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Hybrid Oscillator-Qudit Quantum Processors: Stabilizer States, Stabilizer Codes, Symplectic Operations, and Noncommutative Geometry

Sayan Chakraborty1 and Victor V. Albert2,*

  • *Contact author: vva@umd.edu
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PRX Quantum 7, 020320 – Published 1 May, 2026

DOI: https://doi.org/10.1103/yr64-ypxt

Abstract

We construct stabilizer states and error-correcting codes on hybrid discrete- and continuous-variable systems, generalizing the Gottesman-Kitaev-Preskill (GKP) lattice formalism. Our framework embeds qudit phase space into a hybrid phase space parameterized by continuous variables, with unit cells that grow with qudit dimension—enabling simultaneous measurement of an arbitrarily large range of noncommuting displacements. Simple hybrid states arise from applying a conditional displacement to a GKP state and a Pauli eigenstate, or by encoding qudits of a stabilizer state into a GKP code; their oscillator-qudit entanglement is non-Gaussian, distinguishing them as a resource beyond tensor products of stabilizer states. Simple hybrid codes are subsystem GKP codes whose gauge factor is entangled with a qudit. Numerical experiments suggest that they can outperform GKP codes against physical noise, with decoders tunable to either qudit or oscillator errors. We also relate stabilizer codes to noncommutative tori, showing that a general construction yields multimode multiqudit GKP extensions. We compute the codes’ logical dimension and operators via Morita equivalence.

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Popular Summary

Natural quantum systems, such as light, are typically described by both discrete variables—like polarization—and continuous variables, like the quadratures of the light’s amplitude. Quantum devices also tend to be naturally hybrid: trapped-ion devices, for example, utilize both the spin and vibrational degrees of freedom of ions.

Despite the ubiquity of discrete and continuous degrees of freedom working in tandem, a general theory of states and operations that utilizes both remains underdeveloped. This work aims to fill that gap by developing what one may call “canonical primitives”—a set of simple, easy-to-understand states and operations—for hybrid platforms.

The states we develop are called stabilizer states, which are computationally tractable when defined over discrete variables. The picture is less clear over continuous variables, and our hybrid construction opens further interesting avenues for determining the boundary of classical tractability. This construction also links quantum error correction to classical results in noncommutative geometry. We hope that these connections will bring established mathematical tools to bear on modern quantum problems and open further applications of hybrid systems in quantum science.

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PRX Quantum 7, 020320– Published 1 May, 2026
  • Received 17 September 2025
  • Revised 13 January 2026
  • Accepted 26 February 2026
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