Quantum

Intrinsic locality dimension of quantum codes

Yimin Lu, Esther Xiaozhen Fu, Zi-Wen Liu

Featured June 5, 2026

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Simply

A new math tool called intrinsic dimension helps scientists understand how different quantum error-correcting codes are connected, even if they don't fit on a simple grid, showing how complex their operations can be.

In depth

This paper introduces the intrinsic locality dimension (

β

) for quantum stabilizer codes, a novel geometric parameter derived from fractal geometry that is independent of any background embedding. This dimension unifies the characterization of diverse quantum codes, from topological to highly non-local ones, by quantifying their effective connectivity. The authors demonstrate how this intrinsic dimension fundamentally constrains key code properties, including code parameters, the level of fault-tolerant logical gates in the Clifford hierarchy, and thermal stability.

Key Takeaways

1
The paper defines the intrinsic locality dimension ( $β$ ) for stabilizer codes, based on the Assouad dimension, providing a geometry-independent measure of effective connectivity.
2
The intrinsic dimension establishes fundamental trade-off bounds for code parameters ( $k_{λ}$ , $d_{λ}$ , $n_{λ}$ ) and constrains the achievable level of fault-tolerant logical gates within the Clifford hierarchy.
3
The framework unifies the understanding of various quantum code families, including topological codes, fractal codes, and qLDPC codes, by classifying them according to their intrinsic geometric properties, even allowing for non-integer dimensions.

Conceptual Flow

HIGH LEVEL

Methodology: Unifying Code Geometry

The paper invents a new way to measure how 'connected' a quantum code is, even if it's not on a simple grid, by looking at how its parts grow.

Quantum Code Structure

Analyze Connectivity Growth

Intrinsic Dimension Number

Results: Predicting Code Capabilities

This new 'connectedness number' then tells us how good the code can be at fixing errors and what kind of special operations it can do.

Intrinsic Dimension Number

Predict Limits

Error Correction Power

Allowed Operations