Magic State Distillation using Asymptotically Good Codes on Qudits

TL;DR

This paper introduces a new family of good triorthogonal qudit codes that enable efficient magic state distillation at lower qudit dimensions, advancing quantum error correction and fault-tolerant quantum computing.

Contribution

The authors construct the first family of asymptotically good triorthogonal codes on qudits with dimensions as low as 64, surpassing previous parameter limitations.

Findings

Achieved constant space overhead magic state distillation with qudits of size 64.

Constructed a promising [[42,14,6]] code over qudits of size 64.

Demonstrated reduction of CCZ states across different qudit dimensions with constant-depth circuits.

Abstract

Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension $q\gg 100$ or code length $\mathcal{N}\gg 100$. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the $\mathbb{F}_{2^{2m}}$ alphabet with $m \geq 3$ that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as $q=64$. Further, we identify a promising code with parameters $[[42,14,6]]_{64}$. Finally, we show that a distilled $|{CCZ}\rangle_{2^{2m}}$ can be reduced to a $|{CCZ}\rangle_{2^n}$ state for arbitrary $n$ with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.