Symplectic cone code

Symplectic cone code[1]

Description

Qubit stabilizer code obtained by attaching a CSS ancillary complex to an input stabilizer code represented as a symplectic complex. Gluing and defect maps preserve the symplectic commutation structure, while an embedded column complex controls the logical operators of the output. The construction generalizes the mapping cone framework of quantum code embedding from CSS codes to arbitrary stabilizer codes, thereby extending fault-tolerant logical measurement, geometrically local embedding, and weight reduction to codes and logical operators that admit no CSS structure [1].

Formally, the input code is written as a symplectic complex. A symplectic cone code is a height-two cone whose middle level is the input-code symplectic complex. Its top and bottom levels are an ancillary cochain complex and the corresponding chain complex, respectively. The levels are glued together by a chain map that is compatible with a defect map, and the cone carries a mixed symplectic form pairing the two ancillary sectors. If the ancillary complex \(A\) has \(H_1(A)=0\), the logical space \(H_1(C)\) is naturally isomorphic to that of the embedded column complex \(H^0(A)\to H_1(D)\to H_0(A)\) [1; Thm. 2]. This column-complex homology need not equal the logical space of the input code \(D\). The Layer code and weight-reduction applications choose the column complex to preserve the input logical space. For measurement of one logical operator \(\ell^{\star}\), it instead gives \(H_1(C)\cong[\ell^{\star}]^{\perp}/[\ell^{\star}]\), so the deformed code has one fewer logical qubit. Although the ancilla is CSS in isolation, its checks are generically dressed with extra Pauli support of the opposite type once glued, so the output code is generally non-CSS even when the input code is CSS [1].

Protection

The distance of a symplectic cone code can be lower bounded in terms of the distance of its embedded code via the cleaning lemma of the mapping cone framework [1,2].

Fault Tolerance

A single weight-\(W\) non-CSS logical operator of a QLDPC code can be measured fault-tolerantly using an \(O(W\log W)\)-qubit ancilla [1; Thm. 32]. When the measurement graph has constant expansion, the deformed code retains distance \(\Omega(d)\). A spacetime fault complex establishes a threshold for the procedure. Native parallel measurement of general commuting non-CSS logical operators remains open [1]. The fast-surgery framework [3] extends conditionally to non-CSS logical measurement when a suitable CSS ancilla with meta-syndromes and relative expansion is available [1; Appx. D]. The paper does not construct low-overhead fast-surgery ancillas for arbitrary commuting logical sets that cannot all be made CSS by one tensor product of single-qubit Clifford gates [1].

Cousins

Mapping 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 [1,2].
Qubit QLDPC code— For a QLDPC input and a bounded-degree expanding measurement graph, the symplectic-cone construction remains QLDPC and measures a single non-CSS logical operator without first converting it to a CSS representative by a local Clifford circuit [1].
Kitaev surface code— Measurement of a logical \(Y\) operator of the surface code is an example of non-CSS surgery, with the defect map dressing the attached CSS ancilla such that the deformed code is non-CSS [1; Fig. 1].
Distance-balanced code— Symplectic cones extend weight reduction to non-CSS input codes, producing codes with maximum stabilizer-generator weight nine, total qubit degree at most eight, and block length \(O(w^4q^4)n\) for an \(n\)-qubit input of check weight \(w\) and total qubit degree \(q\) [1; Thm. 53].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Parents

Qubit stabilizer code

A symplectic cone code is a qubit stabilizer code whose symplectic complex has an ancillary cochain level, an input-code level, and an ancillary chain level. Its stabilizer map is lower triangular, and a mixed symplectic form pairs the two ancillary sectors. When \(H_1(A)=0\), the output logical space is naturally isomorphic to the homology of the embedded column complex [1; Thm. 2].

Symplectic cone code

Children

Layer code

Layer codes are height-2 symplectic cones whose embedded column complex is the input QLDPC code; CSS Layer codes reduce to ordinary mapping cones, while the non-CSS construction uses a defect map to preserve symplectic commutation [1,2].

References

[1]: A. C. Yuan and N. Baspin, “Non-CSS Quantum Code Embedding”, (2026) arXiv:2608.16995
[2]: A. C. Yuan, “Unified framework for quantum code embedding”, Physical Review A 113, (2026) arXiv:2507.05361 DOI
[3]: N. Baspin, L. Berent, and L. Z. Cohen, “Fast surgery for quantum LDPC codes”, (2025) arXiv:2510.04521

Page edit log

Victor V. Albert (2026-08-24) — most recent
Victor V. Albert (2026-08-22)

Cite as:

“Symplectic cone code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/symplectic_cone, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/symplectic_cone.yml.