Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality

Abstract

This work analyses the performance of quantum circuits and general processes to transform k uses of an arbitrary unitary operation U into another unitary operation f (U ). When the desired function f a homomorphism, i.e., f (U V ) = f (U )f (V ), it is known that optimal average fidelity is attainable by parallel circuits and indefinite causality does not provide any advantage. Here we show that the situation changes dramatically when considering anti-homomorphisms, i.e., fIn particular, we prove that when f is an anti-homomorphism, sequential circuits could exponentially outperform parallel ones and processes with indefinite causal order could outperform sequential ones. We presented explicit constructions on how to obtain such advantages for the unitary inversion task f (U ) = U −1 and the unitary transposition task f (U ) = U T . We also stablish a one-to-one connection between the problem of unitary estimation and parallel unitary transposition, allowing one to easily translate results from one field to the other. Finally, we apply our results to several concrete problem instances and present a method based on computer-assisted proofs to show optimality. Figure 1: Three different strategies to transform k = 2 uses of an unknown unitary operation U into f (U ). a) parallel circuit, E and D stand for fixed operations (encoder and decoder); b) sequential circuit with multiple encoder operations; and c) general processes acting on U may not have a definite causal order. Here, we analyse the performance of these strategies for different functions f .

Multiparameter simultaneous optimal estimation with an SU(2) coding unitary evolution

“…The utilized sequential scheme even can provide a better estimation performance than the parallel scheme or the scheme of independently estimating each parameter [18]. The various differences among these three scenarios are also discussed in the configuration of quantum circuits [37] and quantum channel discrimination [38]. In this paper, we offer a different perspective for understanding the role of quantum control from the obtainable QFI result.…”

Universal construction of decoders from encoding black boxes

“…In particular, higher-order quantum transformations of unitary operations have been extensively studied for aiming to utilize in quantum information processing (e.g., estimation of group transformations [2], quantum learning of unitary operations [3][4][5][6], cloning of unitary operations [7][8][9][10][11][12], process tomography [13,14], unitary complex conjugation [15], unitary inversion [16][17][18][19] and unitary controllization [21][22][23]). Isometry operations are also frequently used in quantum protocols, as they preserve information of the input state similarly to unitary operations.…”