We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension $\mathrm{d}_\mathrm{L}$ in PI states using physical qudits of dimension $\mathrm{d}_\mathrm{P}$. We extend the Knill--Laflamme (KL) conditions for $d-1$ deletion errors from qubits to qudits and investigate numerically both qubit ($\mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2$) and qudit ($\mathrm{d}_\mathrm{L} > 2$ or $\mathrm{d}_\mathrm{P} > 2$) PI codes. We analyze the scaling of the block length $n$ in terms of the code distance $d$, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to $d-1$ deletion errors have block length $n(d) \geq (3d^2 + 1) / 4$, which implies an upper bound $d \leq \sqrt{12n-3}/3$ on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing $\mathrm{d}_\mathrm{P}$ results in $n$ monotonically decreasing and approaching the quantum Singleton bound $n(d) \geq 2d-1$. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.

翻译：本文研究置换不变（PI）量子纠错码，该码利用维度为 $\mathrm{d}_\mathrm{P}$ 的物理量子比特，通过 PI 态编码维度为 $\mathrm{d}_\mathrm{L}$ 的逻辑量子比特。我们将 Knill–Laflamme（KL）条件从量子比特推广至量子比特以处理 $d-1$ 个删除错误，并对量子比特（$\mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2$）与量子比特（$\mathrm{d}_\mathrm{L} > 2$ 或 $\mathrm{d}_\mathrm{P} > 2$）PI 码进行了数值研究。我们分析了码块长度 $n$ 随码距 $d$ 的标度关系，并与 Ouyang、Aydin–Alekseyev–Barg（AAB）以及 Pollatsek–Ruskai（PR）提出的现有 PI 码族进行了比较。我们的三项主要发现如下：（i）我们推测，能够纠正最多 $d-1$ 个删除错误的量子比特 PI 码的块长度满足 $n(d) \geq (3d^2 + 1) / 4$，这暗示其码距存在上界 $d \leq \sqrt{12n-3}/3$，且 PR 码可达到该上界。（ii）对于编码单个量子比特的量子比特 PI 码，我们数值观测到增大 $\mathrm{d}_\mathrm{P}$ 会导致 $n$ 单调递减并趋近于量子 Singleton 界 $n(d) \geq 2d-1$。（iii）我们提出将量子比特 AAB 构造半解析地推广至量子比特，通过求解线性规划得到显式解。因此，我们的结果为量子比特与量子比特 PI 码的块长度标度下界提供了关键见解，并展示了在 PI 码背景下提升物理局域维度的优势。