Principal eigenstate classical shadows
This addresses a specific problem in quantum information processing for efficiently characterizing quantum states, but it is incremental as it builds on existing classical shadows methods.
The paper tackles the problem of learning a classical description of the principal eigenstate of an unknown quantum state, assuming it has an eigenvalue greater than 1/2, and presents a protocol with sample complexity scaling with this eigenvalue, showing it is optimal among natural approaches and matches pure state classical shadows when the eigenvalue is near 1.
Given many copies of an unknown quantum state $ρ$, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that $ρ$ has an eigenstate $|φ\rangle$ with (unknown) eigenvalue $λ> 1/2$, the goal is to learn a (classical shadows style) classical description of $|φ\rangle$ which can later be used to estimate expectation values $\langle φ|O| φ\rangle$ for any $O$ in some class of observables. We consider the sample-complexity setting in which generating a copy of $ρ$ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue $λ$ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when $λ$ is sufficiently close to $1$, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows.