Efficient Tomography of Non-Interacting Fermion States

Abstract

We give an efficient algorithm that learns a non-interacting fermion state, given copies of the state. For a system of $n$ non-interacting fermions and $m$ modes, we show that $O(m^3{n}^2\log (1/\delta )/{\epsilon }^4)$ copies of the input state and $O(m^4{n}^2\log (1/\delta )/{\epsilon }^4)$ time are sufficient to learn the state to trace distance at most $\epsilon$ with probability at least $1-\delta$. Our algorithm empirically estimates one-mode correlations in $O(m)$ different measurement bases and uses them to reconstruct a succinct description of the entire state efficiently.

arXiv comment18 pages, 1 figure. We strengthen our results by learning the entire state, rather than the distribution, which is accomplished by a more careful error analysis and a slight modification to our algorithm. We also correct an error in the previous version (our analysis assumed the m*m matrix output by our algorithm was rank-n, but it was full-rank). We thank Andrew Zhao for identifying this error