Random dilation superchannel

We present a quantum circuit that implements the random dilation superchannel, transforming parallel queries of an unknown quantum channel into the same number of parallel queries of a randomly chosen dilation isometry of the input channel. This is a natural generalization of the random purification channel, that transforms copies of an unknown mixed state to copies of a randomly chosen purification state. The circuit complexity of our construction is $O(\mathrm{poly}(n, \log d_I, \log d_O))$, where $n$ is the number of queries and $d_I$ and $d_O$ are the input and output dimensions of the input channel, respectively. This random dilation superchannel is extended to the sequential queries approximately, by transforming the parallel random dilation isometry into sequential random dilation unitaries with $O(\mathrm{poly}(d_I))$ overhead in the number of queries. We also show that our results can be further extended to the case of quantum superchannels. On the other hand, we show a no-go theorem on the exact random dilation of sequential queries with $o(\mathrm{poly}(\min\{d_I, d_O\}))$ query overhead, showcasing a fundamental difference between the parallel and sequential cases. As an application, we show an efficient storage-and-retrieval of an unknown quantum channel, which improves the program cost exponentially in the retrieval error $\varepsilon$. For the case where the Kraus rank $r$ is the least possible (i.e., $r = d_I/d_O$), we show quantum circuits that transform $n$ parallel queries of an unknown quantum channel $\Lambda$ to $\Theta(n^\alpha)$ parallel queries of $\Lambda$ for any $\alpha<2$ approximately, and implement its Petz recovery map for the maximally mixed reference state probabilistically and exactly.