Probabilistic exact universal quantum circuits for transforming unitary operations

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 performed 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 analyze 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.