Consistent circuits for indefinite causal order

Augustin Vanrietvelde; Nick Ormrod; Hlér Kristjánsson; Jonathan Barrett

doi:10.22331/q-2025-12-02-1923

Consistent circuits for indefinite causal order

Augustin Vanrietvelde^1,2,3, Nick Ormrod^1,3, Hlér Kristjánsson^1,3, and Jonathan Barrett¹

¹Quantum Group, Department of Computer Science, University of Oxford
²Department of Physics, Imperial College London
³HKU-Oxford Joint Laboratory for Quantum Information and Computation

Published:	2025-12-02, volume 9, page 1923
Eprint:	arXiv:2206.10042v3
Doi:	https://doi.org/10.22331/q-2025-12-02-1923
Citation:	Quantum 9, 1923 (2025).

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Abstract

Over the past decade, a number of quantum processes have been proposed which are logically consistent, yet feature a cyclic causal structure. However, there is no general formal method to construct a process with an exotic causal structure in a way that ensures, and makes clear why, it is consistent. Here we provide such a method, given by an extended circuit formalism. This only requires directed graphs endowed with Boolean matrices, which encode basic constraints on operations. Our framework (a) defines a set of elementary rules for checking the validity of any such graph, (b) provides a way of constructing consistent processes as a circuit from valid graphs, and (c) yields an intuitive interpretation of the causal relations within a process and an explanation of why they do not lead to inconsistencies. We display how several standard examples of exotic processes, including ones that violate causal inequalities, are among the class of processes that can be generated in this way; we conjecture that this class in fact includes all unitarily extendible processes.

Featured image: The routed graphs corresponding to the quantum switch and to the Lugano process.

► BibTeX data

@article{Vanrietvelde2025consistentcircuits,
  doi = {10.22331/q-2025-12-02-1923},
  url = {https://doi.org/10.22331/q-2025-12-02-1923},
  title = {Consistent circuits for indefinite causal order},
  author = {Vanrietvelde, Augustin and Ormrod, Nick and Kristj{\'{a}}nsson, Hl{\'{e}}r and Barrett, Jonathan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1923},
  month = dec,
  year = {2025}
}

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Cited by

[1] Anna Steffinlongo and Hippolyte Dourdent, "Simulating noncausality with quantum control of causal orders", Physical Review Research 8 1, 013127 (2026).

[2] Anna Jenčová, "On the structure of higher order quantum maps", Quantum 10, 2090 (2026).

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[4] Nick Ormrod, Augustin Vanrietvelde, and Jonathan Barrett, "Causal structure in the presence of sectorial constraints, with application to the quantum switch", Quantum 7, 1028 (2023).

[5] Simon Milz and Marco Túlio Quintino, "Characterising transformations between quantum objects, 'completeness' of quantum properties, and transformations without a fixed causal order", Quantum 8, 1415 (2024).

[6] James Hefford and Matt Wilson, "A BV-Category of Spacetime Interventions", arXiv:2502.19022, (2025).

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[9] Eleftherios-Ermis Tselentis and ńmin Baumeler, "Admissible Causal Structures and Correlations", PRX Quantum 4 4, 040307 (2023).

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[12] Matt Wilson and Giulio Chiribella, "Polycategorical Constructions for Unitary Supermaps of Arbitrary Dimension", arXiv:2207.09180, (2022).

[13] Augustin Vanrietvelde, Octave Mestoudjian, and Pablo Arrighi, "Partitions in quantum theory", arXiv:2506.22218, (2025).

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[19] Pablo Arrighi, Amélia Durbec, and Matt Wilson, "Generalised tensors and traces", arXiv:2202.11340, (2022).

[20] Matt Wilson and James Hefford, "Higher-Order Quantum Objects are Strong Profunctors", arXiv:2603.11221, (2026).

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The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 22:49:45) and SAO/NASA ADS (last updated successfully 2026-08-13 22:49:48). The list may be incomplete as not all publishers provide suitable and complete citation data.

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