Architectures and random properties of symplectic quantum circuits

TL;DR

This work initiates a systematic study of quantum circuits implementing symplectic unitaries, a largely neglected subgroup of the unitary group. It provides a universal, nearest-neighbor circuit construction for SP(d/2) using a canonical antisymmetric form, and shows that locally symplectic gates may fail to preserve symplectic structure, yielding universal SU(d) dynamics. Leveraging Weingarten calculus and Brauer algebra, the authors establish that Haar-random (and certain design) symplectic circuits drive Pauli measurements toward Gaussian processes and prove concentration bounds that scale with system size, while also proving anti-concentration for shallow, depth- logarithmic random circuits and offering tensor-network tools to analyze shallow regimes. Together, these results illuminate the statistical properties of symplectic quantum circuits and their potential for quantum-sampling tasks and phase-space simulations on near-term hardware.

Abstract

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 transformations 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 fill this gap. We begin by presenting a universal set of generators $\mathcal{G}$ for the symplectic algebra $\mathfrak{sp}(d/2)$, 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.

Figures (8)

• Figure 1: Schematic representation of our main results. a) When compared against other groups, the compact group $\mathbb{SP}(d/2)$ of $d\times d$ symplectic unitaries has received considerably less attention. b) Here we introduce tools to study $\mathbb{SP}(d/2)$, such as presenting easy-to-implement circuit architectures capable of producing any symplectic evolution. We also review the Weingarten calculus for this group and use it to study properties of random symplectic circuits, like their convergence to Gaussian processes (deep circuits) or the emergence of anti-concentration (shallow circuits).
• Figure 2: Quantum circuits for symplectic unitaries. Example of the basic building block for the implementation of symplectic unitary transformations on a quantum computer. The notation $R_{H_l}$ stands for $e^{i\theta_l H_l}$, with independent $\theta_l$ angles in each gate. As stated in Theorem \ref{['th:symplectic-universal']}, the Lie closure of the generators appearing in this circuit, which are not translationally invariant, produces $\mathfrak{sp}(d/2)$. This implies that any symplectic unitary from the $\mathbb{SP}(d/2)$ group can be implemented by a quantum circuit architecture consisting of blocks of this form.
• Figure 3: Circuits with local symplectic gates are not symplectic. Example of a building block consisting of two-qubit symplectic gates. This architecture is universal for quantum computation, as the Lie closure of all the generators leads to $\mathfrak{su}(d)$, and thus any (special) unitary transformation can be decomposed into a circuit consisting of these gates.
• Figure 4: Elements of the Brauer algebra $\mathfrak{B}_t(-d)$. Here we present all the elements of $\mathfrak{B}_t(-d)$ for $t=1$ and $t=2$, as well as some elements for $t=3$. In all cases we introduce their tensor representation visualization, their decomposition into disjoint pairs, and when convenient, their representation $F_d$. We use the convention whereby we first enumerate all the items on the left-hand side from top to bottom and then the items on the right-hand side (also from top to bottom).
• Figure 5: Computation of the Gram matrix elements. a) Tensor representation of the trace operation. b) We diagrammatically show how to compute all the matrix entries of W for $t=2$, using tensor notation. The computation of these matrix elements leads to Eq. \ref{['eq:gram-t2']}.

Theorems & Definitions (27)

• Proposition 1
• Theorem 1
• Proposition 2
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 1
• Corollary 2
• Theorem 5
• Proposition 1