdivided square complexes [1; Sec. IV.B]. For gauging-based logical measurement, the lemma shows that the \(Z\)-distance of the cone is at least \(\min(h,1)\) times that of the input code [1; Sec. V.A]. Here, \(h\) is the Cheeger constant of the connected graph used to build the cone.Cousins
- Concatenated qubit code— The mapping cone framework can be regarded as a generalization of code concatenation for CSS codes [1].
- Distance-balanced code— The coning step of the weight reduction procedure of Ref. [2] is a height-1 cone [1; Sec. V.A]. The triangulation and thickening steps used in this procedure are regular height-2 cones [1; Sec. V.B].
- Good QLDPC code— Optimal embeddings of good QLDPC codes into \(D\)-dimensional Euclidean space use layer codes [3] or subdivided square complexes [4]. Both constructions yield regular height-2 cones whose embedded code is the input code [1; Sec. IV].
- Kitaev surface code— Surface codes on honeycomb and triangular lattices, with periodic or alternating smooth and rough boundaries, are regular cones whose embedded code is the square-lattice surface code [1; Sec. III.A]. The planar surface code cannot be realized by any 2D CW complex [1; Sec. I].
- Homological code— The barycentric subdivision of an \(n\)-dimensional simplicial complex is a regular height-\(n\) cone whose embedded complex is the original complex [1; Sec. III.B]. The homological codes of the two complexes therefore have isomorphic logical operators.
- Qubit QLDPC code— Gauging-based logical measurement for QLDPC codes [5] is a regular height-1 cone built from a connected graph [1; Sec. V.A]. The \(Z\)-type logical operators of this cone are those of the input code modulo the measured one [1; Sec. V]. Homological measurement is also formulated using height-1 cones [1,6].
- Layer code— CSS Layer codes are height-2 mapping cones whose levels are stacks of 2D surface codes and whose string defects implement the chain homotopy; non-CSS Layer codes require the symplectic cone generalization [1,7].
- Symplectic cone code— The symplectic cone framework generalizes height-two CSS mapping cones to arbitrary stabilizer codes. The full mapping-cone family also contains cones of other heights, so neither family contains the other [1,7].
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]
- D. J. Williamson and N. Baspin, “Layer codes”, Nature Communications 15, (2024) arXiv:2309.16503 DOI
- [4]
- T.-C. Lin, A. Wills, and M.-H. Hsieh, “Geometrically Local Quantum and Classical Codes from Subdivision”, (2024) arXiv:2309.16104
- [5]
- 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
- [6]
- 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
- [7]
- A. C. Yuan and N. Baspin, “Non-CSS Quantum Code Embedding”, (2026) arXiv:2608.16995
Page edit log
- Victor V. Albert (2026-09-24) — most recent
- Victor V. Albert (2026-07-29)
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