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

Marco Túlio Quintino; Daniel Ebler

doi:10.22331/q-2022-03-31-679

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

Marco Túlio Quintino^1,2 and Daniel Ebler^3,4

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

Published:	2022-03-31, volume 6, page 679
Eprint:	arXiv:2109.08202v3
Doi:	https://doi.org/10.22331/q-2022-03-31-679
Citation:	Quantum 6, 679 (2022).

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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.

Featured image: 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 $f$.

Popular summary

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.

► BibTeX data

@article{Quintino2022deterministic,
  doi = {10.22331/q-2022-03-31-679},
  url = {https://doi.org/10.22331/q-2022-03-31-679},
  title = {Deterministic transformations between unitary operations: {E}xponential advantage with adaptive quantum circuits and the power of indefinite causality},
  author = {Quintino, Marco T{\'{u}}lio and Ebler, Daniel},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {679},
  month = mar,
  year = {2022}
}

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[2] Huan-Yu Ku, Kuan-Yi Lee, Po-Rong Lai, Jhen-Dong Lin, and Yueh-Nan Chen, "Coherent activation of a steerability-breaking channel", Physical Review A 107 4, 042415 (2023).

[3] Satoshi Yoshida, Akihito Soeda, and Mio Murao, "Reversing Unknown Qubit-Unitary Operation, Deterministically and Exactly", Physical Review Letters 131 12, 120602 (2023).

[4] Teodor Strömberg, Peter Schiansky, Marco Túlio Quintino, Michael Antesberger, Lee A. Rozema, Iris Agresti, Časlav Brukner, and Philip Walther, "Experimental superposition of a quantum evolution with its time reverse", Physical Review Research 6 2, 023071 (2024).

[5] Matheus Capela, Harshit Verma, Fabio Costa, and Lucas C. Céleri, "Reassessing thermodynamic advantage from indefinite causal order", Physical Review A 107 6, 062208 (2023).

[6] Daniel Ebler, Michał Horodecki, Marcin Marciniak, Tomasz Młynik, Marco Túlio Quintino, and Michał Studziński, "Optimal Universal Quantum Circuits for Unitary Complex Conjugation", IEEE Transactions on Information Theory 69 8, 5069 (2023).

[7] Satoshi Yoshida, Akihito Soeda, and Mio Murao, "Universal construction of decoders from encoding black boxes", Quantum 7, 957 (2023).

[8] Dmitry Grinko, Adam Burchardt, and Maris Ozols, "Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information", arXiv:2310.02252, (2023).

[9] Simon Milz and Marco Túlio Quintino, "Transformations between arbitrary (quantum) objects and the emergence of indefinite causality", arXiv:2305.01247, (2023).

[10] Alastair A. Abbott, Mehdi Mhalla, and Pierre Pocreau, "Quantum Query Complexity of Boolean Functions under Indefinite Causal Order", arXiv:2307.10285, (2023).

[11] Daniel Ebler, Michał Horodecki, Marcin Marciniak, Tomasz Młynik, Marco Túlio Quintino, and Michał Studziński, "Optimal universal quantum circuits for unitary complex conjugation", arXiv:2206.00107, (2022).

[12] Yu Yang, Shihao Ru, Min An, Yunlong Wang, Feiran Wang, Pei Zhang, and Fuli Li, "Multiparameter simultaneous optimal estimation with an SU(2) coding unitary evolution", Physical Review A 105 2, 022406 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-12 12:08:17) and SAO/NASA ADS (last updated successfully 2024-05-12 12:08:18). The list may be incomplete as not all publishers provide suitable and complete citation data.

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