Given a quantum gate implementing a d-dimensional unitary operation U d , 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 U −1 d , 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 U −1 d 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 paper addresses the problem of designing universal quantum circuits to transform k uses of a d-dimensional unitary input-operation into a unitary output-operation in a probabilistic heralded manner. Three classes of protocols are considered, parallel circuits, where the input-operations can be simultaneously, adaptive circuits, where sequential uses of the input-operations are allowed, and general protocols, where the use of the input-operations may be performed without a definite causal order. For these three classes, we develop a systematic semidefinite programming approach that finds a circuit which obtains the desired transformation with the maximal success probability. We then analyse in detail three particular transformations; unitary transposition, unitary complex conjugation, and unitary inversion. For unitary transposition and unitary inverse, we prove that for any fixed dimension d, adaptive circuits have an exponential improvement in terms of uses k when compared to parallel ones. For unitary complex conjugation and unitary inversion we prove that if the number of uses k is strictly smaller than d − 1, the probability of success is necessarily zero. We also discuss the advantage of indefinite causal order protocols over causal ones and introduce the concept of delayed input-state quantum circuits.

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Valid transformations between quantum states are necessarily described by completely positive maps, instead of just positive maps. Positive but not completely positive maps such as the transposition map cannot be implemented due to the existence of entanglement in composite quantum systems, but there are classes of states for which the positivity is guaranteed, e.g., states not correlated to other systems. In this paper, we introduce the concept of N -copy extension of maps to quantitatively analyze the difference between positive maps and completely positive maps. We consider implementations of the action of positive but not completely positive maps on uncorrelated states by allowing an extra resource of consuming multiple copies of the input state and characterize the positive maps in terms of implementability with multiple copies. We show that by consuming multiple copies, the set of implementable positive maps becomes larger, and almost all positive maps are implementable with finite copies of an input state. The number of copies of the input state required to implement a positive map quantifies the degree by which a positive map violates complete positivity. We then analyze the optimal N -copy implementability of a noisy version of the transposition map. arXiv:1808.05788v2 [quant-ph]

The quantum switch is a physical process that creates a coherent control between different unitary operations which is often described as a process which transforms a pair of unitary operations (U 1 , U 2 ) into a controlled unitary operation that coherently applies them in different orders as |0 0|⊗U 1 U 2 +|1 1|⊗U 2 U 1 . This description, however, does not directly define its action on non-unitary operations. The action of quantum switch on non-unitary operations is then chosen to be a "natural" extension of its action on unitary operation. Since, in general, the action of a process on non-unitary operations is not uniquely determined by its action on only unitary operations, in principle, there could be a set of inequivalent extensions of quantum switch for non-unitary operations. In this paper, we prove that there is a unique way to extend the actions of quantum switch to non-unitary operations. In other words, contrary to the general case, the action of quantum switch on non-unitary operations is completely determined by its action on unitary operations. We also discuss the general problem of when the complete description of a quantum process is uniquely determined by its action on unitary operations and identify a set of singleslot processes which are completely defined by their action on unitary operations.