High-threshold magic state distillation with quantum quadratic residue codes

TL;DR

This paper explores the use of quantum quadratic residue codes for magic state distillation, demonstrating their equivalence to known codes and introducing new codes with high thresholds for distilling T and Strange states.

Contribution

It unifies various known and new quantum codes under quadratic residue codes, showing their effectiveness in high-threshold magic state distillation.

Findings

Existing codes are equivalent to quantum quadratic residue codes.

New codes with high thresholds for T and Strange states are introduced.

Infinitely many quadratic residue codes can distill T states with non-trivial thresholds.

Abstract

We present applications of quantum quadratic residue codes in magic state distillation. This includes showing that existing codes which are known to distill magic states, like the $5$-qubit perfect code, the $7$-qubit Steane code, and the $11$-qutrit and $23$-qubit Golay codes, are equivalent to certain quantum quadratic residue codes. We also present new examples of quantum quadratic residue codes that distill qubit $T$ states and qutrit Strange states with high thresholds, and we show that there are infinitely many quantum quadratic residue codes that distill $T$ states with a non-trivial threshold. All of these codes, including the codes with the highest currently known thresholds for $T$ state and Strange state distillation, are unified under the umbrella of quantum quadratic residue codes.