# The role of entropy in topological quantum error correction

## TL;DR

This paper compares two variants of the surface code for quantum error correction, analyzing how lattice orientation affects error resilience and performance tradeoffs across different system sizes and error rates.

## Contribution

It introduces a detailed comparison of square and rotated lattice surface codes, revealing how lattice orientation influences error correction performance and failure mechanisms.

## Key findings

- Larger code distance improves performance at low error rates.
- The rotated lattice code's advantage diminishes near threshold.
- In certain regimes, the square lattice code marginally outperforms the rotated one.

## Abstract

The performance of a quantum error-correction process is determined by the likelihood that a random configuration of errors introduced to the system will lead to the corruption of encoded logical information. In this work we compare two different variants of the surface code with a comparable number of qubits: the surface code defined on a square lattice and the same model on a lattice that is rotated by $\pi / 4$. This seemingly innocuous change increases the distance of the code by a factor of $\sqrt{2}$.However, as we show, this gain can come at the expense of significantly increasing the number of different failure mechanisms that are likely to occur. We use a number of different methods to explore this tradeoff over a large range of parameter space under an independent and identically distributed noise model. We rigorously analyze the leading order performance for low error rates, where the larger distance code performs best for all system sizes. Using an analytical model and Monte Carlo sampling, we find that this improvement persists for fixed sub-threshold error rates for large system size, but that the improvement vanishes close to threshold. Remarkably, intensive numerics uncover a region of system sizes and sub-threshold error rates where the square lattice surface code marginally outperforms the rotated model.