Parametrized and random unitary (or orthogonal) $n$-qubit circuits play a central role in quantum information. As such, one could naturally assume that circuits implementing symplectic transformation would attract similar attention. However, this is not the case, as $\mathbb{SP}(d/2)$ -- the group of $d\times d$ unitary symplectic matrices -- has thus far been overlooked. In this work, we aim at starting to right this wrong. We begin by presenting a universal set of generators $\mathcal{G}$ for the symplectic algebra $i\mathfrak{sp}(d/2)$, consisting of one- and two-qubit Pauli operators acting on neighboring sites in a one-dimensional lattice. Here, we uncover two critical differences between such set, and equivalent ones for unitary and orthogonal circuits. Namely, we find that the operators in $\mathcal{G}$ cannot generate arbitrary local symplectic unitaries and that they are not translationally invariant. We then review the Schur-Weyl duality between the symplectic group and the Brauer algebra, and use tools from Weingarten calculus to prove that Pauli measurements at the output of Haar random symplectic circuits can converge to Gaussian processes. As a by-product, such analysis provides us with concentration bounds for Pauli measurements in circuits that form $t$-designs over $\mathbb{SP}(d/2)$. To finish, we present tensor-network tools to analyze shallow random symplectic circuits, and we use these to numerically show that computational-basis measurements anti-concentrate at logarithmic depth.

翻译：参数化与随机酉（或正交）$n$量子比特电路在量子信息中扮演核心角色。因此，人们自然会认为实现辛变换的电路同样会吸引关注。然而事实并非如此，因为$\mathbb{SP}(d/2)$——即$d\times d$酉辛矩阵群——迄今仍被忽视。本研究旨在纠正这一偏差。我们首先给出辛代数$i\mathfrak{sp}(d/2)$的一组通用生成元$\mathcal{G}$，该生成元由作用于一维晶格相邻位点的单量子比特和两量子比特Pauli算符构成。我们发现该生成元集与酉电路及正交电路的对应生成元集存在两个关键差异：其一，$\mathcal{G}$中的算符无法生成任意局域辛酉变换；其二，这些算符不具备平移不变性。进而我们重新审视辛群与Brauer代数之间的Schur-Weyl对偶性，并运用Weingarten微积分工具证明：Haar随机辛电路输出的Pauli测量可收敛至高斯过程。作为副产品，该分析为$\mathbb{SP}(d/2)$上构成$t$设计的电路中的Pauli测量提供了集中界。最后，我们提出分析浅层随机辛电路的张量网络工具，并利用该工具数值验证计算基测量在深度达到对数级时出现反集中现象。