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

TL;DR

This paper demonstrates that for anti-homomorphic functions like inversion and transposition, adaptive and indefinite causal quantum circuits can exponentially outperform parallel circuits, revealing new advantages in quantum processing.

Contribution

It shows that anti-homomorphisms enable exponential advantages for adaptive and indefinite causal quantum circuits, unlike homomorphisms where parallel circuits are optimal.

Findings

Sequential circuits outperform parallel ones exponentially for anti-homomorphisms.

Processes with indefinite causal order outperform sequential circuits.

Explicit constructions for unitary inversion and transposition tasks demonstrate these advantages.

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.