High-threshold magic state distillation with quantum quadratic residue codes

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.

Figures (6)

• Figure 1: Post-distillation noise $\epsilon'$ as a function of pre-distillation noise $\epsilon$ for quantum quadratic residue codes that distill qubit $T$ states. The $5$-, $23$-, $29$-, $47$-, $53$-, and $71$-qubit codes have non-trivial distillation thresholds.
• Figure 2: Success probability of magic state distillation protocols for qubit $T$ states based on quantum quadratic residue codes.
• Figure 3: Post-distillation noise as a function of pre-distillation noise for codes that distill qutrit Strange states. The $11$-qutrit Golay code, as well as the $17$-, $23$-, and $41$-qutrit codes have non-trivial distillation thresholds.
• Figure 4: Success probability as a function of pre-distillation noise for codes that distill qutrit Strange states.
• Figure 5: Post-distillation noise as a function of predistillation noise for magic state distillation protocols based on expurgated quadratic residue codes for qubit $T$ states. The length $5$, $23$, $29$, $47$, $53$, and $71$ codes have nonzero threshold. The other code lengths tested have a fixed point at the pure magic state but cannot distill them from noisy magic states.

Theorems & Definitions (32)

• Lemma 1
• proof : Proof of Lemma \ref{['Lemma:MTHEqualsHermitian']}
• Definition 1
• Theorem 1
• proof : Proof of Theorem \ref{['Theorem:QRHermitianOrthogonality']}
• Corollary 1
• proof : Proof of Corollary \ref{['Corollary:F4HermitianSelfOrthogonality']}
• Corollary 2
• proof : Proof of Corollary \ref{['Corollary:F9HermitianSelfOrthogonality']}
• Theorem 2