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

Marco Túlio Quintino1,2 and Daniel Ebler3,4

1Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
3Huawei Hong Kong Research Center, Hong Kong SAR, P. R. China
4Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong

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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., $f(UV)=f(U)f(V)$, 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., $f(UV)=f(V)f(U)$. In 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.

In this work, we design quantum circuits that can be used to perform desired transformations on quantum gates (unitary quantum operations). We focus on transformations which preserves the notion of compostability. That is, if A and B represent two quantum gates, we consider functions f such that the composition f(A)f(B) is equivalent to f(AB) or f(BA). We then identify cases where the circuits may parallelised with no loss in performance and cases where parallelisation necessary implies a dramatic loss in performance. We also analyse the power and limitations of quantum processes which may be used to transform operations but do no respect any definite causal order, hence, it may not be viewed as ordinary quantum circuits.

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