We address the question of quantum memory storage for quantum dynamics. In particular, we design an optimal protocol for N→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.
We address the question of quantum memory storage for quantum dynamics. In particular, we design an optimal protocol for N→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.
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. Sedl\'ak, 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 $N\ensuremath{\rightarrow}1$ 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 $({2}^{k}\ensuremath{-}1)\ensuremath{\rightarrow}1$ optimal PSR of phase gates and requires only k controlled-not gates, while having exponentially small failure probability in k.
Storing an unknown quantum computation in a quantum state and retrieving it at a desired later time is a challenging task, hindered by the no-programming theorem of quantum computations. In the previous studies on the task of probabilistic storage-and-retrieval (pSAR) of quantum channels, the maximum probability of exactly retrieving a single unknown unitary channel from a quantum state in which the unknown unitary has been encoded via multiple calls to the unknown unitary channel is derived. In this work, we consider a higher-order version of pSAR, the probabilistic storage-and-retrieval of definite-causal unitary superchannels, which are physically modeled by sequences of unitary channels with open slots where arbitrary channels can be inserted between the unitary channels for intervention. This task requires activating the "retrospective'' intervention functionality on the superchannel, beyond its normal intervention functionality. We propose two protocols: partial teleportation, which is supported by our numerical results for the small queries, and staircase backstitch, which achieves unit success probability asymptotically as the number of queries increases. We also derive a universal inversion protocol for unitary superchannels.
We study the persistent quantum and classical memory required for exact heralded storage and retrieval of quantum measurements. Given only black-box access to the measurement channel, we construct an exact compression of the port-based program into an outcome-type record and a quantum register, with total dimension polynomial in the query count. For every universal retrieval processor we prove the bounds Q+C≥log2n and 2Q+C≥log2((n−1)d2+1) on the numbers Q of qubits and C of bits of the program, for n-outcome POVMs on Cd with d≥2. For n≥2, no finite classical memory removes the need for quantum memory: any such processor requires a quantum program of dimension at least d, at that dimension its classical alphabet must have at least n symbols, and both minima are attained by a single-query protocol with success 1/d2. Combining direct-program compression with a converse from approximate measurement programming gives logarithmic total-memory scaling in 1/(1−p) as the uniform success probability p approaches one, at fixed d,n≥2. At fixed input dimension and target success below one, the optimal total-memory scaling is logarithmic in n. We also construct self-contained programs under branch-decomposition access and compare the quantum and classical costs of all constructions, including where the outcome-type record is kept: for POVMs with n nonzero rank-one effects the direct program admits a purely quantum allocation whose register is never larger than that of the minimal-Naimark unitary program and which succeeds more often whenever n>d.
We address the 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 − 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 nonoptimal measure-and-prepare strategy, highlighting the advantages of our protocol.
Storage and retrieval refer to the task of encoding an unknown quantum channel Λ into a quantum state, known as the program state, such that the channel can later be retrieved. There are two strategies for this task: classical and quantum strategies. The classical strategy uses multiple queries to Λ to estimate Λ and retrieves the channel based on the estimate represented in classical bits. The classical strategy turns out to offer the optimal performance for the storage and retrieval of unitary channels. In this work, we analyze the asymptotic performance of the classical and quantum strategies for the storage and retrieval of isometry channels. We show that the optimal fidelity for isometry estimation is given by F=1-[d(D-d)/n]+O(n^{-2}), where d and D denote the input and output dimensions of the isometry, and n is the number of queries. This result indicates that, unlike in the case of unitary channels, the classical strategy is suboptimal for the storage and retrieval of isometry channels, which requires n=Θ(ε^{-1}) to achieve the diamond-norm error ε. We propose a more efficient quantum strategy based on port-based teleportation, which stores the isometry channel in a program state using only n=Θ(1/sqrt[ε]) queries, achieving a quadratic improvement over the classical strategy. As an application, we extend our approach to general quantum channels, achieving improved program cost compared to prior results by Gschwendtner, Bluhm, and Winter [Quantum 5, 488 (2021)2521-327X10.22331/q-2021-06-29-488].
We address the problem of quantum reading of optical memories, namely the retrieving of classical information stored in the optical properties of a media with minimum energy. We present optimal strategies for ambiguous and unambiguous quantum reading of unitary optical memories, namely when one's task is to minimize the probability of errors in the retrieved information and when perfect retrieving of information is achieved probabilistically, respectively. A comparison of the optimal strategy with coherent probes and homodyne detection shows that the former saves orders of magnitude of energy when achieving the same performances. Experimental proposals for quantum reading which are feasible with present quantum optical technology are reported.