Continuous-variable designs and design-based shadow tomography from random lattices

TL;DR

This paper demonstrates that GKP lattice states form a continuous-variable 2-design and uses this to develop a shadow tomography protocol with derived sample complexity bounds for quantum systems.

Contribution

It introduces the use of GKP lattice states as continuous-variable state designs and constructs a new shadow tomography protocol based on these states.

Findings

GKP states form a rigged continuous-variable 2-design.

Derived sample complexity bounds for GKP shadow tomography.

Provided physical gadgets for implementing the protocol.

Abstract

We investigate state designs for continuous-variable quantum systems using the aid of lattice-like quantum states. These are code states of Gottesman-Kitaev-Preskill (GKP) codes. We show that for an n-mode system, the set of all GKP states forms a rigged continuous-variable state 2-design. We use these lattice state designs to construct a continuous variable shadow tomography protocol, derive sample complexity bounds for both global- and local GKP shadows under reasonable physical assumptions, and provide the physical gadgets needed to implement this protocol.