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.
Simple single-mode single-qudit LCA codewords can be viewed as \(Kc\)-dimensional GKP codewords whose logical subsystem decomposition \(\mathbb{Z}_{Kc}\cong\mathbb{Z}_{K}\times\mathbb{Z}_{c}\) entangles the \(\mathbb{Z}_{c}\) factor with the physical qudit [1].
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].
Rate
Single-mode single-qudit LCA codes have logical dimension \(K=c\theta+d\) for integers \(\theta\geq 0\) and \(d\in\mathbb{Z}_{c}^{\times}\), the multiplicative group of integers modulo \(c\) [1].Encoding
Codewords of a simple 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
Several LCA code families admit logical Clifford gates via Gaussian transformations on the oscillators together with Clifford gates on the qudits; for single-mode \((c,d)\)-LCA codes, this includes a logical Hadamard implemented by an oscillator Fourier transform and a corresponding qudit Clifford operation [1].Adding a conditional oscillator-qudit displacement makes the gate set universal [6].Decoding
Simple LCA codes admit a decoder for pure displacement noise, a decoder for all single-qudit Pauli errors together with a smaller displacement range, and for physical qubits a balanced decoder that trades oscillator against qudit error tolerance by moving syndrome-region boundaries [1].Under Petz (transpose) recovery against photon loss and qubit amplitude damping, \(c=2\) LCA codes can match or outperform comparable GKP codes at low energy for sufficiently large logical dimension [1].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 can be used to construct LCA stabilizer codes [1].
- Group GKP code— Simple single-mode single-qudit LCA codes are Abelian group-GKP codes with \(Kc \mathbb{Z} \subset \mathbb{Z} \subset \mathbb{R} \times \mathbb{Z}_c\), where the logical dimension \(K\) is coprime to the physical qudit dimension \(c\) [1].
Primary Hierarchy
References
- [1]
- S. Chakraborty and V. V. Albert, “Hybrid Oscillator-Qudit Quantum Processors: stabilizer states, stabilizer codes, symplectic operations, and non-commutative geometry”, (2026) 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
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