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Homogeneous-space quantum code

Alternative names: Coset space quantum code, \(G/H\) quantum code.
Root code for the Homogeneous-space quantum Kingdom

Description

Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(L^2\)-normalizable functions on a homogeneous space \(G/H\) or, more generally, induced representations whose base space is \(G/H\). Here, \(G\) is a second-countable unimodular group, and \(H\) is a closed subgroup of \(G\).

Protection

Quantum weight enumerators, linear programming bounds, and Rains shadow enumerators have been extended to quantum codes defined on multiplicity-free two-point homogeneous spaces [1].

Cousin

Member of code lists

Primary Hierarchy

Quantum Domain
Homogeneous-space quantum Kingdom
Parents
Quantum error-correcting code (QECC)Quantum
Homogeneous-space quantum code
Children
Group-based quantum codeFock-state bosonic Stabilizer CSS QLDPC Generalized homological-product Lattice stabilizer Bosonic stabilizer Quantum lattice Analog stabilizer Fracton stabilizer Qubit Hastings-Haah Floquet Fermion Self-complementary qubit Quantum QR code BB Homological Fermion-into-qubit Quantum RM code Color Twist-defect color CDSC Twist-defect surface Kitaev surface
Homogeneous spaces \(G/H\) for trivial \(H\) reduce to group spaces.
Diatomic molecular code
Diatomic molecular codes are defined on the space of orientations of a heterogeneous diatomic molecules. functions on the homogeneous space \(SO(3)/SO(2) = S^2\) (the two-sphere).
Fiber code
Fiber codes are defined on an induced representation \(\text{Ind}_G^{SO(3)} \Gamma\), induced by the irrep \(\Gamma\) of a subgroup \(G\subset SO(3)\).
Hyperbolic tesselation code
Hyperbolic tesselation codes are defined on the space of functions on the hyperbolic plane, the symmetric space \(G/H\) for \(G = SO(2,1)\) the proper Lorentz group and \(H = O(2)\).

References

[1]
R. Okada, “A Quantum Analog of Delsarte’s Linear Programming Bounds”, (2025) arXiv:2502.14165
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Zoo Code ID: homogeneous_space_quantum

Cite as:
“Homogeneous-space quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/homogeneous_space_quantum
BibTeX:
@incollection{eczoo_homogeneous_space_quantum, title={Homogeneous-space quantum code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/homogeneous_space_quantum} }
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Permanent link:
https://errorcorrectionzoo.org/c/homogeneous_space_quantum

Cite as:

“Homogeneous-space quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/homogeneous_space_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/homogeneous/homogeneous_space_quantum.yml.