# Error suppression via complementary gauge choices in Reed-Muller codes

## TL;DR

This paper investigates how concatenating Reed-Muller codes with complementary gauge choices can improve quantum error correction thresholds and logical error rates, advancing fault-tolerant quantum computation.

## Contribution

It demonstrates that concatenating complementary Reed-Muller codes enhances threshold performance and provides insights into logical error behavior under circuit noise.

## Key findings

- Concatenation of Reed-Muller codes increases the code capacity threshold.
- Smaller codes perform better under circuit level noise across various error rates.
- Choosing the inner code with a higher pseudo-threshold reduces logical failure rates.

## Abstract

Concatenation of two quantum error correcting codes with complementary sets of transversal gates can provide a means towards universal fault-tolerant computation. We first show that it is generally preferable to choose the inner code with the higher pseudo-threshold in order to achieve lower logical failure rates. We then explore the threshold properties of a wide range of concatenation schemes. Notably, we demonstrate that the concatenation of complementary sets of Reed-Muller codes can increase the code capacity threshold under depolarizing noise when compared to extensions of previously proposed concatenation models. We also analyze the properties of logical errors under circuit level noise, showing that smaller codes perform better for all sampled physical error rates. Our work provides new insights into the performance of universal concatenated quantum codes for both code capacity and circuit level noise.