Quantum Error Correction in Quaternionic Hilbert Spaces

TL;DR

This work extends quantum error correction into quaternionic Hilbert spaces by defining quaternionic gates and Pauli-like operators and constructing a quaternionic stabilizer code. It develops a quaternionic [[5,1,3]] code, analyzes its syndrome structure, and demonstrates how quaternionic error components can be detected and corrected within the stabilizer formalism. The authors report an improved logical-threshold around $p_{th} \approx 0.015$ and a scaling $p_L \sim p^{2.2}$, indicating enhanced fault tolerance compared with the standard complex-valued code. The study opens a new direction for QEC by leveraging the richer quaternionic algebra to improve error distinguishability and resilience in high-noise quantum environments.

Abstract

We propose quaternion-based strategies for quantum error correction by extending quantum mechanics into quaternionic Hilbert spaces. Building on the properties of quaternionic quantum states, we define quaternionic analogues of Pauli operators and quantum gates, ensuring inner product preservation and Hilbert space conditions. A simple encoding scheme maps logical qubits into quaternionic systems, introducing natural redundancy and enhanced resilience against noise. We construct a quaternionic extension of the five-qubit code, introducing a framework of 15 syndrome measurements to detect quaternionic errors, including quaternionically rotated error components. Numerical estimates show that the quaternionic five-qubit code achieves a logical error threshold of approximately $p_{th} \approx 0.015$, demonstrating improved performance compared to the standard complex-valued code. These results suggest a new pathway for quantum error correction in high-noise environments, leveraging the richer structure of quaternionic quantum mechanics to improve fault tolerance.

Figures (1)

• Figure 1: Comparison of the logical error rate versus physical error rate for the standard [[5,1,3]] code and its quaternionic extension.