Improved classical shadows from local symmetries in the Schur basis

TL;DR

This paper introduces a new joint measurement protocol for classical shadows that scales with the rank of the unknown quantum state, offering significant efficiency improvements in low-rank regimes.

Contribution

The authors develop the first joint measurement protocol for classical shadows with sample complexity depending on the state's rank, utilizing local symmetries in the Schur basis.

Findings

Sample complexity scales as O(√(rB)/ε²), improving efficiency for low-rank states.

Introduces a solution for classical shadows with non-identical input states.

Uses local symmetries in the Schur basis to simplify analysis and avoid complex calculations.

Abstract

We study the sample complexity of the classical shadows task: what is the fewest number of copies of an unknown state you need to measure to predict expected values with respect to some class of observables? Large joint measurements are likely required in order to minimize sample complexity, but previous joint measurement protocols only work when the unknown state is pure. We present the first joint measurement protocol for classical shadows whose sample complexity scales with the rank of the unknown state. In particular we prove $\mathcal O(\sqrt{rB}/\epsilon^2)$ samples suffice, where $r$ is the rank of the state, $B$ is a bound on the squared Frobenius norm of the observables, and $\epsilon$ is the target accuracy. In the low-rank regime, this is a nearly quadratic advantage over traditional approaches that use single-copy measurements. We present several intermediate results that may be of independent interest: a solution to a new formulation of classical shadows that captures functions of non-identical input states; a generalization of a ``nice'' Schur basis used for optimal qubit purification and quantum majority vote; and a measurement strategy that allows us to use local symmetries in the Schur basis to avoid intractable Weingarten calculations in the analysis.