Magic State Distillation using Asymptotically Good Codes on Qudits

TL;DR

This work develops a lifting-based framework to construct triorthogonal TVZ-classical codes over square qudit alphabets $q=2^{2m}$ with $m\ge3$, yielding asymptotically good quantum codes that admit transversal CCZ gates and enable constant-space overhead MSD at relatively small qudit dimensions (notably $q=64$). By proving that the lifting operation preserves triorthogonality, the authors produce a family of codes over $\mathbb{F}_{r^2}$ with $r=2^m$ that surpass the TVZ bound, and from these derive quantum codes with transversal CCZ and concrete parameters, including the explicit $[[42,14,6]]_{64}$ code with $\gamma\approx0.613$. They also show how a distilled $|CCZ\rangle_{2^{2m}}$ state can be reduced to $|CCZ\rangle_{2^n}$ using a constant-depth Clifford circuit, requiring at most 12 single-qudit and 9 two-qudit Clifford gates, enabling scalable MSD across arbitrary $n$. Collectively, these results point to practical MSD implementations on near-term high-dimensional qudit platforms, with TVZ-bound-exceeding classical codes feeding into robust quantum codes and state-reduction procedures.

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.

Figures (5)

• Figure 1: The blue curve corresponds to a function field $F_j$. Then $n$ pairwise distinct points $P_i$ correspond to rational places of $F_j$. The divisor $G_j$ specifies the Riemann-Roch space $\mathcal{L}(G_j)$. The points $P_i$ on the curves and the divisor $G_j$ specify an AG code of which the codewords are the vectors $(f(P_1), {...}, f(P_n))$ where $f \in \mathcal{L}(G_j)$. The yellow curve $F_{j+1}$ is an extension of $F_j$. An arrow $P_i \rightarrow P'_j$ indicates that $P'_j$ lies above $P_i$. This relation allows us to lift the code on $F_j$ into a code on $F_{j+1}$ while preserving triorthogonality. Then, we lift the Riemann-Roch space similarly by taking the conorm map of $G_{j+1} = \operatorname{Con}_{F_{j+1}/F_j}(G_j)$ to specify the lifted code. This builds sequences of triorthogonal codes by successive liftings.
• Figure 2: Bounds on $\gamma$ vs. $\mathcal{R}$ for quantum codes $\mathcal{Q}_j$ at different levels of the lifting for $q=2^6=64$. For $\mathcal{Q}_0$, we present two-sided bounds, while for the rest we have only upper bounds.
• Figure 3: Bounds on $\gamma$ vs. $\mathcal{R}$ for quantum codes $\mathcal{Q}_j$ at different levels of the lifting for $q=2^8=256$. For $\mathcal{Q}_0$, we present two-sided bounds, while for the rest we have only upper bounds.
• Figure 4: Bounds on $\gamma$ vs. $\mathcal{R}$ for quantum codes $Q_j$ at different levels of the lifting for $q=2^{10}=1024$. For $\mathcal{Q}_0$, we present two-sided bounds, while for the rest we have only upper bounds.
• Figure 5: Reduction of state $\ket{CCZ}_{r^2}$ to a state $\ket{CCZ^{\gamma}}_r$ where $\gamma := 1 + \eta$, $\eta := \theta^{r + 1}$ and $\{\theta, \theta^r\}$ is a normal basis of $\mathbb{F}_{r^2}$ over $\mathbb{F}_r$ such that $\theta + \theta^r = 1$.

Theorems & Definitions (31)

• Theorem 1: Informal version of the main results
• Definition 1: Algebraic Function field
• Definition 2: Places
• Definition 3: Divisors
• Definition 4: Riemann-Roch Space
• Definition 5: Algebraic geometry code
• Definition 6: Dual code
• Proposition 3.1: Dual of $C_{\mathcal{L}}(D,G)$ STICHTENOTH1988199
• Definition 7: Triorthogonal code
• Proposition 3.2: Some useful properties of AG codes