Probabilistic storage and retrieval (PSR) of unitary quantum dynamics is possible with exponentially small failure probability with respect to the number of systems used as a quantum memory [M. Sedlák, A. Bisio, and M. Ziman, Phys. Rev. Lett. 122, 170502 (2019)]. Here we study improvements due to a priori knowledge about the unitary transformation to be stored. In particular, we study $N\to 1$ PSR of qubit phase gates, i.e., qubit rotations around the $Z$ axis with an unknown angle, and show that if we access the gate only $N$ times the optimal probability of perfect retrieving of its single use is $N/(N+1)$. We propose a quantum circuit realization for the optimal protocol and show that a programmable phase gate [G. Vidal, L. Masanes, and J. I. Cirac, Phys. Rev. Lett. 88, 047905 (2002)] can be turned into a $(2^k-1)\to 1$ optimal PSR of phase gates and requires only $k$ controlled-not gates, while having exponentially small failure probability in $k$.

• Received 5 May 2020
• Accepted 21 August 2020

DOI:https://doi.org/10.1103/PhysRevA.102.032618