AI 中文总结
本文提出基于随机匹配的浅Clifford电路架构，在最优O(log n)深度下构造稳定子码，达到量子Gilbert-Varshamov速率-距离权衡，匹配线性距离编码器的光锥下界。
AI 中文摘要
用低电路开销编码量子信息是容错量子计算中的核心挑战。随机电路提供了一种自然机制，可通过并行应用简单门快速扩展逻辑信息。Brown和Fawzi证明，基于两量子比特Clifford门的随机电路可实现量子Gilbert-Varshamov速率-距离权衡，其深度为O(log³n)。本文证明，在支持更受限的门分布下，相同的渐近权衡可在最优O(log n)深度下实现。对于任意固定δ>0和足够大的n，若k/n < 1 - H(d/n) - (d/n)log₂3 - δ，我们可构造深度为O(log n)的随机电路，其以高概率定义一个[n,k]稳定子码，距离至少为d+1，这与线性距离编码器的Ω(log n)光锥下界匹配。我们的集合采用随机匹配电路架构，由T个独立的置换不变层组成。每层中，量子比特通过均匀随机完美匹配配对，每对应用一个随机独立的两量子比特Clifford门。门分布无需在两量子比特Clifford群上均匀，甚至无需具有完全支持；相反，我们允许满足三个正则性条件的Clifford门的非常一般分布。特别地，该构造可在每层中使用n/2个随机匹配对上的CNOT门，结合并行单量子比特Clifford旋转来实现。这些正则性条件使我们能将随机电路的二阶矩动力学简化为二元支撑串上的可逆马尔可夫链。我们建立该马尔可夫链的对数击中时间界，并比较其平稳分布，以证明电路的编码性质。
英文摘要
Encoding quantum information with low circuit overhead is a fundamental challenge in fault-tolerant quantum computation. Random circuits provide a natural mechanism for rapidly spreading logical information through simple gates applied in parallel. Brown and Fawzi showed that random Clifford circuits on two-qubit Clifford gates provide such encoders that achieve the quantum Gilbert-Varshamov rate-distance tradeoff with depth $O(\log^3 n)$. We show that the same asymptotic tradeoff is attained in optimal $O(\log n)$ depth under a gate distribution with a more restricted support. For every fixed $δ>0$ and sufficiently large $n$, if $\frac kn < 1 - H(\frac{d}{n}) - \frac{d}{n}\log_2 3 - δ$, we can construct random circuits of depth $O(\log n)$ which define, with high probability, an $[n,k]$ stabilizer code of distance at least $d+1$, which matches the $Ω(\log n)$ light-cone lower bound for linear distance encoders. Our ensemble employs a random matching circuit architecture consisting of $T$ independent permutation-invariant layers. In each layer, the qubits are paired up by a uniformly random perfect matching, and a random independent two-qubit Clifford gate is applied to each pair. The gate distribution need not be uniform over, or even have full support on, the two-qubit Clifford group; rather, we allow for very general distributions on Clifford gates satisfying three regularity conditions. In particular, the construction can be implemented using $n/2$ CNOT gates on randomly matched pairs in each layer, with parallel one-qubit Clifford twirls. These regularity conditions allow us to reduce the second-moment dynamics of our random circuits to a reversible Markov chain on binary support strings. We establish logarithmic hitting-time bounds for this Markov chain and comparisons of its stationary distribution to prove the coding properties of the circuits.
Comments33 pages, 2 figures, 1 table
