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 \(G/H\) [1–5]. 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 [6].Cousin
- Homogeneous-space code— Homogeneous-space quantum codes are quantum counterparts of homogeneous-space codes.
Member of code lists
Primary Hierarchy
Parents
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. A group-\(G\) space can also be thought of as a multiplicity-free homogeneous space \((G\times G) / G\) [7; pg. 60].
Diatomic molecular codes are defined on the space of orientations of a heterogeneous diatomic molecules. This is equivalent to the space of normalizable functions on the two-sphere, i.e., the symmetric space \(SO(3)/SO(2) = S^2\).
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 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]
- G. W. Mackey, “Induced Representations of Locally Compact Groups I”, The Annals of Mathematics 55, 101 (1952) DOI
- [2]
- A. J. COLEMAN, “Induced and Subduced Representations”, Group Theory and its Applications 57 (1968) DOI
- [3]
- Isham, C. J. (1984). Topological and global aspects of quantum theory (pp. p. 1059–1290.). North-Holland.
- [4]
- N. P. Landsman and N. Linden, “The geometry of inequivalent quantizations”, Nuclear Physics B 365, 121 (1991) DOI
- [5]
- F. Scarabotti and F. Tolli, “Induced representations and harmonic analysis on finite groups”, (2013) arXiv:1304.3366
- [6]
- R. Okada, “A Quantum Analog of Delsarte’s Linear Programming Bounds”, (2025) arXiv:2502.14165
- [7]
- Diaconis, Persi. “Group representations in probability and statistics.” Lecture notes-monograph series 11 (1988): i-192.
Page edit log
- Victor V. Albert (2025-10-27) — most recent
Cite as:
“Homogeneous-space quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/homogeneous_space_quantum