Agnostic Product Mixed State Tomography via Robust Statistics
Alvan Arulandu, Ilias Diakonikolas, Daniel Kane, Jerry Li
TL;DR
This work addresses agnostic tomography for product mixed states by linking quantum tomography to classical robust statistics. It proves that agnostic tomography can be solved in polynomial time using only single-qubit measurements by reducing to robustly learning binary product distributions and then solving that robust problem with new closeness certificates and a convex relaxation. The authors present a near-optimal classical robust learner for binary products, achieving $d_{tv}(p,\hat{p}) \le O(\epsilon \log(1/\epsilon))$ under $\epsilon$-corruption with poly$(n,1/\epsilon)$ samples, which feeds into a quantum tomography algorithm that outputs a product state $\hat{\pi}$ with $d_{tr}(\pi,\hat{\pi}) \le O(\epsilon \log(1/\epsilon))$. They also show that adaptivity is information-theoretically necessary for any efficient algorithm using only non-adaptive single-qubit measurements, via a non-adaptive lower bound. Overall, the paper delivers the first efficient agnostic tomography guarantee for a nontrivial class of mixed quantum states and connects it to robust statistics with concrete algorithmic gains and limits.
Abstract
We consider the problem of agnostic tomography with \emph{mixed state} ansatz, and specifically, the natural ansatz class of product mixed states. In more detail, given $N$ copies of an $n$-qubit state $ρ$ which is $ε$-close to a product mixed state $π$, the goal is to output a nearly-optimal product mixed state approximation to $ρ$. While there has been a flurry of recent work on agnostic tomography, prior work could only handle pure state ansatz, such as product states or stabilizer states. Here we give an algorithm for agnostic tomography of product mixed states which finds a product state which is $O(ε\log 1 / ε)$ close to $ρ$ which uses polynomially many copies of $ρ$, and which runs in polynomial time. Moreover, our algorithm only uses single-qubit, single-copy measurements. To our knowledge, this is the first efficient algorithm that achieves any non-trivial agnostic tomography guarantee for any class of mixed state ansatz. Our algorithm proceeds in two main conceptual steps, which we believe are of independent interest. First, we demonstrate a novel, black-box efficient reduction from agnostic tomography of product mixed states to the classical task of \emph{robustly learning binary product distributions} -- a textbook problem in robust statistics. We then demonstrate a nearly-optimal efficient algorithm for the classical task of robustly learning a binary product, answering an open problem in the literature. Our approach hinges on developing a new optimal certificate of closeness for binary product distributions that can be leveraged algorithmically via a carefully defined convex relaxation. Finally, we complement our upper bounds with a lower bound demonstrating that adaptivity is information-theoretically necessary for our agnostic tomography task, so long as the algorithm only uses single-qubit two-outcome projective measurements.