Locally compact Abelian (LCA) stabilizer code[1]
Description
A mixed oscillator stabilizer code whose codewords are quantum lattice states defined on any number of qudits and a nonzero number of oscillators. Its stabilizers are countably infinite subgroups of the qudit Pauli and oscillator displacement groups. Codewords are entangled across the qudit-oscillator bipartition.
The simplest LCA state is a Bell state of a single physical qubit and a GKP-encoded qubit, \begin{align} |\text{LCA}\rangle&=\sum_{\ell\in\mathbb{Z}}|x=\ell\sqrt{\pi}\rangle\left|\ell\text{ mod }2\right\rangle \tag*{(1)}\\&=\sum_{s\in\mathbb{Z}}{|{x=(2s)\sqrt{\pi}}\rangle}\left|0\right\rangle +{|{x=(2s+1)\sqrt{\pi}}\rangle}\left|1\right\rangle ~. \tag*{(2)}\end{align} LCA stabilizers with such codewords are called simple.
Protection
Simple LCA stabilizers can protect against either a larger set of displacements than GKP codes or a smaller set along with all single-qudit Pauli errors [1].
The syndrome space of a simple LCA stabilizer can be characterized entirely by a unit cell of displacements. For a single \(c\)-dimensional qudit and single-oscillator code, the area of this cell is \(2\pi c\). Displacement values that keep the codewords inside this unit cell can be measured simultaneously, meaning that one can simultaneously measure an arbitrary range of values of two non-commuting displacements given sufficient qudit dimension [1].
Encoding
Codewords of a single-qudit single-oscillator code can be initialized by applying a conditional oscillator-qudit displacement to a GKP state and a qudit \(|+\rangle\) state [1]. Alternatively, one can prepare a two-qudit Bell state and encode one subsystem into a GKP qudit code [1].General LCA stabilizers can be created by using a general embedding of their stabilizer algebra, a non-commutative torus [2–5], into LCA groups [1]. Logical operators can be obtained via Morita equivalence [1].Gates
Logical Clifford gates should be realizable by performing Gaussian gates on the oscillators and Clifford gates on the qudits [1]. Adding a conditional oscillator-qudit displacement makes the gate set universal [6].Cousins
- Linear binary code— Linear binary codes can be used to construct LCA stabilizer codes [1].
- \(E_8\) Gosset lattice— Integer symplectic matrices like the symplectic \(E_8\) generator matrix [7; Appx. 2] can be used to construct LCA stabilizer codes [1].
Member of code lists
Primary Hierarchy
References
- [1]
- S. Chakraborty and V. V. Albert, “Hybrid oscillator-qudit quantum processors: stabilizer states and symplectic operations”, (2025) arXiv:2508.04819
- [2]
- M. A. Rieffel, “Projective Modules over Higher-Dimensional Non-Commutative Tori”, Canadian Journal of Mathematics 40, 257 (1988) DOI
- [3]
- A. Schwarz, “Morita equivalence and duality”, Nuclear Physics B 534, 720 (1998) arXiv:hep-th/9805034 DOI
- [4]
- A. Connes, M. R. Douglas, and A. Schwarz, “Noncommutative geometry and Matrix theory”, Journal of High Energy Physics 1998, 003 (1998) arXiv:hep-th/9711162 DOI
- [5]
- M. Rieffel and A. Schwarz, “Morita equivalence of multidimensional noncommutative tori”, (1998) arXiv:math/9803057
- [6]
- L. Brenner, B. Dias, and R. Koenig, “Trading modes against energy”, (2025) arXiv:2509.18854
- [7]
- P. Buser and P. Sarnak, “On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane)”, Inventiones Mathematicae 117, 27 (1994) DOI
Page edit log
- Victor V. Albert (2025-10-24) — most recent
Cite as:
“Locally compact Abelian (LCA) stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/lca_stabilizer