Efficient quantum tomography

In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/ε) copies suffice to obtain an estimate ρ that satisfies ||ρ − ρ||F2 ≤ ε (with high probability). An immediate consequence is that O((ρ) · d/ε2) ≤ O(d2/ε2) copies suffice to obtain an ε-accurate estimate in the standard trace distance. This improves on the best known prior result of O(d3/ε2) copies for full tomography, and even on the best known prior result of O(d2log(d/ε)/ε2) copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension. Next, we generalize these results to show that one can perform efficient principal component analysis on ρ. Our main result is that O(k d/ε2) copies suffice to output a rank-k approximation ρ whose trace-distance error is at most ε more than that of the best rank-k approximator to ρ. This subsumes our above trace distance tomography result and generalizes it to the case when ρ is not guaranteed to be of low rank. A key part of the proof is the analogous generalization of our spectrum-learning results: we show that the largest k eigenvalues of ρ can be estimated to trace-distance error ε using O(k2/ε2) copies. In turn, this result relies on a new coupling theorem concerning the Robinson–Schensted–Knuth algorithm that should be of independent combinatorial interest.