⚛️ quantum physics

Subsystem Symmetries and Fracton Models in Quantum Error Correction

This thesis establishes the high resilience of fracton-based quantum error-correcting codes by utilizing statistical-mechanical mappings and Kramers-Wannier-type dualities to analyze classical Ising models, ultimately determining that the Checkerboard code achieves the theoretical maximum error threshold for three-dimensional CSS codes.

Original authors: Giovanni Canossa

Published 2026-08-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Giovanni Canossa

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

To build a computer that can solve problems impossible for today's machines, scientists must first solve a problem of its own making: how to keep fragile information safe. Quantum computers rely on the strange behavior of subatomic particles, which can exist in multiple states at once, but this same fragility makes them incredibly sensitive to the slightest disturbance. A whisper of heat or a stray magnetic field can scramble the data, causing the machine to fail. To prevent this, researchers use quantum error correction, a method that spreads a single piece of information across many physical particles. If one particle gets corrupted, the system can detect the error and fix it without ever looking directly at the data, which would destroy it. The goal is to find a way to store this information so robustly that the computer can run for a long time without collapsing.

For decades, the most promising designs for these error-correcting codes have been based on topology, a branch of mathematics that studies shapes and how they hold together. Imagine a knot on a string; you can stretch or twist the string, but you cannot untie the knot without cutting the string. Similarly, topological codes store information in the global shape of the system rather than in the local state of individual particles. This makes the information hard to destroy because a local error cannot change the global shape. However, while these topological codes are excellent at protecting data, they have a limit to how much noise they can tolerate before the protection breaks down. Scientists have been searching for new types of codes that can withstand even more chaos, looking for structures that are not just topologically protected, but also constrained by deeper, more rigid rules.

A recent doctoral thesis by Giovanni Canossa at the Ludwig-Maximilians-Universität München explores a new frontier in this search, focusing on a class of exotic materials known as fracton models. These models describe a state of matter where the particles, or excitations, are not free to move around like gas molecules or even like electrons in a wire. Instead, they are stuck in place or can only move in very specific, restricted ways, as if they were trapped in a cage that only opens for certain moves. This extreme restriction, which arises from complex symmetries in the underlying physics, suggests that these systems might be incredibly resistant to errors. Canossa's work investigates whether these fracton models can serve as the foundation for the next generation of quantum memory, and if so, how well they perform compared to the best codes we have today.

The research begins by looking at the classical side of the problem, studying simplified models of magnetic spins that behave according to specific symmetry rules. In these models, the spins are arranged in three-dimensional grids, and the rules governing them involve flipping entire planes or complex, self-repeating fractal patterns of spins simultaneously. Canossa and his team simulated these systems to see how they behave when heated up. They found that these systems undergo a dramatic change, shifting from a disordered state to an ordered one, but this change happens in a very sharp, sudden way, unlike the gradual transitions seen in ordinary materials. This sharp transition is a sign of a highly stable ordered phase, which is a good omen for a memory device. Crucially, the team discovered that the way these systems organize themselves is tied to their geometry in a unique way, creating a vast number of possible stable states that are difficult to confuse with one another.

The next step was to translate these classical findings into the language of quantum error correction. By applying a mathematical process that turns the classical rules into quantum constraints, the researchers constructed two specific quantum codes: the Checkerboard model and Haah's code. These codes are designed so that errors create defects that are stuck in place, just like the particles in the fracton models. Because the errors cannot move freely to spread across the system, they are less likely to cause a catastrophic failure. The team then asked the critical question: how much noise can these codes actually handle before they fail? To answer this, they used a powerful technique that maps the problem of decoding errors onto a problem of thermal physics, allowing them to use computer simulations to predict the limits of the codes.

The results were striking. For the Checkerboard model, the researchers determined that it can tolerate an error rate of approximately 0.107, meaning it can function correctly even if roughly 10.7 percent of its physical components are corrupted. This is a remarkably high number, sitting just below the theoretical maximum limit for this type of code, which is around 11 percent. In fact, this is the highest error threshold ever found for a three-dimensional quantum code, surpassing previous records held by other topological codes. The study suggests that the Checkerboard model is not just a theoretical curiosity but a highly practical candidate for building robust quantum memories. Furthermore, the researchers found that the mathematical relationship between the classical models and the quantum codes is so strong that it allows them to predict the performance of other, more complex codes, such as Haah's code, with high confidence. They suspect that Haah's code will also perform near this theoretical limit, though proving this requires even more difficult simulations.

While the results are promising, the work also highlights the challenges that remain. The codes studied here are designed for a specific type of noise where errors happen independently and randomly. In a real-world quantum computer, errors are often correlated in time and space, making the problem much harder to solve. The researchers note that while their findings establish a strong foundation, the full potential of these fracton codes against realistic, complex noise patterns is still an open question. Nevertheless, by linking the behavior of classical magnetic models to the performance of quantum codes, this research provides a clear roadmap for designing better error correction. It shows that by harnessing the rigid, immobile nature of fracton phases, we can build quantum memories that are far more resilient than previously thought possible, bringing the dream of a stable, large-scale quantum computer one step closer to reality.

Technical Summary: Subsystem Symmetries and Fracton Models in Quantum Error Correction

Problem Statement
The development of robust quantum memories requires the construction of quantum error-correcting (QEC) codes with high error thresholds and favorable scaling properties. While topological codes like the toric code and color code are well-established, they often suffer from low error thresholds or limited encoding rates in three dimensions. A promising avenue involves "fracton" models, which exhibit topological order with excitations that have restricted mobility, potentially offering enhanced resilience against local noise. However, the connection between the statistical mechanics of the classical models underlying these fracton phases and the performance of the resulting quantum codes remains incompletely understood. Specifically, the behavior of classical Ising models with subsystem symmetries (symmetries acting on lower-dimensional submanifolds) and their relation to the error thresholds of associated fracton codes requires systematic investigation.

Methodology
The thesis employs a multi-faceted approach combining classical statistical mechanics, lattice gauge theory, and numerical simulation:

Classical Model Analysis: The author investigates two specific three-dimensional classical Ising models with subsystem symmetries: the Tetrahedral Ising Model (TIM) and the Fractal Ising Model (FIM). These models are analyzed to understand their thermal phase transitions, symmetry-breaking patterns, and ground-state degeneracies.
Gauging Procedure: The classical subsystem symmetries are "gauged" to construct corresponding $\mathbb{Z}_2$ lattice gauge theories. This procedure maps the classical TIM to the Checkerboard model and the classical FIM to Haah's cubic code. These quantum models serve as the candidate QEC codes.
Statistical-Mechanical (SM) Mapping: The decoding problem for the quantum codes under independent and identically distributed (i.i.d.) noise is mapped to the thermodynamics of classical Ising models with quenched disorder (Random-Bond Ising models). This allows the error threshold of the quantum code to be identified with the phase transition point of the corresponding disordered classical model.
Numerical Simulations:
- Disorder-Free: Multicanonical Monte Carlo (MUCA) methods are used to overcome the severe metastability associated with the strong first-order phase transitions observed in the disorder-free TIM and FIM.
- Disordered: Parallel Tempering (PT) Monte Carlo simulations are employed to study the random-bond versions of these models. This is necessary because the strong first-order transitions soften in the presence of disorder, and PT is effective for sampling across the resulting energy landscapes.
Finite-Size Scaling: The transition points are determined by analyzing the scaling of energy histograms, correlation lengths, and Binder cumulants across various system sizes ( $L$ ) to distinguish between first-order transitions, crossovers, and continuous transitions.

Key Contributions and Results

Characterization of Subsystem Symmetry Breaking:
- Both the TIM and FIM exhibit a single, strongly first-order phase transition separating a high-temperature disordered phase from a low-temperature phase with long-range subsystem order (planar for TIM, fractal for FIM).
- The finite-size scaling of the pseudo-critical temperature $\beta_c(L)$ follows an anomalous law $\beta_c(L) - \beta_c(\infty) \propto L^{-(D-d)}$ , where $D=3$ is the system dimension and $d$ is the dimension of the subsystem symmetry support. This scaling is driven by the subextensive ground-state degeneracy ( $\text{GSD} \sim 2^{L^{D-d}}$ ) inherent to these models.
- The study confirms that the breaking of $\mathbb{Z}_2$ subsystem symmetries in short-range 3D models typically favors discontinuous (first-order) transitions, a phenomenology distinct from standard global symmetry breaking.
Error Threshold Determination:
- Using the SM mapping, the author determines the optimal code-capacity threshold for the Checkerboard model (derived from the TIM).
- Numerical analysis of the disordered TIM identifies a threshold of $p_{\text{th}} = 0.107(3)$ .
- This value nearly saturates the theoretical upper bound for optimal thresholds in mutually dual CSS codes, predicted by a generalized Kramers-Wannier duality fixed-point condition ( $p_{\text{th}} \approx 0.11$ ).
- The Checkerboard model thus represents the highest known optimal error threshold among 3D CSS codes, surpassing the 3D toric code (~~3.3%), 3D color code (~~1.9%), and the non-self-dual X-cube model (~7.5%).
Extension to Haah's Code:
- While a direct numerical determination of the threshold for Haah's code (derived from the FIM) is hindered by severe finite-size effects and the complex size-dependence of the FIM's symmetry group, the author provides a heuristic argument.
- Based on the structural similarities between the TIM and FIM (both being self-dual, exhibiting strong first-order transitions, and lacking spin-glass phases), it is argued that Haah's code should also satisfy the principal-factor condition and possess a threshold close to the theoretical limit of $0.11$.

Significance and Claims
The paper claims to establish a unified framework connecting classical subsystem-symmetric spin physics, fracton topological order, and quantum error correction.

Validation of Duality-Based Predictions: The work provides strong numerical evidence that the generalized Kramers-Wannier duality fixed-point condition accurately predicts the error thresholds of fracton codes. This supports the hypothesis that mutually dual classical spin models (even those with subsystem symmetries) lead to quantum codes that saturate the optimal threshold bound.
Superiority of Fracton Codes: The results identify the Checkerboard model as a highly resilient candidate for quantum memory, offering both a near-optimal error threshold and a superior encoding rate compared to other known 3D topological and fracton codes.
Methodological Power: The thesis demonstrates the efficacy of the statistical-mechanical mapping combined with duality arguments as a powerful tool for analyzing and constructing robust QEC codes, potentially bypassing the need for prohibitively expensive direct decoding simulations for complex 3D models.
Open Questions: The author modestly notes that while the threshold under static, uncorrelated noise is well-characterized, the behavior of these codes under realistic circuit-level noise (involving temporal correlations) remains an open problem. The mapping for circuit-level noise would require simulating a 4D disordered Ising model, which is currently computationally intractable.

In summary, the dissertation argues that the interplay between subsystem symmetries, geometric constraints, and duality relations in classical spin models provides a robust foundation for designing quantum error-correcting codes with near-optimal performance.

More like this