Hybrid Oscillator-Qudit Quantum Processors: stabilizer states, stabilizer codes, symplectic operations, and non-commutative geometry

TL;DR

This work develops oscillator–qudit stabilizer states and error-correcting codes (LCA states and LCA codes) by embedding discrete qudit phase spaces into a continuous-variable oscillator lattice, yielding a hybrid phase space with unit-cell area $2\pi c$ and enhanced displacement capabilities. Simple LCA states are entangled across the oscillator–qudit bipartition and cannot be generated by Gaussian-Clifford operations, a property analyzed via non-commutative torus and Morita equivalence formalisms; the authors provide explicit state constructions and decoding strategies. The paper introduces simple and general LCA codes, derives their logical dimensions via Pfaffians and determinants, and develops two error-correction decoders (pure-dispacement and qudit-error) with a tunable balanced decoder, supported by numerical evidence showing potential advantages over GKP codes in certain noise regimes. By linking stabilizer codes to non-commutative geometry, the work opens a pathway to fault-tolerant hybrid quantum protocols and connects lattice-based quantum information to Morita-equivalent tori, offering versatile multi-mode and multi-qudit extensions and new directions for quantum cryptography and lattice-based computation.

Abstract

We construct stabilizer states and error-correcting codes on combinations of discrete- and continuous-variable systems, generalizing the Gottesman-Kitaev-Preskill (GKP) quantum lattice formalism. Our framework absorbs the discrete phase space of a qudit into a hybrid phase space parameterizable entirely by the continuous variables of a harmonic oscillator. The unit cell of a hybrid quantum lattice grows with the qudit dimension, yielding a way to simultaneously measure an arbitrarily large range of non-commuting position and momentum displacements. Simple hybrid states can be obtained by applying a conditional displacement to a Gottesman-Kitaev-Preskill (GKP) state and a Pauli eigenstate, or by encoding some of the physical qudits of a stabilizer state into a GKP code. The states' oscillator-qudit entanglement cannot be generated using symplectic (i.e., Gaussian-Clifford) operations, distinguishing them as a resource from tensor products of oscillator and qudit stabilizer states. Simple hybrid codes can be thought of as subsystem GKP codes whose gauge factor is entangled with a qudit. Our numerical investigations suggest that such codes can sometimes outperform GKP codes against physical noise, and their decoders can be tuned to accommodate either more qudit or more oscillator errors. We also relate stabilizer codes to non-commutative tori, identifying that a general construction of such tori yields multi-mode multi-qudit extensions of GKP codes. We explicitly calculate these codes' logical dimension and logical operators by utilizing the Morita equivalence between their stabilizer and logical tori. We provide examples using commutation matrices, integer symplectic matrices, and binary codes.

Figures (5)

• Figure 1: (a) Ordinary Bell states and (b) encoded GKP Bell states are defined on qudit (thin line) and oscillator (thick line) systems, respectively. (c) Simple locally compact Abelian (LCA) states interpolate between these purely $\textsc{dv}$ and $\textsc{cv}$ states since they can be obtained by encoding one half of a Bell state into a GKP code. LCA states cannot be obtained from separable oscillator-qudit states via symplectic (i.e., Gaussian-Clifford) operations, underscoring the non-Gaussianity of the GKP encoder (cf. baragiola2019all). (d) The LCA framework absorbs the qudit into the oscillator to form a hybrid phase space that is parameterizable by the oscillator's original variables. The minimal area enclosed by two commuting displacements on this phase space is $2\pi c$, where $c$ is the qudit dimension. This means position and momentum can be simultaneously measured modulo $\sqrt{2\pi c}$ for arbitrarily large $c$.
• Figure 2: (a) A GKP qunaught state $|\varnothing\rangle$\ref{['eq:gkp']} is a superposition, or comb, of oscillator position vectors spaced uniformly at a distance of $d_{\text{GKP}} = \sqrt{2\pi}$. (b) A simple oscillator-qubit LCA state \ref{['eq:lca-qubit-state']} is a superposition of two such combs, each tensored with the qubit state $|0\rangle$ or $|1\rangle$. The spacing within each comb is a factor of $\sqrt{2}$ larger than that of the qunaught state, while the spacing between combs is smaller by the same factor. (c) Simple oscillator-qudit LCA states \ref{['eq:qudit-lca-state']}, depicted here for qudit dimension $c=5$, are a superposition of $c$ combs tensored with the $c$ qudit basis states. The intra-comb spacing is a factor $\sqrt{c}$ larger than that of the qunaught state.
• Figure 3: Depictions of logical codewords of LCA codes encoding (a) a logical qutrit ($K=3$) into a mode and a physical qubit ($c=2$) [see Eq. \ref{['eq:qutrit']}], and (b) a logical qubit ($K=2$) into a mode and a physical qutrit ($c=3$) [see Eq. \ref{['eq:qubit']}]. These are the smallest encodings of a logical qutrit and a logical qubit, respectively. Both utilize the same six oscillator combs, paired up in two different ways to the physical qubit and qutrit basis states. The shortest undetectable displacement is $\sqrt{c}d_{\text{GKP}}$, where $d_{\text{GKP}} = \sqrt{2\pi/K}$\ref{['eq:code-distance']}. A decoder correcting against all single-qudit Pauli-$\hat{X}$ errors along with small position displacements can be constructed by splitting up the syndrome space into segments $\Circled{1},\Circled{2},\cdots,\Circled{c}$, as shown (see Sec. \ref{['sec:qec']}). The same picture holds against Pauli-$\hat{Z}$ errors and momentum displacements in the Fourier domain.
• Figure 4: Entanglement fidelity versus energy \ref{['eq:energy']} under the transpose recovery of GKP and oscillator-qubit LCA codes against photon loss and qubit amplitude damping at the same noise rate $\gamma = 0.05$ and various logical dimensions $K$. The four dotted curves correspond to fidelities of GKP codes of logical dimension $K=3$, $5$, $7$, and $9$, shown from left to right. Similarly, the four solid curves correspond to oscillator-qubit LCA codes with the same logical dimensions. The $K=3,5$ encodings perform similarly, but LCA code fidelities at lower energy are higher than GKP fidelities for $K=7,9$. To make this plot, we have implemented a cutoff of the oscillator occupation-number, and have numerically made an orthonormal basis out of the normalized LCA states. We attach a Mathematica notebook with our numerical derivations to the arXiv submission of this manuscript.
• Figure 5: Four-qubit circuit obtained from interpreting the $E_8$ symplectic generator matrix modulo 2 as a binary symplectic matrix. This circuit is invariant under cyclic permutations and maps $\hat{\sigma}_{\mathsf{x}}^{j}\to \hat{\sigma}_{\mathsf{x}}^{j-1}\hat{\sigma}_{\mathsf{z}}^{j}\hat{\sigma}_{\mathsf{x}}^{j+1}$ and $\hat{\sigma}_{\mathsf{z}}^{j}\to \hat{\sigma}_{\mathsf{z}}^{j-1}\hat{\sigma}_{\mathsf{x}}^{j}\hat{\sigma}_{\mathsf{z}}^{j+1}$, where superscripts are evaluated modulo 4. The two sets of CZ gates are examples of circuits creating cluster states son2011quantumson2012topological.

Theorems & Definitions (8)

• Theorem 1: Informal.
• Remark 1
• Definition 1
• Theorem 2: Theorem \ref{['thm:main_text']}, detailed restatement
• proof
• Remark 2
• Lemma 3
• proof