发表机构
Stanford University; IIT Bombay; TTIC（斯坦福大学; 印度理工学院孟买分校; 东京工业研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
我们构造了逼近量子Gilbert-Varshamov界的随机QLDPC码系综，通过控制简并度并级联随机内码，在多个量子信道上达到随机稳定子码的性能。
AI 中文摘要
我们证明，对于任意$R,\epsilon >0$和任意素数$p\geq 2$，存在一个无限族的$p$元量子低密度奇偶校验（QLDPC）码，其码率$R$，校验权重为$O_\epsilon(1)$，归一化距离至少为$\delta_{\mathrm{GV}}(p,R)-\epsilon$。这里，$\delta_{\mathrm{GV}}(p,R)$表示$p$元稳定子码的量子Gilbert-Varshamov（GV）界。事实上，我们构造了这样的QLDPC码的一个系综，使得从该系综中随机选取的码以高概率接近GV界。此外，该系综在多个量子信道上的性能与随机稳定子码相匹配。具体而言，它逼近擦除信道的量子容量以及无记忆Pauli信道（包括去极化信道）的哈希界。处理QLDPC码的一个重大挑战是它们必然是简并的，即包含许多低权重稳定子。我们工作的一个关键贡献是构造了对其简并度具有定量控制的QLDPC码。这些码通过将已知的渐近好QLDPC码的构造与Alon、Edmonds和Luby [FOCS'95]的基于扩展器的距离放大过程相结合而获得。我们的随机系综通过从这些接近量子Singleton界的低简并QLDPC码出发，并将每个坐标与一个随机内码级联来构造。这可以看作是Thommesen构造的量子类比，也是Ouyang的一个结果（使用适当设计的外码）的LDPC版本。
英文摘要
We show that for any $R,ε>0$ and any prime $p\geq 2$, there exists an infinite family of $p$-ary quantum low-density parity-check (QLDPC) codes, rate $R$, checks of weight $O_ε(1)$, and normalized distance at least $δ_{\mathrm{GV}}(p,R)-ε$. Here, $δ_{\mathrm{GV}}(p,R)$ denotes the quantum Gilbert-Varshamov (GV) bound for $p$-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.
