Agnostic Tomography of Stabilizer Product States

Abstract

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 $n^{O(\log (2/\tau ))}/{\epsilon }^2$, which is $\mathsf{poly}(n/\epsilon )$ for any constant $\tau$.

20 pages. V2: minor corrections. V3: addition of new references. V4: reworked the algorithm and presentation. V5: accepted to Quantum

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