# Improved rate-distance trade-offs for quantum codes with restricted connectivity

**Authors:** Nouédyn Baspin, Venkatesan Guruswami, Anirudh Krishna, Ray Li  
**arXiv:** [2307.03283](https://arxiv.org/abs/2307.03283)  
**Date:** 2023-07-06  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2307.03283v1/

## Key points

- Quantum error-correcting codes rely on limited connectivity of qubits
- Geometric locality is crucial in constraining code properties
- Quantum codes on a D-dimensional lattice must adhere to the inequality $k d^{2/(D-1)} \le O(n)$
- Connectivity graph influences code parameters beyond geometrically-local codes
- Baspin et al. establish a tighter dimension-distance trade-off based on size of separators
- Advancement enhances understanding of connectivity's impact on quantum code properties
- Research extends distance bound to stabilizer codes with specific separation profile, beyond LDPC codes
- Work broadens scope of applicable codes and refines trade-offs for optimizing quantum error-correction mechanisms

## Summary

In the realm of quantum error-correcting codes, the feasibility of these codes hinges on the limited connectivity exhibited by the qubits subject to the code constraints. Pioneering works by Bravyi & Terhal (BT) and Bravyi, Poulin & Terhal (BPT) have shed light on how geometric locality plays a crucial role in constraining code properties. Specifically, they have shown that $[[n,k,d]]$ quantum codes defined by local checks on a $D$-dimensional lattice must adhere to the inequality $k d^{2/(D-1)} \le O(n)$. Building upon this foundation, Baspin and Krishna delved into a more general exploration of how the connectivity graph associated with a quantum code influences the code parameters. Their findings extend beyond the scope of geometrically-local codes captured by BPT and BT bounds, encompassing a richer class of codes. In their recent work titled "Improved rate-distance trade-offs for quantum codes with restricted connectivity," Nouédyn Baspin, Venkatesan Guruswami, Anirudh Krishna, and Ray Li delve deeper into this intricate relationship between connectivity and code parameters. They refine existing knowledge by establishing a tighter dimension-distance trade-off that is contingent upon the size of separators in the connectivity graph. This advancement not only enhances our understanding of how connectivity impacts quantum code properties but also paves the way for more efficient and effective error correction strategies. Moreover, their research yields a distance bound that extends to all stabilizer codes exhibiting a specific separation profile, transcending previous limitations that were confined to LDPC codes. By broadening the scope of applicable codes and refining trade-offs based on connectivity considerations, Baspin et al. 's work represents a significant step forward in optimizing quantum error-correction mechanisms. Through their meticulous analysis and innovative insights, they contribute valuable knowledge that can potentially enhance the reliability and performance of quantum computing systems in practical applications.