Improved classical shadows from local symmetries in the Schur basis

Understand

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.