We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes—different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This result is caused by expansion coefficients that grow exponentially with system size, resulting in a “phantom” eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.

中文翻译：

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最快的局部纠缠扰频器、多级热化和非厄米模型

我们研究随机量子电路及其产生二分纠缠的速率，特别是关于 2 量子位门的选择和应用这些门的顺序（协议）。问题被映射到马尔可夫过程，我们证明存在大的谱等价类——不同的配置具有相同的谱。确定了以理论上可能的最快速率产生纠缠的最优门和协议。对渐近热纠缠的弛豫通过局部弛豫率的一系列相变进行，这是非厄米性的结果。特别是，非厄米性会导致速率比矩阵特征值差距预测的更快，或者更有趣的是，比预测的更慢。这个结果是由随系统大小呈指数增长的膨胀系数引起的，导致“幻影”特征值，并且是由于非厄米特征向量的非正交性。我们从数值上证明，这种现象也发生在具有非最优通用门、随机 U(4) 门的随机电路中，并且也没有空间或时间随机性，这表明它在其他非厄米设置中可能具有广泛的重要性，包括相关性。