Hybrid oscillator-qudit quantum processors: stabilizer states and symplectic operations

2025-08-06

AI Summary

This work addresses the challenge of constructing stabilizer states and designing error-correcting codes for hybrid discrete–continuous variable quantum systems. Methodologically, it introduces an extended Gottesman–Kitaev–Preskill (GKP) lattice framework that embeds the discrete phase space of a qudit into the continuous-variable phase space of a harmonic oscillator, yielding a unified noncommutative hybrid phase space. Leveraging noncommutative torus geometry and Morita equivalence theory, logical operators are constructed; combined with GKP encoding, conditional displacements, symplectic transformations, and integer-valued symmetric matrices, experimentally feasible hybrid stabilizer codes are designed. Key contributions include: (i) the first generation of oscillator–qudit entangled resource states not preparable via Gaussian Clifford operations; (ii) support for joint measurements of a broad class of noncommuting observables; and (iii) a hardware-compatible, scalable framework for hybrid quantum error correction and entanglement generation.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

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. We construct general hybrid error-correcting codes by relating stabilizer codes to non-commutative tori and obtaining logical operators via Morita equivalence. We provide examples using commutation matrices, integer symplectic matrices, and binary codes.

Problem

• Construct stabilizer states on hybrid discrete-continuous quantum systems
• Generalize GKP formalism for hybrid oscillator-qudit phase space
• Develop hybrid error-correcting codes via non-commutative tori

Innovation

• Hybrid oscillator-qudit phase space framework
• Non-Gaussian stabilizer states via entanglement
• Error-correcting codes using non-commutative tori