[Jump to code hierarchy]

Mixed oscillator code

Alternative names: LCA code.

Description

Encodes a logical Hilbert space into some number of modular qudits, some number of rotors, and a nonzero number of oscillators, i.e., the Hilbert space of \(L^2\)-normalizable functions on a locally compact Abelian (LCA) group. In photonic systems, photonic states of multiple degrees of freedom of a photon (e.g., frequency, amplitude, and polarization) are called hyper-entangled states [1].

Gates

Symplectic transformations (i.e., transformations that preserve the qudit Pauli and oscillator displacement group structure) are tensor products of qudit Clifford and oscillator Gaussian operations, and there are no entangling symplectic operations [2–4].Adding a conditional oscillator-qudit displacement makes the symplectic gate set universal [5].

Notes

See reviews [6–8] for introductions to mixed oscillator platforms.

Cousins

Member of code lists

Primary Hierarchy

Quantum Domain
Group Kingdom
Group-based quantum codeQECC Quantum
Parents
Group-based quantum code
Group quantum codes whose physical spaces are constructed using some number of modular qudits, some number of rotors, and a nonzero number of oscillators, which together constitute a general locally compact Abelian (LCA) group, are mixed oscillator codes.
Mixed oscillator code
Children
Hybrid cat code
Locally compact Abelian (LCA) stabilizer codeQuantum lattice
Bosonic codeFock-state bosonic Bosonic stabilizer Quantum lattice Analog stabilizer
Mixed oscillator codes defined only on oscillators reduce to oscillator codes.

References

[1]
P. G. Kwiat, “Hyper-entangled states”, Journal of Modern Optics 44, 2173 (1997) DOI
[2]
A. Prasad and M. K. Vemuri, “Decomposition of phase space and classification of Heisenberg groups”, (2008) arXiv:0806.4064
[3]
J. Bermejo-Vega, “Normalizer Circuits and Quantum Computation”, (2016) arXiv:1611.09274
[4]
S. Chakraborty and V. V. Albert, “Hybrid oscillator-qudit quantum processors: stabilizer states and symplectic operations”, (2025) arXiv:2508.04819
[5]
L. Brenner, B. Dias, and R. Koenig, “Trading modes against energy”, (2025) arXiv:2509.18854
[6]
U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Hybrid discrete- and continuous-variable quantum information”, Nature Physics 11, 713 (2015) arXiv:1409.3719 DOI
[7]
A. Furusawa and P. van Loock, Quantum Teleportation and Entanglement (Wiley, 2011) DOI
[8]
Y. Liu et al., “Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications”, (2025) arXiv:2407.10381
[9]
M. M. Wilde and D. B. Uskov, “Linear-optical hyperentanglement-assisted quantum error-correcting code”, Physical Review A 79, (2009) arXiv:0807.4906 DOI
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: hybrid_qudit_oscillator

Cite as:
“Mixed oscillator code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/hybrid_qudit_oscillator
BibTeX:
@incollection{eczoo_hybrid_qudit_oscillator, title={Mixed oscillator code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hybrid_qudit_oscillator} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/hybrid_qudit_oscillator

Cite as:

“Mixed oscillator code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/hybrid_qudit_oscillator

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/lca/hybrid_qudit_oscillator.yml.