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

25+12 pages, 8 figures. Some parts of the text were improved, some typos were corrected. Accepted in Quantum. Code available at https://github.com/mtcq/deterministic_unitary_transformation

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• Reversing Unknown Quantum Processes via Virtual Combs for Channels with Limited Information
• Optimal universal quantum circuits for unitary complex conjugation
• Universal construction of decoders from encoding black boxes
• Strategies for single-shot discrimination of process matrices
• Linear programming with unitary-equivariant constraints
• Quantum Query Complexity of Boolean Functions under Indefinite Causal Order
• Universal adjointation of isometry operations using conversion of quantum supermaps
• Simulating the quantum switch with quantum circuits is computationally hard
• Parameterized quantum comb and simpler circuits for reversing unknown qubit-unitary operations
• Characterising transformations between quantum objects, 'completeness' of quantum properties, and transformations without a fixed causal order
• On the distinguishability of geometrically uniform quantum states
• A resource theory of asynchronous quantum information processing
• Analytical Lower Bound on Query Complexity for Transformations of Unknown Unitary Operations
• No quantum advantage for violating fixed-order inequalities?