Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.15739

Non-Gaussianity in cat codes: global incompatibility and local geometric alignment with the magic resource

Yuwei Zhu, Seungbeom Chin, William J. Munro, Kae Nemoto

quant-phSubmitted Jul 17, 2026• Announced Jul 20, 2026• 0 citations
arXivHTMLPDF

Abstract

Non-Gaussianity is an essential resource for genuine quantum advantages in continuous-variable quantum systems and is regarded as a counterpart of the magic resource in discrete-variable systems. Recently, an exact correspondence between non-Gaussianity (NG) and the magic resource was identified within the Gottesman--Kitaev--Preskill (GKP) encoding framework. Whether such a relation persists beyond GKP encoding, however, remains unclear. Here, we address this question in the cat-code setting. By comparing the Wigner logarithmic negativity (WLN) and a magic measure defined from a phase-operator basis, we analyze the resource geometry of non-degenerate $d$-peaked cat states. We find that cat codes do not inherit the global value-preserving GKP magic--NG equivalence, but their asymptotic WLN geometry allows a constructive local alignment with the magic-measure geometry. Specifically, under a distinguished SU($d$) asymptotic cat code, we establish a sector-dependent alignment between WLN level sets and magic-measure level sets based on their intrinsic local geometries. These results identify both the global incompatibility and the locally constructive relation between magic and non-Gaussian resources in cat codes, suggesting a geometry-based approach to resource correspondences beyond the GKP framework.

Related papers

Bridging Magic and Non-Gaussian Resources via Gottesman-Kitaev-Preskill Encoding

Oliver Hahn, Giulia Ferrini, Ryuji Takagi

Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.

arXiv:2406.06418•PRX Quantum•Feb 18, 2025•15 citations

The logical set of Gaussian states under the stabilizer subsystem decomposition

Santiago Zamora, Kaustav Chatterjee, Ulrik Lund Andersen +4 more

Universal quantum computation requires non-Gaussianity in continuous-variable systems and non-stabilizerness in discrete-variable systems. Yet, mapping a Gaussian state into the Gottesman-Kitaev-Preskill logical subspace can yield a logical qubit with non-stabilizer resources. To understand how the continuous-variable structure determines logical resources, we characterize the image of single-mode Gaussian states under the corresponding extraction channel, known as the stabilizer subsystem decomposition. Specifically, we derive series expressions for the logical Bloch vector and characterize the resulting reachable set, enabling an analytical treatment of the robustness of magic of the logical states. We then turn this geometric framework into a certification tool: by mapping a discrete-variable qubit witness to a continuous-variable squeezing witness, we obtain lower bounds on the squeezing of prepared states. We also show that ensembles of Gaussian states can violate both stabilizer and classical bounds. Violation of the stabilizer bound certifies the relational resource known as "set-magic," while violation of the classical bound enables the certification of cryptographic randomness at moderate squeezing.

quant-pharXiv:2609.26898•Sep 24, 2026•0 citations

Quantifying Qubit Magic Resource with Gottesman-Kitaev-Preskill Encoding

Oliver Hahn, Alessandro Ferraro, Lina Hultquist +2 more

Quantum resource theories are a powerful framework for characterizing and quantifying relevant quantum phenomena and identifying processes that optimize their use for different tasks. Here, we define a resource measure for magic, the sought-after property in most fault-tolerant quantum computers. In contrast to previous literature, our formulation is based on bosonic codes, well-studied tools in continuous-variable quantum computation. Particularly, we use the Gottesman-Kitaev-Preskill code to represent multiqubit states and consider the resource theory for the Wigner negativity. Our techniques are useful in finding resource lower bounds for different applications as state conversion and gate synthesis. The analytical expression of our magic measure allows us to extend current analysis limited to small dimensions, easily addressing systems of up to 12 qubits.

arXiv:2109.13018•Physical Review Letters•May 23, 2022•34 citations

Improved GKP magic states from error-corrected non-Gaussian quantum states

Sharon David, Jack Davis, Ulysse Chabaud +1 more

Gate teleportation, together with magic state distillation, is a promising route towards fault-tolerant, universal computation. In the context of bosonic quantum computation, Baragiola et al. PRL 123(20).200502 (2019) showed that within the framework of Gottesman--Kitaev--Preskill codes, encoded magic states suitable for distillation can be produced by error correcting Gaussian states, such as the vacuum. Here, we show that applying the same framework to simple non-Gaussian input states can significantly improve the quality of the magic states obtained, reducing the overall resources for the complete distillation procedure. We focus on superpositions of coherent states or Fock states, showing that many can lead to improvements in the generation of high-quality encoded magic states, which in some cases reduces the resources required for magic state distillation by about a factor $3$. We also investigate the primary source of these improvements and find that, unlike what was previously conjectured, the suitability of input states is not fully explained by symmetry arguments. Instead, the best states seem to avoid projection near stabilizer states as a result of the error correction procedure.

quant-pharXiv:2607.07833•Jul 10, 2026•0 citations

Near-Optimal Bell Nonlocality with GKP and Entangled Cat States

Özlem Erkılıç, Aritra Das, S. Nibedita Swain +2 more

Gottesman-Kitaev-Preskill (GKP) states are widely studied as a bosonic encoding for fault-tolerant quantum computing because small displacement errors can be identified and corrected through syndrome measurements. However, fault-tolerant operation requires substantially greater GKP squeezing than is currently available in optical platforms. It is therefore important to identify quantum-information tasks that can be realised with finite-squeezing GKP states. Here, we study Bell nonlocality in finite-energy GKP Bell states under experimentally motivated measurement constraints. Although finite-energy GKP Bell states support Bell nonlocality even at low squeezing, logical Pauli measurements alone cannot reveal it. Accessing the maximum CHSH violation supported by the state generally requires logical non-Clifford rotations. We instead show that the required non-Gaussian resource can reside in the measurement itself. Photon-number-resolving detection combined with Gaussian preprocessing and adaptive feed-forward closely approaches the maximum violation of the CHSH Bell inequality without implementing logical non-Clifford gates. The same measurement principle strongly enhances Bell violations for cat-code Bell pairs and entangled coherent states, demonstrating that its usefulness extends beyond the GKP lattice. Our results identify non-Gaussian measurements as a practical route to revealing Bell nonlocality in finite-energy bosonic states.

quant-pharXiv:2609.27640•Sep 24, 2026•0 citations

Resource theory of quantum non-Gaussianity and Wigner negativity

Francesco Albarelli, Marco G. Genoni, Matteo G. A. Paris +1 more

We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state set---i.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone---the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states---e.g., photon-added, photon-subtracted, cubic-phase, and cat states---and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to subuniversal and universal quantum information processing over continuous variables.

arXiv:1804.05763•Physical Review A•Nov 28, 2018•249 citations
Non-Gaussianity is an essential resource for genuine quantum advantages in continuous-variable quantum systems and is regarded as a counterpart of the magic resource in discrete-variable systems. Recently, an exact correspondence between non-Gaussianity (NG) and the magic resource was identified within the Gottesman--Kitaev--Preskill (GKP) encoding framework. Whether such a relation persists beyond GKP encoding, however, remains unclear. Here, we address this question in the cat-code setting. By comparing the Wigner logarithmic negativity (WLN) and a magic measure defined from a phase-operator basis, we analyze the resource geometry of non-degenerate ddd-peaked cat states. We find that cat codes do not inherit the global value-preserving GKP magic--NG equivalence, but their asymptotic WLN geometry allows a constructive local alignment with the magic-measure geometry. Specifically, under a distinguished SU(ddd) asymptotic cat code, we establish a sector-dependent alignment between WLN level sets and magic-measure level sets based on their intrinsic local geometries. These results identify both the global incompatibility and the locally constructive relation between magic and non-Gaussian resources in cat codes, suggesting a geometry-based approach to resource correspondences beyond the GKP framework.
Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.
Universal quantum computation requires non-Gaussianity in continuous-variable systems and non-stabilizerness in discrete-variable systems. Yet, mapping a Gaussian state into the Gottesman-Kitaev-Preskill logical subspace can yield a logical qubit with non-stabilizer resources. To understand how the continuous-variable structure determines logical resources, we characterize the image of single-mode Gaussian states under the corresponding extraction channel, known as the stabilizer subsystem decomposition. Specifically, we derive series expressions for the logical Bloch vector and characterize the resulting reachable set, enabling an analytical treatment of the robustness of magic of the logical states. We then turn this geometric framework into a certification tool: by mapping a discrete-variable qubit witness to a continuous-variable squeezing witness, we obtain lower bounds on the squeezing of prepared states. We also show that ensembles of Gaussian states can violate both stabilizer and classical bounds. Violation of the stabilizer bound certifies the relational resource known as "set-magic," while violation of the classical bound enables the certification of cryptographic randomness at moderate squeezing.
Quantum resource theories are a powerful framework for characterizing and quantifying relevant quantum phenomena and identifying processes that optimize their use for different tasks. Here, we define a resource measure for magic, the sought-after property in most fault-tolerant quantum computers. In contrast to previous literature, our formulation is based on bosonic codes, well-studied tools in continuous-variable quantum computation. Particularly, we use the Gottesman-Kitaev-Preskill code to represent multiqubit states and consider the resource theory for the Wigner negativity. Our techniques are useful in finding resource lower bounds for different applications as state conversion and gate synthesis. The analytical expression of our magic measure allows us to extend current analysis limited to small dimensions, easily addressing systems of up to 12 qubits.
Gate teleportation, together with magic state distillation, is a promising route towards fault-tolerant, universal computation. In the context of bosonic quantum computation, Baragiola et al. PRL 123(20).200502 (2019) showed that within the framework of Gottesman--Kitaev--Preskill codes, encoded magic states suitable for distillation can be produced by error correcting Gaussian states, such as the vacuum. Here, we show that applying the same framework to simple non-Gaussian input states can significantly improve the quality of the magic states obtained, reducing the overall resources for the complete distillation procedure. We focus on superpositions of coherent states or Fock states, showing that many can lead to improvements in the generation of high-quality encoded magic states, which in some cases reduces the resources required for magic state distillation by about a factor 333. We also investigate the primary source of these improvements and find that, unlike what was previously conjectured, the suitability of input states is not fully explained by symmetry arguments. Instead, the best states seem to avoid projection near stabilizer states as a result of the error correction procedure.
Gottesman-Kitaev-Preskill (GKP) states are widely studied as a bosonic encoding for fault-tolerant quantum computing because small displacement errors can be identified and corrected through syndrome measurements. However, fault-tolerant operation requires substantially greater GKP squeezing than is currently available in optical platforms. It is therefore important to identify quantum-information tasks that can be realised with finite-squeezing GKP states. Here, we study Bell nonlocality in finite-energy GKP Bell states under experimentally motivated measurement constraints. Although finite-energy GKP Bell states support Bell nonlocality even at low squeezing, logical Pauli measurements alone cannot reveal it. Accessing the maximum CHSH violation supported by the state generally requires logical non-Clifford rotations. We instead show that the required non-Gaussian resource can reside in the measurement itself. Photon-number-resolving detection combined with Gaussian preprocessing and adaptive feed-forward closely approaches the maximum violation of the CHSH Bell inequality without implementing logical non-Clifford gates. The same measurement principle strongly enhances Bell violations for cat-code Bell pairs and entangled coherent states, demonstrating that its usefulness extends beyond the GKP lattice. Our results identify non-Gaussian measurements as a practical route to revealing Bell nonlocality in finite-energy bosonic states.
We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state set---i.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone---the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states---e.g., photon-added, photon-subtracted, cubic-phase, and cat states---and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to subuniversal and universal quantum information processing over continuous variables.