Storage and retrieval of two unknown unitary channels

TL;DR

This paper investigates optimal methods for storing and retrieving one of two unknown unitary transformations, demonstrating strategies that maximize fidelity and success probability, with practical optical experiments validating the approach.

Contribution

It introduces the optimal sequential storage strategy and measure-and-prepare retrieval method for two unknown unitaries, with improved success scaling and experimental validation.

Findings

Sequential application of unitaries yields optimal storage states.

Measure-and-prepare retrieval maximizes fidelity.

Success probability scales as 1 - O(n^{-2}) for large fidelity.

Abstract

We address the fundamental task of converting $n$ uses of an unknown unitary transformation into a quantum state (i.e., storage) and later retrieval of the transformation. Specifically, we consider the case where the unknown unitary is selected with equal prior probability from two options. First, we prove that the optimal storage strategy involves the sequential application of the $n$ uses of the unknown unitary, and it produces the optimal state for discrimination between the two possible unitaries. Next, we show that incoherent "measure-and-prepare" retrieval achieves the maximum fidelity between the retrieved operation and the original (qubit) unitary. We then identify the retrieval strategy that maximizes the probability of successfully and perfectly retrieving the unknown transformation. In the regime in which the fidelity between the two possible unitaries is large the probability of success scales as $ P_{succ} = 1 - \mathcal{O}(n^{-2} ) $, which is a quadratic improvement with respect to the case in which the unitaries are drawn from the entire unitary group $U(d)$ with uniform prior probability. Finally, we present an optical experiment for this approach and assess the storage and retrieval quality using quantum tomography of states and processes. The results are discussed in relation to non-optimal measure-and-prepare strategy, highlighting the advantages of our protocol.