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Homological Thresholds in Randomly Monitored Quantum Error-Correcting Codes

This paper investigates the robustness of stabilizer quantum error-correcting codes under single-round random Pauli measurements, demonstrating that their threshold behavior falls into distinct geometric and homological percolation categories and establishing a framework to determine critical exponents and phase diagrams for various code families to guide decoding strategies.

Original authors: Yoshito Watanabe, Daiki Sasamoto, Bo Han, Simon Trebst

Published 2026-10-05

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CC BY 4.0

Original authors: Yoshito Watanabe, Daiki Sasamoto, Bo Han, Simon Trebst

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

Quantum computers promise to solve problems that would take classical machines millennia, but they are notoriously fragile. The slightest disturbance from the environment can scramble the delicate information they hold, a phenomenon known as decoherence. To combat this, scientists use quantum error-correcting codes, which spread a single piece of information across many physical particles. By constantly checking the relationships between these particles without looking at the information itself, the system can detect and fix errors before they destroy the calculation. However, a new challenge has emerged in the study of these systems: what happens if we actively measure the particles to learn about the system, rather than just protecting it? This question sits at the intersection of quantum information and statistical physics, exploring how much we can learn about a quantum state before the act of learning destroys the very information we seek.

In a recent study, researchers investigated the limits of this "monitoring" process using a specific family of quantum codes called stabilizer codes. They simulated a scenario where individual particles in a quantum system are randomly measured one by one, asking a simple but profound question: at what point does the act of measurement become so frequent that the encoded information is irretrievably lost? The team discovered that the answer depends entirely on the geometry of the code being used. They found that the transition from a protected state to a collapsed one falls into two distinct categories. For some codes, the loss of information happens when the measured particles form a connected path across the system, much like a bridge forming across a river. For others, the loss occurs through a more complex, branching network of connections that does not rely on simple paths.

The researchers focused their investigation on two famous examples to illustrate this difference: the toric code and the color code. Both are designed to protect quantum information on a grid, but they behave very differently when subjected to random measurements. In the toric code, the information is lost only when the measured particles create a continuous, unbroken loop that wraps around the system. This is a straightforward geometric problem, similar to asking how many tiles need to be painted on a floor before a path connects one side to the other. The team confirmed that the point at which this happens follows the well-known rules of geometric percolation, a concept used to describe how fluids flow through porous materials.

In contrast, the color code behaves in a way that defies simple geometric intuition. Here, the information is lost not when a single path forms, but when the measured particles create a complex, branching structure that can split and rejoin at junctions. This process, which the authors call homological percolation, is governed by the algebraic rules of the code rather than simple connectivity. Even though the color code and the toric code lose their information at the exact same measurement frequency in certain directions, the way they lose it is fundamentally different. The color code's transition involves a critical point that is distinct from the standard geometric rules, suggesting a new type of phase transition in physics.

To reach these conclusions, the team performed massive computer simulations involving over one million qubits, a scale far beyond what is typically possible for these types of calculations. They tracked how the logical information survived as they increased the probability of measuring each particle. By analyzing the data with high precision, they were able to measure the "critical exponents" that describe how the system behaves near the tipping point. For the toric code, these numbers matched the standard predictions for geometric percolation. For the color code, the numbers were different, confirming that the underlying physics was unique. This distinction is crucial because it shows that two systems can fail at the same rate but for entirely different reasons, requiring different strategies to understand and predict their behavior.

The study also explored how these findings apply to more complex and practical quantum codes, including those used in modern proposals for scalable quantum computers. They found that the principles of homological percolation extend to these advanced systems as well, providing a unified language to describe when and how quantum information collapses under observation. The researchers also showed that understanding these monitoring thresholds can help predict the performance of these codes in real-world error correction scenarios, where the goal is to decode information from noisy data. By mapping out the boundaries where information survives, they provided a guide for designing more robust quantum systems.

Ultimately, this work clarifies the landscape of quantum error correction by revealing that the path to information loss is not a single road. Depending on the structure of the code, the system may fail because a simple connection is made, or because a complex, branching network forms. This insight helps physicists distinguish between different types of quantum phase transitions and offers a clearer picture of how to protect quantum information in the face of constant observation. The results suggest that while some codes are vulnerable to simple geometric connections, others are protected by more subtle algebraic structures, opening new avenues for designing quantum computers that can withstand the rigors of measurement and noise.

Technical Summary: Homological Thresholds in Randomly Monitored Quantum Error-Correcting Codes

Problem Statement
This work investigates the robustness of stabilizer quantum error-correcting (QEC) codes under a single round of random, independent single-qubit Pauli measurements. The study addresses the transition where encoded logical information is either preserved or collapsed by the measurement record. While previous research has linked decoherence thresholds to statistical mechanics (e.g., the random-bond Ising model) and inference transitions to weak measurements, this paper focuses on the specific phenomenology of projective monitoring. The central problem is to determine whether the threshold behavior for logical collapse in these monitored systems falls into the category of ordinary geometric percolation or a distinct "homological" percolation, and to map out the phase diagrams for various code families, including non-CSS, subsystem, and Floquet codes.

Methodology
The authors define a random measurement protocol where each physical qubit is measured in the $X$ , $Y$ , or $Z$ basis with probabilities $p_X, p_Y, p_Z$ , or left unmeasured with probability $1 - \sum p_\alpha$ . The phase boundaries are determined by calculating the coherent information $I_c(R|QC)$ between reference qubits and the code, conditioned on the full classical measurement record. Logical information is considered lost if the measurement record fixes a nontrivial logical Pauli operator (up to stabilizers).

To analyze these transitions, the authors employ two primary frameworks:

Stabilizer Tableau Simulations: They use optimized Clifford stabilizer simulations to track the evolution of the code state. For large-scale systems (up to $N \approx 1.4 \times 10^6$ qubits), they implement algorithmic improvements by working in "face-syndrome space" and utilizing binary elimination, reducing memory usage by a factor of $\approx 32$ compared to full tableau methods.
Homological Connectivity Analysis: Instead of tracking full quantum states, they analyze the algebraic structure of the measurement record. They define "homological-connectivity classes" based on the linear span of syndrome vectors generated by measured operators. This allows them to characterize logical collapse not merely by geometric path connectivity, but by whether the measured set supports a nontrivial logical class in the chain complex.

Key Contributions and Results

Distinction Between Geometric and Homological Percolation:
The paper establishes that logical survival transitions fall into two distinct universality classes:
- Geometric Percolation: Exemplified by the Toric Code in the pure- $Z$ limit. Here, logical collapse occurs when measured edges form a non-contractible loop. The critical exponent $\nu \approx 1.334(43)$ matches the standard 2D bond percolation value ( $4/3$ ).
- Homological Percolation: Exemplified by the Color Code. Here, logical collapse is determined by the ability of stabilizer deformations to split and reconnect logical strings into branched networks (string-nets) that span the system. This is a non-geometric connectivity problem. The critical exponent is found to be $\nu \approx 1.2073(15)$ , distinct from ordinary percolation.
Phase Diagrams and Thresholds:
The authors map out spherical phase diagrams for various codes, parameterized by measurement probabilities.
- Toric Code: The phase boundary is largely determined by geometric percolation thresholds ( $p_c = 1/2$ for pure axes), with subtle cooperative effects from $Y$ -measurements extending the collapse region slightly below the pure-axis thresholds.
- Color Code: Due to $X/Y/Z$ symmetry and the complementary-set structure, the threshold is fixed at $p_c = 1/2$ for all pure axes. The phase diagram exhibits "bites" along all three axes, corresponding to the condensation of different anyon types.
- Gross Code (BB Codes): The finite $[[144, 12, 12]]$ Gross code shows behavior intermediate between the Toric and Color codes. While finite-size scaling suggests a drift toward the geometric value, the authors note that the asymptotic thermodynamic limit remains unresolved due to slow convergence associated with the mobility sublattice of anyons.
Generalization to Other Codes:
The framework is extended to:
- 3D Codes: The 3D Toric code exhibits geometric percolation for string-like logical operators ( $\nu \approx 0.866$ ), while the 3D Color code exhibits homological percolation with membrane-like logical operators.
- Non-CSS and Subsystem Codes: The Wen plaquette model (non-CSS) shows unit thresholds ( $p_c=1$ ) for pure axes due to sublattice constraints, resulting in a phase diagram without inward "bites." Subsystem codes (e.g., Bacon-Shor) and Floquet codes are analyzed by fixing gauge qubits or measurement cycles to define a subspace, revealing diverse critical behaviors.
Algorithmic Efficiency:
The paper demonstrates that face-only linear algebra and binary elimination allow for simulations of systems with over one million qubits, a scale previously inaccessible to standard stabilizer tableau methods.

Significance
The paper claims to provide a unifying language—homological percolation—for logical-survival transitions across a broad range of randomly monitored stabilizer codes. It demonstrates that identical threshold locations (e.g., $p_c = 1/2$ ) do not necessarily imply identical critical behavior; the underlying algebraic structure (geometric vs. homological connectivity) dictates the universality class.

Furthermore, the work establishes that the relatively tractable problem of projective monitoring provides quantitative and qualitative guidance for more computationally demanding problems, such as:

Tunable Learning: Bounding the Nishimori thresholds for weak measurements.
Maximum-Likelihood Decoding: Providing bounds for decoding thresholds in the presence of incoherent noise.

By characterizing the phase diagrams of these monitored systems, the authors offer a framework to predict the robustness of logical information against measurement-induced collapse, linking topological order, anyon condensation, and statistical mechanics in the context of quantum error correction.

Technical Summary: Homological Thresholds in Randomly Monitored Quantum Error-Correcting Codes

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