Optimal Probabilistic Storage and Retrieval of Unitary Channels

We address the question of quantum memory storage for quantum dynamics. In particular, we design an optimal protocol for $N\to 1$ probabilistic storage and retrieval of unitary channels on $d$-dimensional quantum systems. If we access the unknown unitary gate only $N$ times, the optimal success probability of perfect single-use retrieval is $N/(N-1+d^2)$. The derived size of the memory system exponentially improves the known upper bound on the size of the program register needed for probabilistic programmable quantum processors. Our results are closely related to probabilistic perfect alignment of reference frames and probabilistic port-based teleportation.

Figure 1

Optimal $1\to 1$ PSAR of unitary channels.

Figure 2

Illustration of $N\to 1$ PSAR. (a) PSAR with the most general strategy. (b) PSAR with parallel use of unitary channels.

Figure 3

A modified optimal $1\to 1$ PSAR in which $\mathcal{U}$ is stored and the inverse transformation ${\mathcal{U}}^{†}$ is retrieved [$SU(2)$ case]. The generalization to the $N\to 1$ case is straightforward.

Figure 4

Use of port-based teleportation scheme for PSAR.