We propose a learning method for estimating unknown pure quantum states. The basic idea of our method is to learn a unitary operation $\widehat{U}$ that transforms a given unknown state $|{\psi }_{\tau }\rangle$ to a known fiducial state $|f\rangle$. Then, after completion of the learning process, we can estimate and reproduce $|{\psi }_{\tau }\rangle$ based on the learned $\widehat{U}$ and $|f\rangle$. To realize this idea, we cast a random-based learning algorithm, called “single-shot measurement learning,” in which the learning rule is based on an intuitive and reasonable criterion: the greater the number of success (or failure), the less (or more) changes are imposed. Remarkably, the learning process occurs by means of a single-shot measurement outcome. We demonstrate that our method works effectively, i.e., the learning is completed with a finite number, say $N$, of unknown-state copies. Most surprisingly, our method allows the maximum statistical accuracy to be achieved for large $N$, namely $\simeq O\left(N^{-1}\right)$ scales of average infidelity. It highlights a nontrivial message, that is, a random-based strategy can potentially be as accurate as other standard statistical approaches.

• Received 16 May 2018

DOI:https://doi.org/10.1103/PhysRevA.98.052302