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On the structure of higher order quantum maps

Quantum Science and Technology (arXiv overlay)

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Anna Jenčová’s May 2026 study introduces a framework linking higher-order quantum maps to Boolean functions, called "type functions," within -autonomous categories of affine subspaces. The work reveals that these type functions inherit algebraic structures from Boolean logic, enabling systematic classification of complex quantum operations through combinatorial methods. Using the Möbius transform, the paper maps each type function to a partially ordered set (poset) labeled by subspace indices, establishing a direct correspondence between quantum processes and combinatorial structures. A key finding shows that "comb-type" quantum maps—critical for sequential operations—align with posets forming linear chains, simplifying their characterization and potential implementation. The study also proposes a decomposition method, breaking posets into basic chains whose maxima (affine mixtures) and minima (intersections) reconstruct the original type function, offering a toolkit for quantum algorithm design.

On the structure of higher order quantum maps

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AbstractWe study higher order quantum maps in the context of a *-autonomous category of affine subspaces. We show that types of higher order maps can be identified with certain Boolean functions that we call type functions. By an extension of this identification, the algebraic structure of Boolean functions is inherited by some sets of quantum objects including higher order maps. Using the Mobius transform, we assign to each type function a poset whose elements are labelled by subsets of indices of the involved spaces. We then show that the type function corresponds to a comb type if and only if the poset is a chain. We also devise a procedure for decomposition of the poset to a set of basic chains from which the type function is constructed by taking maxima and minima of concatenations of the basic chains in different orders. On the level of higher order maps, maxima and minima correspond to affine mixtures and intersections, respectively.► BibTeX data@article{Jencova2026structureofhigher, doi = {10.22331/q-2026-05-05-2090}, url = {https://doi.org/10.22331/q-2026-05-05-2090}, title = {On the structure of higher order quantum maps}, author = {Jen{\v{c}}ov{\'{a}}, Anna}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2090}, month = may, year = {2026} }► References [1] Luca Apadula, Alessandro Bisio, and Paolo Perinotti. ``No-signalling constrains quantum computation with indefinite causal structure''. Quantum 8, 1241 (2024). arXiv:2202.10214. https://doi.org/10.22331/q-2024-02-05-1241 arXiv:2202.10214 [2] G. Chiribella, G. M. D’Ariano, and P. Perinotti. ``Transforming quantum operations: Quantum supermaps''. EPL (Europhysics Letters) 83, 30004 (2008). arXiv:0804.0180. https://doi.org/10.1209/0295-5075/83/30004 arXiv:0804.0180 [3] G. Chiribella, G. M. D'Ariano, and P. Perinotti. ``Quantum circuit architecture''. Phys. Rev. Lett. 101, 060401 (2008). arXiv:0712.1325. https://doi.org/10.1103/PhysRevLett.101.060401 arXiv:0712.1325 [4] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Theoretical framework for quantum networks''. Phys. Rev. A 80, 022339 (2009). arXiv:0904.4483. https://doi.org/10.1103/PhysRevA.80.022339 arXiv:0904.4483 [5] Gus Gutoski and John Watrous. ``Toward a general theory of quantum games''. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing. Page 565–574. STOC07. ACM (2007). arXiv:quant-ph/0611234. https://doi.org/10.1145/1250790.1250873 arXiv:quant-ph/0611234 [6] Giulio Chiribella, Giacomo M. D'Ariano, and Paolo Perinotti. ``Memory effects in quantum channel discrimination''. Phys. Rev. Lett. 101, 180501 (2008). arXiv:0803.3237. https://doi.org/10.1103/PhysRevLett.101.180501 arXiv:0803.3237 [7] Mário Ziman. ``Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments''. Phys. Rev. A 77, 062112 (2008). arXiv:0802.3862. https://doi.org/10.1103/PhysRevA.77.062112 arXiv:0802.3862 [8] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. ``Quantum correlations with no causal order''. Nature Communications 3, 1092 (2012). arXiv:1105.4464v3. https://doi.org/10.1038/ncomms2076 arXiv:1105.4464v3 [9] Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoit Valiron. ``Quantum computations without definite causal structure''. Physical Review A 88, 022318 (2013). arXiv:0912.0195. https://doi.org/10.1103/PhysRevA.88.022318 arXiv:0912.0195 [10] Paolo Perinotti. ``Causal structures and the classification of higher order quantum computations''.

In Renato Renner and Sandra Stupar, editors, Time in Physics. Page 103–127.

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Cambridge University Press. (2011). 2nd edition. https://doi.org/10.1017/CBO9781139058520Cited by[1] James Hefford and Matt Wilson, "A BV-Category of Spacetime Interventions", arXiv:2502.19022, (2025). [2] Anna Jenčová, "Order structure and signalling in higher order quantum maps", arXiv:2604.09192, (2026). [3] Matt Wilson, James Hefford, and Timothée Hoffreumon, "Supermaps on generalised theories", arXiv:2602.23865, (2026). [4] Matt Wilson and James Hefford, "Higher-Order Quantum Objects are Strong Profunctors", arXiv:2603.11221, (2026). [5] Matt Wilson, "Agent policies from higher-order causal functions", arXiv:2512.10937, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-05-05 09:17:56). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-05-05 09:17:54: Could not fetch cited-by data for 10.22331/q-2026-05-05-2090 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe study higher order quantum maps in the context of a *-autonomous category of affine subspaces. We show that types of higher order maps can be identified with certain Boolean functions that we call type functions. By an extension of this identification, the algebraic structure of Boolean functions is inherited by some sets of quantum objects including higher order maps. Using the Mobius transform, we assign to each type function a poset whose elements are labelled by subsets of indices of the involved spaces. We then show that the type function corresponds to a comb type if and only if the poset is a chain. We also devise a procedure for decomposition of the poset to a set of basic chains from which the type function is constructed by taking maxima and minima of concatenations of the basic chains in different orders. On the level of higher order maps, maxima and minima correspond to affine mixtures and intersections, respectively.► BibTeX data@article{Jencova2026structureofhigher, doi = {10.22331/q-2026-05-05-2090}, url = {https://doi.org/10.22331/q-2026-05-05-2090}, title = {On the structure of higher order quantum maps}, author = {Jen{\v{c}}ov{\'{a}}, Anna}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2090}, month = may, year = {2026} }► References [1] Luca Apadula, Alessandro Bisio, and Paolo Perinotti. ``No-signalling constrains quantum computation with indefinite causal structure''. Quantum 8, 1241 (2024). arXiv:2202.10214. https://doi.org/10.22331/q-2024-02-05-1241 arXiv:2202.10214 [2] G. Chiribella, G. M. D’Ariano, and P. Perinotti. ``Transforming quantum operations: Quantum supermaps''. EPL (Europhysics Letters) 83, 30004 (2008). arXiv:0804.0180. https://doi.org/10.1209/0295-5075/83/30004 arXiv:0804.0180 [3] G. Chiribella, G. M. D'Ariano, and P. Perinotti. ``Quantum circuit architecture''. Phys. Rev. Lett. 101, 060401 (2008). arXiv:0712.1325. https://doi.org/10.1103/PhysRevLett.101.060401 arXiv:0712.1325 [4] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Theoretical framework for quantum networks''. Phys. Rev. A 80, 022339 (2009). arXiv:0904.4483. https://doi.org/10.1103/PhysRevA.80.022339 arXiv:0904.4483 [5] Gus Gutoski and John Watrous. ``Toward a general theory of quantum games''. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing. Page 565–574. STOC07. ACM (2007). arXiv:quant-ph/0611234. https://doi.org/10.1145/1250790.1250873 arXiv:quant-ph/0611234 [6] Giulio Chiribella, Giacomo M. D'Ariano, and Paolo Perinotti. ``Memory effects in quantum channel discrimination''. Phys. Rev. Lett. 101, 180501 (2008). arXiv:0803.3237. https://doi.org/10.1103/PhysRevLett.101.180501 arXiv:0803.3237 [7] Mário Ziman. ``Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments''. Phys. Rev. A 77, 062112 (2008). arXiv:0802.3862. https://doi.org/10.1103/PhysRevA.77.062112 arXiv:0802.3862 [8] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. ``Quantum correlations with no causal order''. Nature Communications 3, 1092 (2012). arXiv:1105.4464v3. https://doi.org/10.1038/ncomms2076 arXiv:1105.4464v3 [9] Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoit Valiron. ``Quantum computations without definite causal structure''. Physical Review A 88, 022318 (2013). arXiv:0912.0195. https://doi.org/10.1103/PhysRevA.88.022318 arXiv:0912.0195 [10] Paolo Perinotti. ``Causal structures and the classification of higher order quantum computations''.

In Renato Renner and Sandra Stupar, editors, Time in Physics. Page 103–127.

Cambridge University Press. (2004). https://doi.org/10.1017/CBO9780511525896 [34] Richard P Stanley. ``Enumerative combinatorics, vol. 1''. Number 49 in Cambridge Studies in Advanced Mathematics.

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On the structure of higher order quantum maps

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