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 optimal for a small number of storage 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 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.
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(poly(n,logdI,logdO)), where
n is the number of queries and
dI and
dO 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(poly(dI)) 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(poly(min{dI,dO})) 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
ε. For the case where the Kraus rank
r is the least possible (i.e.,
r=dI/dO), we show quantum circuits that transform
n parallel queries of an unknown quantum channel
Λ to
Θ(nα) parallel queries of
Λ for any
α<2 approximately, and implement its Petz recovery map for the maximally mixed reference state probabilistically and exactly.
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.
Given a quantum gate implementing a
d-dimensional unitary operation
Ud, without any specific description but
d, and permitted to use
k times, we present a universal probabilistic heralded quantum circuit that implements the exact inverse
Ud−1, whose failure probability decays, exponentially in
k. The protocol employs an adaptive strategy, proven necessary for the exponential performance. It requires
k≥d−1, proven necessary for exact implementation of
Ud−1 with quantum circuits. Moreover, even when quantum circuits with indefinite causal order are allowed,
k≥d−1 uses are required. We then present a finite set of linear and positive semidefinite constraints characterizing universal unitary inversion protocols and formulate a convex optimization problem whose solution is the maximum success probability for given
k and
d. The optimal values are computed using semidefinite programming solvers for
k≤3 when
d=2 and
k≤2 for
d=3. With this numerical approach we show for the first time that indefinite causal order circuits provide an advantage over causally ordered ones in a task involving multiple uses of the same unitary operation.
This work considers a teleportation task for Alice and Bob in a scenario where Bob cannot perform corrections. In particular, we analyse the task of multicopy state teleportation , where Alice has k identical copies of an arbitrary unknown d -dimensional qudit state | ψ ⟩ to teleport a single copy of | ψ ⟩ to Bob using a maximally entangled two-qudit state shared between Alice and Bob without Bob's correction. Alice may perform a joint measurement on her half of the entangled state and the k copies of | ψ ⟩ . We prove that the maximal probability of success for teleporting the exact state | ψ ⟩ to Bob is p ( d , k ) = k d ( k − 1 + d ) and present an explicit protocol to attain this performance. Then, by utilising k copies of an arbitrary target state | ψ ⟩ , we show how the multicopy state teleportation protocol can be employed to enhance the success probability of storage and retrieval of quantum programs, which aims to universally retrieve the action of an arbitrary quantum channel that is stored in a state. Our proofs make use of group representation theory methods, which may find applications beyond the problems addressed in this work.