quantum-computing

A decompositional framework for process theories in spacetime

Matthias Salzger and John H. Selby

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Matthias Salzger and John H. Selby propose a framework integrating spacetime structure into quantum process theories, addressing a critical gap where quantum circuits traditionally ignore physical qubit connectivity and causal constraints. The study introduces "process implementations"—decompositions of quantum processes—and defines embeddability as requiring localized components with wires following timelike paths, though verifying this remains computationally intractable in general cases. Researchers identify a simplified subset of implementations to determine embeddability and distinguishable causal structures, reducing complexity while preserving theoretical rigor for quantum-gravity-aligned process theories. A key discovery reveals countably infinite "zigzag" causal structures, irrelevant in classical physics but essential for quantum operations like the CNOT gate, which only functions in these non-standard configurations. The findings suggest zigzag structures may enable novel quantum resources and advance causal modeling, bridging quantum information with general relativity’s spacetime constraints.

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AbstractThere has been a recent surge of interest within the field of quantum foundations regarding incorporating ideas from general relativity and quantum gravity. However, many quantum information tools remain agnostic to the underlying spacetime. For instance, whenever we draw a quantum circuit the effective spacetime imposed by the connectivity of the physical qubits which will realize this circuit is not taken into account. In this work, we aim to address this limitation by extending the framework of process theories to include a background spacetime structure. We introduce the notion of process implementations, i.e., decompositions of a process. A process is then embeddable if and only if one of its implementations can be embedded in such a way that all the component processes are localized and all wires follow timelike paths. While conceptually simple, checking for embeddability is generally computationally intractable. We therefore work towards simplifying this problem as much as possible, identifying a canonical subset of implementations that determine both the embeddability of a process and the causal structures distinguishable at least in some process theory. Notably, we discover countably infinite ''zigzag'' causal structures beyond those typically considered. While these can be ignored in classical theory, they seem to be essential in quantum theory, as the quantum CNOT gate can be implemented by all zigzag structures but not in a standard causal structure, except in the trivial undecomposed way. These zigzags could be significant for quantum causal modeling and the study of novel quantum resources.Featured image: The range of inequivalent decompositional structure for processes with two inputs and two outputs► BibTeX data@article{Salzger2026decompositional, doi = {10.22331/q-2026-01-07-1959}, url = {https://doi.org/10.22331/q-2026-01-07-1959}, title = {A decompositional framework for process theories in spacetime}, author = {Salzger, Matthias and Selby, John H.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1959}, month = jan, year = {2026} }► References [1] Nicola Pinzani, Stefano Gogioso, and Bob Coecke, 34th Annual ACM/IEEE Symposium On Logic In Computer Science (LICS), pp. 1-20 (2019), 10.1109/LICS.2019.8785670. https://doi.org/10.1109/LICS.2019.8785670 [2] V. Vilasini, Christopher Portmann, and Lídia del Rio, New Journal Of Physics, 21, 043057 (2019), 10.1088/1367-2630/ab0e3b. https://doi.org/10.1088/1367-2630/ab0e3b [3] Sandu Popescu, and Daniel Rohrlich, Foundations Of Physics, 24 pp. 379-385 (1994), 10.1007/BF02058098. https://doi.org/10.1007/BF02058098 [4] Christopher Portmann, Christian Matt, Ueli Maurer, Renato Renner, and Björn Tackmann, IEEE Transactions On Information Theory, 63, 3277-3305 (2017), 10.1109/TIT.2017.2675482. https://doi.org/10.1109/TIT.2017.2675482 [5] David B Malament, Journal Of Mathematical Physics, 18, 1399-1404 (1977), https://doi.org/10.1063/1.523436. https://doi.org/10.1063/1.523436 [6] Stefano Gogioso, Maria E Stasinou, and Bob Coecke, Frontiers In Physics, 9, pp. 534265 (2021), 10.3389/fphy.2021.534265. https://doi.org/10.3389/fphy.2021.534265 [7] G. M. D’Ariano, Franco Manessi, and Paolo Perinotti, Physica Scripta, 2014, 014013 (2014), 10.1088/0031-8949/2014/T163/014013. https://doi.org/10.1088/0031-8949/2014/T163/014013 [8] Adrian Kent, Physical Review Letters, 109, 130501 (2012), 10.1103/PhysRevLett.109.130501. https://doi.org/10.1103/PhysRevLett.109.130501 [9] Jedrzej Kaniewski, Marco Tomamichel, Esther Hanggi, and Stephanie Wehner, IEEE Transactions On Information Theory, 59, 4687-4699 (2013), 10.1109/TIT.2013.2247463. https://doi.org/10.1109/TIT.2013.2247463 [10] Tommaso Lunghi, Jedrzej Kaniewski, Felix Bussières, Raphael Houlmann, Marco Tomamichel, Stephanie Wehner, and Hugo Zbinden, Physical Review Letters, 115, 030502 (2015), 10.1103/PhysRevLett.115.030502. https://doi.org/10.1103/PhysRevLett.115.030502 [11] Emily Adlam, and Adrian Kent, International Journal Of Quantum Information, 13, 1550029 (2015), 10.1142/S021974991550029X. https://doi.org/10.1142/S021974991550029X [12] Harry Buhrman, Nishanth Chandran, Serge Fehr, Ran Gelles, Vipul Goyal, Rafail Ostrovsky, and Christian Schaffner, SIAM Journal On Computing, 43, 150-178 (2014), 10.1137/130913687. https://doi.org/10.1137/130913687 [13] John H Selby, Carlo Maria Scandolo, and Bob Coecke, Quantum, 5, pp. 445 (2021), 10.22331/q-2021-04-28-445. https://doi.org/10.22331/q-2021-04-28-445 [14] Dominique Unruh, 34th Annual Cryptology Conference, Santa Barbara, CA, USA, August 17–21, 2014, Proceedings, Part II.

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Quantum Information Processing-From Theory To Experiment, 199, pp. 137-158 (2006).Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-07 18:42:02: Could not fetch cited-by data for 10.22331/q-2026-01-07-1959 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-07 18:42:02: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractThere has been a recent surge of interest within the field of quantum foundations regarding incorporating ideas from general relativity and quantum gravity. However, many quantum information tools remain agnostic to the underlying spacetime. For instance, whenever we draw a quantum circuit the effective spacetime imposed by the connectivity of the physical qubits which will realize this circuit is not taken into account. In this work, we aim to address this limitation by extending the framework of process theories to include a background spacetime structure. We introduce the notion of process implementations, i.e., decompositions of a process. A process is then embeddable if and only if one of its implementations can be embedded in such a way that all the component processes are localized and all wires follow timelike paths. While conceptually simple, checking for embeddability is generally computationally intractable. We therefore work towards simplifying this problem as much as possible, identifying a canonical subset of implementations that determine both the embeddability of a process and the causal structures distinguishable at least in some process theory. Notably, we discover countably infinite ''zigzag'' causal structures beyond those typically considered. While these can be ignored in classical theory, they seem to be essential in quantum theory, as the quantum CNOT gate can be implemented by all zigzag structures but not in a standard causal structure, except in the trivial undecomposed way. These zigzags could be significant for quantum causal modeling and the study of novel quantum resources.Featured image: The range of inequivalent decompositional structure for processes with two inputs and two outputs► BibTeX data@article{Salzger2026decompositional, doi = {10.22331/q-2026-01-07-1959}, url = {https://doi.org/10.22331/q-2026-01-07-1959}, title = {A decompositional framework for process theories in spacetime}, author = {Salzger, Matthias and Selby, John H.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1959}, month = jan, year = {2026} }► References [1] Nicola Pinzani, Stefano Gogioso, and Bob Coecke, 34th Annual ACM/IEEE Symposium On Logic In Computer Science (LICS), pp. 1-20 (2019), 10.1109/LICS.2019.8785670. https://doi.org/10.1109/LICS.2019.8785670 [2] V. Vilasini, Christopher Portmann, and Lídia del Rio, New Journal Of Physics, 21, 043057 (2019), 10.1088/1367-2630/ab0e3b. https://doi.org/10.1088/1367-2630/ab0e3b [3] Sandu Popescu, and Daniel Rohrlich, Foundations Of Physics, 24 pp. 379-385 (1994), 10.1007/BF02058098. https://doi.org/10.1007/BF02058098 [4] Christopher Portmann, Christian Matt, Ueli Maurer, Renato Renner, and Björn Tackmann, IEEE Transactions On Information Theory, 63, 3277-3305 (2017), 10.1109/TIT.2017.2675482. https://doi.org/10.1109/TIT.2017.2675482 [5] David B Malament, Journal Of Mathematical Physics, 18, 1399-1404 (1977), https://doi.org/10.1063/1.523436. https://doi.org/10.1063/1.523436 [6] Stefano Gogioso, Maria E Stasinou, and Bob Coecke, Frontiers In Physics, 9, pp. 534265 (2021), 10.3389/fphy.2021.534265. https://doi.org/10.3389/fphy.2021.534265 [7] G. M. D’Ariano, Franco Manessi, and Paolo Perinotti, Physica Scripta, 2014, 014013 (2014), 10.1088/0031-8949/2014/T163/014013. https://doi.org/10.1088/0031-8949/2014/T163/014013 [8] Adrian Kent, Physical Review Letters, 109, 130501 (2012), 10.1103/PhysRevLett.109.130501. https://doi.org/10.1103/PhysRevLett.109.130501 [9] Jedrzej Kaniewski, Marco Tomamichel, Esther Hanggi, and Stephanie Wehner, IEEE Transactions On Information Theory, 59, 4687-4699 (2013), 10.1109/TIT.2013.2247463. https://doi.org/10.1109/TIT.2013.2247463 [10] Tommaso Lunghi, Jedrzej Kaniewski, Felix Bussières, Raphael Houlmann, Marco Tomamichel, Stephanie Wehner, and Hugo Zbinden, Physical Review Letters, 115, 030502 (2015), 10.1103/PhysRevLett.115.030502. https://doi.org/10.1103/PhysRevLett.115.030502 [11] Emily Adlam, and Adrian Kent, International Journal Of Quantum Information, 13, 1550029 (2015), 10.1142/S021974991550029X. https://doi.org/10.1142/S021974991550029X [12] Harry Buhrman, Nishanth Chandran, Serge Fehr, Ran Gelles, Vipul Goyal, Rafail Ostrovsky, and Christian Schaffner, SIAM Journal On Computing, 43, 150-178 (2014), 10.1137/130913687. https://doi.org/10.1137/130913687 [13] John H Selby, Carlo Maria Scandolo, and Bob Coecke, Quantum, 5, pp. 445 (2021), 10.22331/q-2021-04-28-445. https://doi.org/10.22331/q-2021-04-28-445 [14] Dominique Unruh, 34th Annual Cryptology Conference, Santa Barbara, CA, USA, August 17–21, 2014, Proceedings, Part II.

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A decompositional framework for process theories in spacetime

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