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  • Letter
  • Open Access

Quantum query complexity of Boolean functions under indefinite causal order

Alastair A. Abbott1, Mehdi Mhalla2, and Pierre Pocreau1,2

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Phys. Rev. Research 6, L032020 – Published 26 July, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L032020

Abstract

The standard model of quantum circuits assumes operations are applied in a fixed sequential “causal” order. In recent years, the possibility of relaxing this constraint to obtain causally indefinite computations has received significant attention. The quantum switch, for example, uses a quantum system to coherently control the order of operations. Several ad hoc computational and information-theoretical advantages have been demonstrated, raising questions as to whether advantages can be obtained in a more unified complexity theoretic framework. In this paper, we approach this problem by studying the query complexity of Boolean functions under general higher-order quantum computations. To this end, we generalize the framework of query complexity from quantum circuits to quantum supermaps to compare different models on an equal footing. We show that the recently introduced class of quantum circuits with quantum control of causal order cannot lead to any reduction in query complexity, and that any potential advantage arising from causally indefinite supermaps can be bounded by the polynomial method, as is the case with quantum circuits. Nevertheless, we find some functions for which the minimum error with which they can be computed using two queries is strictly lower when exploiting causally indefinite supermaps.

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Phys. Rev. Research 6, L032020– Published 26 July, 2024
  • Received 26 October 2023
  • Revised 13 February 2024
  • Accepted 1 July 2024
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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