Architectures and random properties of symplectic quantum circuits

TL;DR

This paper explores the properties and behaviors of symplectic quantum circuits, introducing a generator set, analyzing their randomness and measurement outcomes, and providing tools for their study.

Contribution

It presents a universal generator set for symplectic algebra, analyzes the limitations of symplectic circuits, and develops tensor-network methods for shallow circuit analysis.

Findings

Pauli measurements in Haar random symplectic circuits can converge to Gaussian processes.

Concentration bounds for Pauli measurements in t-designs over symplectic group.

Computational-basis measurements anti-concentrate at logarithmic depth in shallow circuits.

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.