We study entanglement transitions in Clifford (stabilizer) random tensor networks (RTNs) and monitored quantum circuits, by introducing an exact mapping onto a (replica) statistical mechanics model. For RTNs and monitored quantum circuits with random Haar unitary gates, entanglement properties can be computed using statistical mechanics models whose fundamental degrees of freedom (“spins”) are permutations, because all operators commuting with the action of the unitaries on a tensor product Hilbert space are (linear combinations of) permutations of the tensor factors (“Schur-Weyl duality”). When the unitary gates are restricted to the smaller group of Clifford unitaries, the set of all operators commuting with this action, called the commutant, will be larger, and no longer form a group. We use the recent full characterization of this commutant by Gross et al. [Commun. Math. Phys. 385, 1325 (2021)] to construct statistical mechanics models for both Clifford RTNs and monitored quantum circuits, for on-site Hilbert-space dimensions which are powers of a prime number $p$. The elements of the commutant form the spin degrees of freedom of these statistical mechanics models, and we show that the Boltzmann weights are invariant under a symmetry group involving orthogonal matrices with entries in the finite number field ${\mathbf{F}}_p$ (“Galois field”) with $p$ elements. This implies that the symmetry group and consequently all universal properties of entanglement transitions in Clifford circuits and RTNs will, respectively, in general depend on and only on the prime $p$. We show that Clifford monitored circuits with on-site Hilbert-space dimension $d=p^M$ are described by percolation in the limits $d\to \infty$ at (a) $p=$ fixed but $M\to \infty$, and at (b) $M=1$ but $p\to \infty$. In the limit (a) we calculate the effective central charge, and in the limit (b) we derive the following universal minimal cut entanglement entropy $S_A=(\sqrt{3}/\pi )lnpln{L}_A$ for $d=p$ large at the transition. We verify those predictions numerically, and present extensive numerical results for critical exponents at the transition in monitored Clifford circuits for prime number on-site Hilbert-space dimension $d=p$ for a variety of different values of $p$, finding that projective and forced measurement schemes yield the same critical exponents and that they approach percolation values at large $p$. We clearly establish multifractal scaling of the purity, reflected in a continuous spectrum of critical exponents, while the typical exponent is the prefactor of the logarithm in the entanglement entropy. As a technical result, we generalize the notion of the Weingarten function, previously known for averages involving the Haar measure, to averages over the Clifford group.

中文翻译：

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Clifford 随机张量网络和监控量子电路的统计力学模型

我们通过引入精确映射到（复制）统计力学模型来研究 Clifford（稳定器）随机张量网络 (RTN) 和监控量子电路中的纠缠跃迁。对于 RTN 和具有随机 Haar 酉门的受监控量子电路，可以使用其基本自由度（“自旋”）是排列的统计力学模型来计算纠缠属性，因为所有算子都与张量积希尔伯特空间上酉系的作用进行交换是张量因子的排列（“Schur-Weyl 对偶性”）（的线性组合）。当酉门被限制为较小的 Clifford酉群时，所有与此动作交换的算子的集合（称为交换子）将会更大，并且不再形成一个群。我们使用 Gross等人最近对该交换子的完整表征。 [交流。数学。物理。 385 , 1325 (2021)] 为 Clifford RTN 和受监控的量子电路构建统计力学模型，用于现场希尔伯特空间维度（素数的幂）$p$。交换子的元素形成了这些统计力学模型的自旋自由度，并且我们证明了玻尔兹曼权重在涉及有限数域中的正交矩阵的对称群下是不变的${\mathbf{F}}_p$（“伽罗瓦域”）$p$元素。这意味着 Clifford 电路和 RTN 中纠缠跃迁的对称群和所有通用属性通常分别取决于且仅取决于素数$p$。我们展示了 Clifford 监测具有现场希尔伯特空间维数的电路$d=p^{\mathrm{中号}}$通过限制中的渗透来描述$d\to \mathrm{无穷大}$在（一）$p=$固定但是$\mathrm{中号}\to \mathrm{无穷大}$，并且在 (b)$\mathrm{中号}=1$但$p\to \mathrm{无穷大}$。在极限（a）中我们计算有效中心电荷，在极限（b）中我们推导出以下通用最小割纠缠熵$S_A=（\sqrt{3}/\pi ）因p因L_A$为了$d=p$过渡时较大。我们以数值方式验证了这些预测，并针对质数现场希尔伯特空间维数的受监控 Clifford 电路中的转变处的关键指数提供了广泛的数值结果$d=p$对于各种不同的值$p$，发现投影和强制测量方案产生相同的临界指数，并且它们在很大程度上接近渗透值$p$。我们清楚地建立了纯度的多重分形标度，反映在临界指数的连续谱中，而典型指数是纠缠熵中对数的前因子。作为技术结果，我们将 Weingarten 函数的概念（以前以涉及 Haar 度量的平均值而闻名）推广到 Clifford 组的平均值。