Storage and retrieval of two unknown unitary channels

TL;DR

This work investigates storing and later retrieving an unknown unitary drawn from two options with equal prior, showing that optimal storage is a sequential use of the $n$ available calls and yields a state optimal for discriminating between the two unitaries. It then analyzes retrieval under two regimes: an approximate deterministic case where a measure-and-prepare strategy achieves maximal average process fidelity, and a perfect probabilistic case where the retrieval succeeds with maximal probability, scaling as $P_{succ}=1-\mathcal{O}(n^{-2})$ when the two unitary fidelities are high. The results are instantiated with a qubit analysis, a compact quantum circuit using a single CNOT gate for perfect probabilistic storage and retrieval, and a photonic demonstration validating the protocol via quantum process tomography. By framing retrieval as a programmable quantum processor, the paper connects fundamental limits of transforming unknown unitary channels to practical architectures, and demonstrates a quadratic improvement over Haar-random unitary priors in the two-unitary scenario. This work thus advances both the theoretical limits and experimental practice of quantum transformation processing with direct implications for programmable quantum devices.

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.

Figures (10)

• Figure 1: Illustration of a general storage and retrieval protocol for two $d$-dimensional unitary transformations $U_0$, $U_1$. These operations are used $n$ times during the storage phase (a). Retrieval can be deterministic (b) and hence yielding only approximation to $\mathcal{U}_{0\!/\!1}(\xi)$, or probabilistic (c), yielding perfect application of the stored unitary to any state $\xi$.
• Figure 2: Optimal quantum circuit for storage of two d-dimensional unitary transformations $U_0$,$U_1$ used $n$ times. The circuit is the same for both considered types of retrieval.
• Figure 3: Optimal approximate deterministic storage and retrieval of two $d$-dimensional unitary transformations can be realized via optimal discrimination among the two unitaries in the storage phase and conditional preparation of slightly modified unitaries in the retrieval phase.
• Figure 4: Optimal success probability of perfect storage and retrieval of two $d$-dimensional unitary transformations as a function of $n$ - the number of black box uses and of the angle $\alpha$, which equals half of the angular spread of $(U_1)^\dagger U_0$. Dotted and dashed lines represent performance achievable by non-optimal strategies based on results from Ref. PhysRevA.102.032618 and based on unambiguous discrimination, respectively. The blue circles on the lines mark the transition between small/large $\alpha$ regime of the success probability.
• Figure 5: Maximum achievable quantum process fidelity $F_e$ in the approximate deterministic retrieval as a function of the angle of the program states $\beta$ and angle $\alpha$ representing one fourth of the angular spread of the eigenvalues of the relative unitary $U_1^{\dag } U_0$ for unitary transformations $U_0$, $U_1$.

Theorems & Definitions (14)

• Proposition 1: Optimal storage
• Lemma 1: Symmetric retrieval is optimal
• proof
• Lemma 2: Block diagonal retrieval
• proof
• Proposition 2
• proof
• Proposition 3
• Lemma 3
• proof