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Bosonic codes from compact phase spaces

This paper establishes the algebraic structure of bosonic quantum error-correcting codes on genus-two Riemann surfaces by constructing code words as automorphic forms, while proving a fundamental no-go theorem that non-amenable stabilizer groups on higher-genus surfaces preclude the existence of normalizable exact code states, thereby distinguishing them from standard GKP codes.

Original authors: David Roberts, Aaron Slipper, Alireza Parhizkar, Victor V. Albert, Mohammad Hafezi

Published 2026-09-01

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CC BY 4.0

Original authors: David Roberts, Aaron Slipper, Alireza Parhizkar, Victor V. Albert, Mohammad Hafezi

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, scientists face a fundamental problem: quantum information is incredibly fragile. A single bit of data, encoded in a particle of light or a vibration in a crystal, can be scrambled by the slightest whisper of heat or noise. To protect this data, researchers use error-correcting codes, which spread a single piece of information across a larger system so that if part of it gets corrupted, the whole can be recovered. One of the most promising methods for storing this information involves using bosonic modes, such as the electromagnetic fields inside a microwave cavity. These systems are continuous, meaning they can hold an infinite amount of data in theory, but in practice, they need a way to be tamed into a finite, manageable shape.

For years, the leading approach has been the Gottesman-Kitaev-Preskill, or GKP, code. This method works by arranging the possible states of the system on a flat, repeating grid, much like the squares on a chessboard. This grid is formed by a specific set of symmetries that shift the system back and forth. Because the underlying mathematics of this flat grid is "amenable"—a technical term meaning the symmetries behave in a predictable, well-behaved way—scientists can create approximate versions of the code that are good enough to work, even if they aren't perfect. The question that has lingered is what happens if we try to build these codes on a different kind of shape. Instead of a flat plane, what if the space where the information lives is curved and closed, like the surface of a sphere or a more complex, multi-holed shape?

A team of researchers has now explored this question by constructing quantum codes on a specific type of curved surface known as a genus-two Riemann surface. In simple terms, imagine a shape with two holes, like a double-holed doughnut, but where the geometry is hyperbolic, meaning it curves away from itself in every direction, creating a vast, saddle-like landscape. The researchers treated this surface as the phase space for their quantum system, using the symmetries of this curved shape to define the rules of the code. They successfully built the mathematical framework for these codes, showing how to generate the specific quantum states, or code words, that would live on this surface. These states are not just abstract ideas; they are constructed using specific mathematical functions called automorphic forms, which are the natural language for describing patterns on such curved surfaces. The team also demonstrated that the logical operations, the "gates" used to process the information, could be performed using standard quantum tools like squeezing and rotation, and that the group of these operations could be incredibly rich, capable of representing any finite group of symmetries.

However, the most significant finding of this work is a sharp and definitive limitation. While the researchers could build the mathematical structure of the code, they proved that it is impossible to create a physical, stable quantum state that satisfies all the rules of this code. In the flat GKP code, the symmetries allow for states that get arbitrarily close to the perfect solution, even if they require a lot of energy. On this curved, two-holed surface, the geometry is fundamentally different. The symmetries of this shape are "non-amenable," meaning they are chaotic and do not settle down in the same way. This chaos forces a strict barrier: there is a minimum amount of energy, or a spectral gap, that any state must have. Because of this gap, no normalizable quantum state can ever satisfy all the stabilizer conditions simultaneously. It is not just that the states are hard to find; it is mathematically impossible for them to exist in a physical system.

The researchers confirmed this theoretical obstruction with numerical simulations. They modeled the system and calculated the energy levels, finding a clear gap that prevented the system from settling into a stable code state. This result stands in stark contrast to the flat GKP code, where such a gap does not exist, allowing for approximate solutions. The study concludes that while the rich geometry of these higher-genus surfaces offers a beautiful and powerful way to organize quantum gates, that same geometry acts as a wall against the existence of the code states themselves. The very feature that makes the logical operations so versatile is what makes the storage of information impossible in this specific configuration. This discovery draws a clear line in the sand for the field of quantum error correction, showing that not all geometric ideas, no matter how elegant, can be realized as physical quantum memories.

Technical Summary: Bosonic Codes from Compact Phase Spaces

Problem Statement
The Gottesman–Kitaev–Preskill (GKP) code encodes quantum information in a single bosonic mode by imposing a lattice of displacement symmetries on a flat phase space (the complex plane). The quotient of this plane by the lattice is a torus (genus $g=1$ ), and the code words correspond to theta functions. A fundamental question arises: what bosonic codes emerge when the phase space is a compact Riemann surface of genus $g \ge 2$ ? Such surfaces possess non-trivial topology and negative curvature (hyperbolic geometry), fundamentally altering the algebraic and spectral properties of the stabilizer group compared to the flat torus case.

Methodology and Framework
The authors construct bosonic quantum error-correcting codes on compact hyperbolic Riemann surfaces by utilizing the geometric quantization of the phase space.

Phase Space and Stabilizers: The phase space is modeled as a compact surface $C = \Gamma \backslash \mathbb{D}$ , where $\mathbb{D}$ is the Poincaré disk and $\Gamma$ is a cocompact Fuchsian group. The physical system consists of two bosonic modes ( $\hat{a}, \hat{b}$ ) carrying a representation of $SU(1,1)$ . The stabilizer group $\Gamma$ is lifted to a subgroup of $SU(1,1)$ , acting as squeezing transformations (the hyperbolic counterpart to GKP displacements).
Stellar Representation: The authors employ the stellar representation, mapping quantum states to holomorphic sections of a line bundle over the phase space. For a compact surface of genus $g$ , the stellar functions are not entire functions but rather automorphic forms (specifically, holomorphic $k$ -differentials) that transform under the stabilizer group $\Gamma$ .
Code Space Dimension: Using the Riemann–Roch theorem, the dimension of the code space $\mathcal{H}_\Gamma$ is derived as $\dim \mathcal{H}_\Gamma = (2k-1)(g-1)$ for integer weight $k \ge 2$ , where $k$ relates to the total boson number sector.
Logical Gates: The logical gate group is identified as the automorphism group of the surface, $\text{Aut}(C)$ , which preserves the stabilizer lattice. By Greenberg's theorem, any finite group can be realized as the logical gate group of such a code on some hyperbolic surface.

Key Contributions and Results

Explicit Construction of Code Words:
The authors explicitly construct code words as automorphic forms. For the genus-two Bolza surface (the most symmetric genus-two curve), they generate the full tower of code spaces at all integer weights.
- At weight $k=2$ , the code space is spanned by Poincaré theta series associated with the three shortest closed geodesics (the "pants" decomposition of the surface).
- These theta series satisfy a conic relation $Y^2 = XZ$ , allowing the extraction of square roots to form a basis for the $k=1$ code space (a logical qubit).
- The logical gates for the Bolza code are explicitly computed as $48$-element Gaussian operations (squeezing and rotations) forming the group $GL(2, \mathbb{F}_3)$ .
The Spectral Gap Obstruction (No-Go Theorem):
The paper proves a fundamental no-go theorem regarding the existence of normalizable code states for $g \ge 2$ .
- Non-Amenability: For genus $g \ge 2$ , the surface group $\pi_1(\Sigma_g)$ is non-amenable.
- Spectral Gap: This non-amenability forces a strictly positive spectral gap $\delta > 0$ in the stabilizer Hamiltonian $\hat{H}_{\text{stab}} = 1 - \frac{1}{N} \sum (\hat{D}_i + \hat{D}_i^\dagger)$ .
- Consequence: Because of this gap, no normalizable quantum state can satisfy all stabilizer conditions simultaneously, nor can a sequence of normalizable states approximate the code space with arbitrary precision. This stands in sharp contrast to the $g=1$ GKP code, where the stabilizer group is amenable ( $\mathbb{Z}^2$ ), the spectral gap is zero, and approximate code words of arbitrary precision exist.
Amenability Dichotomy:
The authors establish a sharp dividing line based on the amenability of the stabilizer group:
- $g=0$ (Sphere): Finite stabilizer group; exact code words exist.
- $g=1$ (Torus): Amenable infinite stabilizer group; approximate code words of arbitrary precision exist.
- $g \ge 2$ (Hyperbolic): Non-amenable stabilizer group; no normalizable exact or approximate code words exist.
Experimental Platform and Dissipative Stabilization:
The paper proposes a circuit-QED implementation using two microwave cavities coupled to a transmon ancilla.
- The effective Hamiltonian includes cross-Kerr interactions and parametric two-mode squeezing, enabling the necessary Gaussian gates.
- A "conditional-kick" protocol is described for stabilizing the code. Numerical simulations show that while a single-generator protocol successfully drives the system toward the code space, multi-generator protocols (required for $g \ge 2$ ) fail to converge to zero stabilizer energy, instead plateauing at a value determined by the spectral gap. This provides numerical evidence for the theoretical obstruction.

Significance and Claims
The paper claims to complete the classification of continuous-variable quantum codes on compact, two-dimensional phase spaces by curvature and genus.

It resolves the question of higher-genus bosonic codes by demonstrating that the rich geometric structure enabling complex logical gate groups (any finite group) simultaneously obstructs the existence of the code states themselves.
The work highlights a fundamental tension between the geometry required for rich logical operations (non-amenability) and the stability required for error correction (amenability).
The authors note that while exact stabilizer eigenstates are impossible for $g \ge 2$ , finite-energy analogues (where unitaries are deformed by non-unitary envelopes) might bypass this obstruction, presenting an open challenge for hardware realization.

The paper does not claim that these codes are immediately viable for fault-tolerant quantum computing in their current form due to the spectral gap obstruction. Instead, it provides a rigorous mathematical framework and a "no-go" result that defines the limits of bosonic codes on hyperbolic phase spaces, contrasting them with the successful flat-space GKP codes.

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