Efficient Local Classical Shadow Tomography with Number Conservation

TL;DR

This paper introduces a new local shadow tomography protocol tailored for quantum systems with number conservation, enabling efficient reconstruction of few-body observables with minimal samples and fast post-processing.

Contribution

The authors propose the 'All-Pairs' protocol that uses two-body gates and permutation symmetry to efficiently reconstruct observables in number-conserving quantum systems.

Findings

Requires only polynomial samples for reconstruction

Linear time post-processing algorithm developed

Demonstrated on paired Luttinger liquid of hardcore bosons

Abstract

Shadow tomography aims to build a classical description of a quantum state from a sequence of simple random measurements. Physical observables are then reconstructed from the resulting classical shadow. Shadow protocols which use single-body random measurements are simple to implement and capture few-body observables efficiently, but do not apply to systems with fundamental number conservation laws, such as ultracold atoms. We address this shortcoming by proposing and analyzing a new local shadow protocol adapted to such systems. The "All-Pairs" protocol requires one layer of two-body gates and only $\textrm{poly}(V)$ samples to reconstruct arbitrary few body observables. Moreover, by exploiting the permutation symmetry of the protocol, we derive a linear time post-processing algorithm. We provide a proof-of-principle reference implementation and demonstrate the reconstruction of 2- and 4-point functions in a paired Luttinger liquid of hardcore bosons.