Probabilistic storage and retrieval of qubit phase gates

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. Sedlak, 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 -> 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) -> 1 optimal PSR of phase gates and requires only k controlled-Nor gates, while having exponentially small failure probability in k.