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Stellar Representation of Gottesman Kitaev Preskill States on the Double Discrete Cylinder

This paper develops a covariant integral quantization framework on the double discrete cylinder phase space to characterize Gottesman-Kitaev-Preskill (GKP) states via their stellar representations, revealing distinct zero patterns that differentiate ideal logical states from superpositions and finitely squeezed variants, thereby elucidating the vacuum's role as a computational resource.

Original authors: Aidan Zlotak, Nicolas Fabre, Romain Murenzi

Published 2026-10-30

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CC BY 4.0

Original authors: Aidan Zlotak, Nicolas Fabre, Romain Murenzi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to build a quantum computer that does not break down under its own complexity, scientists are turning to a strange and powerful idea: encoding information not in simple on-off switches, but in the continuous, flowing waves of light or sound. These systems, known as continuous-variable quantum systems, offer a vast playground for processing data, but they are notoriously difficult to control. A major hurdle is the "noise" that creeps in, scrambling the delicate quantum information. To solve this, researchers developed a special type of error-correcting code called the Gottesman–Kitaev–Preskill, or GKP, code. Imagine a grid drawn on a sheet of paper; the code works by placing quantum information at specific, repeating points on this grid. If the information gets nudged slightly off its spot, the system can detect the shift and pull it back, much like a ball rolling back to the bottom of a bowl. However, while these ideal codes are mathematically perfect, they are physically impossible to create because they would require infinite energy. Real-world versions must be "squeezed" into finite shapes, and understanding exactly how this squeezing changes the quantum state is crucial for building machines that can actually work.

A team of physicists has now mapped out the hidden landscape of these states using a new mathematical lens. They focused on a specific geometric shape called a double discrete cylinder, which serves as a unique map for the quantum variables involved. In standard quantum mechanics, we often look at a flat plane to see where a particle is and how fast it is moving. But for these grid-based codes, the rules are different. The researchers developed a method to translate the quantum state into a probability map on this cylindrical surface, revealing features that are invisible on a flat plane. Their work centers on a concept called the "stellar representation," which counts the "zeros" or empty spots in the quantum map. In the world of quantum computing, the presence and arrangement of these zeros tell us how complex and "non-classical" a state is, which is a key measure of its power.

The researchers applied this new map to several key examples, starting with the simplest possible states. They found that the ideal, perfect logical states used in the GKP code have no zeros on this cylindrical map at all. This absence of zeros aligns with the fact that these ideal states are surprisingly simple in their quantum behavior, despite looking complex in other ways. However, when they looked at superpositions—states that are a mix of two different logical possibilities—the picture changed dramatically. These mixed states were found to have an infinite number of zeros arranged in a specific pattern on the map. This abundance of zeros corresponds to a "negative" probability in other types of maps, a signature of the deep quantum weirdness required for a computer to outperform a classical one.

Perhaps the most surprising discovery concerned the vacuum, the state of absolute emptiness. In standard flat maps, the vacuum is a smooth, featureless hill with no zeros. But on this new cylindrical map, the vacuum is not so simple. The researchers found that the vacuum actually possesses zeros, and their positions depend on how the map is constructed. This is a critical finding because it suggests that the vacuum itself, often thought of as a passive background, is an active resource that makes these quantum circuits hard to simulate on ordinary computers. The presence of these zeros in the vacuum state is what introduces the necessary complexity to break classical simulations.

The team then moved from the theoretical ideal to the physical reality of "finitely squeezed" states, which are the only kind we can actually build in a lab. They discovered that as the peaks of the quantum state become wider and less sharp, the zeros on the map begin to move. For the ideal states, the zeros sit in a perfect, fixed pattern. But for real, physical states, these zeros drift away from their perfect positions. They found that for the zeros to land exactly on the grid points of the map again, the width of the state must be tuned to a very specific, precise value. If the width is even slightly off, the zeros miss the grid entirely. This sensitivity highlights the delicate balance required to maintain the quantum resources needed for error correction.

Furthermore, the researchers examined what happens when two logical states are combined. In the ideal case, the zeros of the combined state form a rigid, infinite lattice. But once the state is squeezed into a finite shape, this lattice begins to dissolve. The zeros drift away from the grid, except for a small, protected group of four zeros that remain fixed due to a fundamental symmetry of the system. This drift signifies a loss of the perfect quantum structure, yet the fact that some zeros remain protected suggests that the core resource for quantum advantage is robust. The study concludes that the transition from the ideal, mathematical world to the messy, physical world is marked by the movement of these zeros. The vacuum and the finite squeezing of the state are not just imperfections; they are the very elements that generate the complex zero patterns which make GKP-based quantum computers so powerful and so difficult for classical machines to mimic. By charting these zeros on the double discrete cylinder, the researchers have provided a clear, geometric way to see exactly where the quantum power lies and how it changes as we move from theory to reality.

Technical Summary: Stellar Representation of Gottesman–Kitaev–Preskill States on the Double Discrete Cylinder

Problem Statement
The Gottesman–Kitaev–Preskill (GKP) code is a cornerstone of fault-tolerant continuous-variable (CV) quantum computing, encoding logical qubits into the phase space of an oscillator. While ideal GKP states are non-Gaussian, circuits composed of ideal GKP states, Gaussian operations, and Gaussian measurements can be efficiently simulated classically. However, the inclusion of the vacuum state renders these circuits hard to simulate, suggesting the vacuum acts as a critical resource. The origin of this quantum resource remains subtle: ideal GKP states exhibit negative Wigner distributions in the standard plane but positive distributions on the torus phase space used for GKP qudits. There is a need for a phase-space representation adapted to the translational symmetry of GKP states to precisely locate the non-classical resources (specifically, the zeros of the distribution) that distinguish simulable from non-simulable scenarios.

Methodology
The authors develop a framework based on covariant integral quantization on a specific phase space: the double discrete cylinder ( $\Gamma = \mathbb{Z}^2_\ell \times \mathbb{T}^2_\ell$ ).

Phase Space Construction: The configuration space is defined as a torus $\mathbb{T}^2_\ell$ , formed by the product of two circles with circumferences $\ell$ and $2\pi/\ell$ . This geometry arises from decomposing continuous position ( $x$ ) and momentum ( $p$ ) variables into integer and modular parts (modular variables), where the modular variables are bounded and commuting.
Quantization: A weight function $\varpi$ $ϖ$ on the phase space defines a quantization operator $\hat{M}_\varpi$ $\hat{M}_{ϖ}$ as a weighted superposition of displacement operators.
- Choosing a constant weight yields the Wigner distribution (expectation value of the displaced parity operator).
- Choosing a weight defined by the overlap with a fiducial state yields the Husimi distribution (expectation value of the displaced projector).
Stellar Representation: The fiducial state is chosen as a periodized Gaussian (a "wrapped Gaussian"), generating wrapped coherent states on the torus. The stellar function is defined as the overlap between a quantum state and these coherent states (removing the Gaussian prefactor). The zeros of this entire function on the lattice of heights (integer values of the modular variables) define a stellar rank on the double discrete cylinder.

Key Contributions and Results
The paper applies this formalism to position eigenstates, the vacuum, and GKP states (both ideal and physical), analyzing the location and behavior of the stellar function's zeros.

Ideal GKP Logical States: Ideal logical states ( $|0\rangle_x$ and $|1\rangle_x$ ) correspond to position eigenstates on the torus. Their stellar functions have zeros located strictly at half-integer heights. Consequently, they possess no zeros on the integer lattice of the phase space. This absence of lattice zeros is consistent with their positive Wigner distribution on the double cylinder.
Superpositions of Ideal States: Superpositions of ideal logical states (e.g., $|+\rangle_x$ ) possess infinitely many zeros on the integer lattice. This correlates with the negativity of their Wigner distribution on the double cylinder.
The Vacuum State: Although the vacuum is a Gaussian state with no zeros in the standard plane, its Husimi distribution on the double discrete cylinder does possess zeros on the lattice. These zeros appear beyond the single-cell approximation and depend on the choice of the unit cell. This finding supports the hypothesis that the vacuum introduces non-classical features (zeros) in this specific geometry, acting as a resource that complicates classical simulation.
Physical (Finitely Squeezed) GKP States:
- In the "large-comb" approximation (many peaks under a Gaussian envelope), the modular wavefunction factorizes.
- The zeros of the stellar function form continuous circles at specific heights. For generic squeezing widths, these circles do not intersect the integer lattice.
- Fine-tuned widths: Zeros reach the lattice only when the peak width ( $\sigma$ ) and envelope width ( $\kappa$ ) satisfy specific rational relationships (fine-tuning).
- Drift: As the peaks acquire finite width, the lattice zeros of superposition states drift away from the lattice, except for four specific zeros in the $|-\rangle_x$ superposition, which are protected by parity and remain on the lattice for all widths.

Significance and Claims
The paper claims that the stellar representation on the double discrete cylinder provides a natural tool to locate the quantum resources of GKP-based architectures. The emergence of zeros on the lattice for the vacuum and for finitely squeezed states (under specific conditions) mirrors the role of the vacuum in making GKP circuits hard to simulate. The work suggests that the "non-Gaussianity" required for quantum advantage in these systems is linked to the appearance of zeros on the lattice of the double discrete cylinder, rather than just negativity in the standard plane. The authors note that the stellar rank on this geometry quantifies the non-Gaussian character of the state in a way that respects the translational symmetry of the GKP code.

The paper concludes by identifying several extensions, including analyzing the coupling of modular variables beyond the large-comb approximation (where zeros may form isolated points rather than circles), relating stellar rank to simulation costs, and applying the formalism to non-rectangular lattices and rotor codes. The authors explicitly state that the construction currently depends on the choice of the unit cell, which remains an open question for future work.

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