Agnostic Tomography of Stabilizer Product States

We define a quantum learning task called agnostic tomography, where given copies of an arbitrary state $\rho$ and a class of quantum states $\mathcal{C}$, the goal is to output a succinct description of a state that approximates $\rho$ at least as well as any state in $\mathcal{C}$ (up to some small error $\epsilon$). This task generalizes ordinary quantum tomography of states in $\mathcal{C}$ and is more challenging because the learning algorithm must be robust to perturbations of $\rho$. We give an efficient agnostic tomography algorithm for the class $\mathcal{C}$ of $n$-qubit stabilizer product states. Assuming $\rho$ has fidelity at least $\tau$ with a stabilizer product state, the algorithm runs in time $n^{O(\log (2\mathrm{/}\tau ))}\mathrm{/}{\epsilon }^2$, which is $\mathsf{p}\mathsf{o}\mathsf{l}\mathsf{y}(n\mathrm{/}\epsilon )$ for any constant $\tau$.