Consistent circuits for indefinite causal order
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AbstractOver 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} }► References [1] D. E. Deutsch, ``Quantum computational networks,'' Proceedings of the Royal Society of London: A. Mathematical and Physical Sciences 425 no. 1868, (1989) 73–90. https://doi.org/10.1098/rspa.1989.0099 [2] D. Aharonov, A. Kitaev, and N. Nisan, ``Quantum circuits with mixed states,'' in Proceedings of the thirtieth annual ACM symposium on Theory of computing, pp. 20–30. 1998. arXiv:quant-ph/9806029. https://doi.org/10.1145/276698.276708 arXiv:quant-ph/9806029 [3] S. Abramsky and B. Coecke, ``A categorical semantics of quantum protocols,'' in Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004., pp. 415–425, IEEE. 2004. arXiv:quant-ph/0402130. https://doi.org/10.1109/LICS.2004.1319636 arXiv:quant-ph/0402130 [4] M. A. Nielsen and I. L. Chuang, Quantum information and quantum computation.
Cambridge University Press, 2000. https://doi.org/10.1017/CBO9780511976667 [5] B. Coecke and A. Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning.
Cambridge University Press, 2017. https://doi.org/10.1017/9781316219317 [6] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Transforming quantum operations: Quantum supermaps,'' EPL (Europhysics Letters) 83 no. 3, (2008) 30004, arXiv:0804.0180 [quant-ph]. https://doi.org/10.1209/0295-5075/83/30004 arXiv:0804.0180 [7] L. Hardy, ``Probability theories with dynamic causal structure: A New framework for quantum gravity,'' arXiv:gr-qc/0509120. arXiv:gr-qc/0509120 [8] G. Chiribella, G. D’Ariano, P. Perinotti, and B. Valiron, ``Beyond quantum computers,'' arXiv:0912.0195v1 [quant-ph]. arXiv:0912.0195v1 [9] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure,'' Physical Review A 88 no. 2, (2013) 022318, arXiv:0912.0195 [quant-ph]. https://doi.org/10.1103/PhysRevA.88.022318 arXiv:0912.0195 [10] O. Oreshkov, F. Costa, and C. Brukner, ``Quantum correlations with no causal order,'' Nature Communications 3 (2012) 1092, arXiv:1105.4464 [quant-ph]. https://doi.org/10.1038/ncomms2076 arXiv:1105.4464 [11] Ä. Baumeler, A. Feix, and S. Wolf, ``Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios,'' Physical Review A 90 no. 4, (2014) 042106, arXiv:1403.7333 [quant-ph]. https://doi.org/10.1103/PhysRevA.90.042106 arXiv:1403.7333 [12] Ä. Baumeler and S. Wolf, ``The space of logically consistent classical processes without causal order,'' New Journal of Physics 18 no. 1, (2016) 013036, arXiv:1507.01714 [quant-ph]. https://doi.org/10.1088/1367-2630/18/1/013036 arXiv:1507.01714 [13] J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, ``Quantum circuits with classical versus quantum control of causal order,'' PRX Quantum 2 no. 3, (2021) 030335, arXiv:2101.08796 [quant-ph]. https://doi.org/10.1103/PRXQuantum.2.030335 arXiv:2101.08796 [14] T. Colnaghi, G. M. D'Ariano, S. Facchini, and P. Perinotti, ``Quantum computation with programmable connections between gates,'' Physics Letters A 376 no. 45, (2012) 2940–2943, arXiv:1109.5987 [quant-ph]. https://doi.org/10.1016/j.physleta.2012.08.028 arXiv:1109.5987 [15] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, ``A purification postulate for quantum mechanics with indefinite causal order,'' Quantum 1 (2017) 10, arXiv:1611.08535 [quant-ph]. https://doi.org/10.22331/q-2017-04-26-10 arXiv:1611.08535 [16] A. Vanrietvelde, H. Kristjánsson, and J. Barrett, ``Routed quantum circuits,'' Quantum 5 (Jul, 2021) 503, arXiv:2011.08120 [quant-ph]. https://doi.org/10.22331/q-2021-07-13-503 arXiv:2011.08120 [17] J. Barrett, R. Lorenz, and O. Oreshkov, ``Quantum causal models,'' arXiv:1906.10726 [quant-ph]. arXiv:1906.10726 [18] R. Lorenz and J. Barrett, ``Causal and compositional structure of unitary transformations,'' Quantum 5 (2021) 511, arXiv:2001.07774 [quant-ph]. https://doi.org/10.22331/q-2021-07-28-511 arXiv:2001.07774 [19] A. Vanrietvelde and G. Chiribella, ``Universal control of quantum processes using sector-preserving channels,'' Quantum Information and Computation 21 no. 15-16, (Dec, 2021) 1320–1352, arXiv:2106.12463 [quant-ph]. https://doi.org/10.26421/QIC21.15-16-5 arXiv:2106.12463 [20] M. Wilson and A. Vanrietvelde, ``Composable constraints,'' arXiv:2112.06818 [math.CT]. arXiv:2112.06818 [21] A. Kissinger and S. Uijlen, ``A categorical semantics for causal structure,'' Logical Methods in Computer Science Volume 15, Issue 3 (2019) . https://lmcs.episciences.org/5681. https://doi.org/10.23638/LMCS-15(3:15)2019 https://lmcs.episciences.org/5681 [22] O. Oreshkov, ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics,'' Quantum 3 (2019) 206. http://dx.doi.org/10.22331/q-2019-12-02-206. https://doi.org/10.22331/q-2019-12-02-206 [23] G. Chiribella and H. Kristjánsson, ``Quantum Shannon theory with superpositions of trajectories,'' Proceedings of the Royal Society A 475 (2019) , arXiv:1812.05292 [quant-ph]. https://doi.org/10.1098/rspa.2018.0903 arXiv:1812.05292 [24] Ämin Baumeler and E. Tselentis, ``Equivalence of grandfather and information antinomy under intervention,'' Electronic Proceedings in Theoretical Computer Science 340 (Sep, 2021) 1–12, arXiv:2004.12921 [quant-ph]. https://doi.org/10.4204/eptcs.340.1 arXiv:2004.12921 [25] J. Barrett, R. Lorenz, and O. Oreshkov, ``Cyclic quantum causal models,'' Nature Communications 12 no. 1, (2021) 1–15, arXiv:2002.12157 [quant-ph]. https://doi.org/10.1038/s41467-020-20456-x arXiv:2002.12157 [26] A. Joyal, R. Street, and D. Verity, ``Traced monoidal categories,'' Mathematical Proceedings of the Cambridge Philosophical Society 119 no. 3, (1996) 447–468. https://doi.org/10.1017/S0305004100074338 [27] M. Araújo, P. A. Guérin, and Ä. Baumeler, ``Quantum computation with indefinite causal structures,'' Physical Review A 96 no. 5, (2017) 052315, arXiv:1706.09854 [quant-ph]. https://doi.org/10.1103/PhysRevA.96.052315 arXiv:1706.09854 [28] W. Yokojima, M. T. Quintino, A. Soeda, and M. Murao, ``Consequences of preserving reversibility in quantum superchannels,'' Quantum 5 (2021) 441, arXiv:2003.05682 [quant-ph]. https://doi.org/10.22331/q-2021-04-26-441 arXiv:2003.05682 [29] S. Abramsky and A. Brandenburger, ``The sheaf-theoretic structure of non-locality and contextuality,'' New Journal of Physics 13 no. 11, (2011) 113036, arXiv:1102.0264 [quant-ph]. https://doi.org/10.1088/1367-2630/13/11/113036 arXiv:1102.0264 [30] T. Fritz, ``Possibilistic physics,'' 2009. fqxi.org/community/forum/topic/569. http://fqxi.org/community/forum/topic/569 [31] C. Brukner, ``Bounding quantum correlations with indefinite causal order,'' New J. Phys. 17 no. 8, (2015) 083034, arXiv:1404.0721 [quant-ph]. https://doi.org/10.1088/1367-2630/17/8/083034 arXiv:1404.0721 [32] Z. Liu and G. Chiribella, ``Tsirelson bounds for quantum correlations with indefinite causal order,'' Nature Communications 16 no. 1, (2025) 3314, arXiv:2403.02749 [quant-ph]. https://doi.org/10.1038/s41467-025-58508-9 arXiv:2403.02749 [33] E.-E. Tselentis and Ä. Baumeler, ``Admissible Causal Structures and Correlations,'' PRX Quantum 4 no. 4, (2023) 040307, arXiv:2210.12796 [quant-ph]. https://doi.org/10.1103/PRXQuantum.4.040307 arXiv:2210.12796 [34] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, ``The simplest causal inequalities and their violation,'' New Journal of Physics 18 no. 1, (2015) 013008, arXiv:1508.01704 [quant-ph]. https://doi.org/10.1088/1367-2630/18/1/013008 arXiv:1508.01704 [35] I. Mejdoub and A. Vanrietvelde, ``Unilateral determination of causal order in a cyclic process,'' arXiv:2506.18540 [quant-ph]. arXiv:2506.18540 [36] A. Jamiołkowski, ``Linear transformations which preserve trace and positive semidefiniteness of operators,'' Reports on Mathematical Physics 3 no. 4, (1972) 275–278. https://doi.org/10.1016/0034-4877(72)90011-0 [37] M.-D. Choi, ``Completely positive linear maps on complex matrices,'' Linear algebra and its applications 10 no. 3, (1975) 285–290. https://doi.org/10.1016/0024-3795(75)90075-0 [38] N. Ormrod, A. Vanrietvelde, and J. Barrett, ``Causal structure in the presence of sectorial constraints, with application to the quantum switch,'' Quantum 7 (2023) 1028, arXiv:2204.10273 [quant-ph]. https://doi.org/10.22331/q-2023-06-01-1028 arXiv:2204.10273Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-02 13:46:55: Could not fetch cited-by data for 10.22331/q-2025-12-02-1923 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-02 13:46:55: 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. AbstractOver 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} }► References [1] D. E. Deutsch, ``Quantum computational networks,'' Proceedings of the Royal Society of London: A. Mathematical and Physical Sciences 425 no. 1868, (1989) 73–90. https://doi.org/10.1098/rspa.1989.0099 [2] D. Aharonov, A. Kitaev, and N. Nisan, ``Quantum circuits with mixed states,'' in Proceedings of the thirtieth annual ACM symposium on Theory of computing, pp. 20–30. 1998. arXiv:quant-ph/9806029. https://doi.org/10.1145/276698.276708 arXiv:quant-ph/9806029 [3] S. Abramsky and B. Coecke, ``A categorical semantics of quantum protocols,'' in Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004., pp. 415–425, IEEE. 2004. arXiv:quant-ph/0402130. https://doi.org/10.1109/LICS.2004.1319636 arXiv:quant-ph/0402130 [4] M. A. Nielsen and I. L. Chuang, Quantum information and quantum computation.
Cambridge University Press, 2000. https://doi.org/10.1017/CBO9780511976667 [5] B. Coecke and A. Kissinger, Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning.
Cambridge University Press, 2017. https://doi.org/10.1017/9781316219317 [6] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Transforming quantum operations: Quantum supermaps,'' EPL (Europhysics Letters) 83 no. 3, (2008) 30004, arXiv:0804.0180 [quant-ph]. https://doi.org/10.1209/0295-5075/83/30004 arXiv:0804.0180 [7] L. Hardy, ``Probability theories with dynamic causal structure: A New framework for quantum gravity,'' arXiv:gr-qc/0509120. arXiv:gr-qc/0509120 [8] G. Chiribella, G. D’Ariano, P. Perinotti, and B. Valiron, ``Beyond quantum computers,'' arXiv:0912.0195v1 [quant-ph]. arXiv:0912.0195v1 [9] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure,'' Physical Review A 88 no. 2, (2013) 022318, arXiv:0912.0195 [quant-ph]. https://doi.org/10.1103/PhysRevA.88.022318 arXiv:0912.0195 [10] O. Oreshkov, F. Costa, and C. Brukner, ``Quantum correlations with no causal order,'' Nature Communications 3 (2012) 1092, arXiv:1105.4464 [quant-ph]. https://doi.org/10.1038/ncomms2076 arXiv:1105.4464 [11] Ä. Baumeler, A. Feix, and S. Wolf, ``Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios,'' Physical Review A 90 no. 4, (2014) 042106, arXiv:1403.7333 [quant-ph]. https://doi.org/10.1103/PhysRevA.90.042106 arXiv:1403.7333 [12] Ä. Baumeler and S. Wolf, ``The space of logically consistent classical processes without causal order,'' New Journal of Physics 18 no. 1, (2016) 013036, arXiv:1507.01714 [quant-ph]. https://doi.org/10.1088/1367-2630/18/1/013036 arXiv:1507.01714 [13] J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, ``Quantum circuits with classical versus quantum control of causal order,'' PRX Quantum 2 no. 3, (2021) 030335, arXiv:2101.08796 [quant-ph]. https://doi.org/10.1103/PRXQuantum.2.030335 arXiv:2101.08796 [14] T. Colnaghi, G. M. D'Ariano, S. Facchini, and P. Perinotti, ``Quantum computation with programmable connections between gates,'' Physics Letters A 376 no. 45, (2012) 2940–2943, arXiv:1109.5987 [quant-ph]. https://doi.org/10.1016/j.physleta.2012.08.028 arXiv:1109.5987 [15] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, ``A purification postulate for quantum mechanics with indefinite causal order,'' Quantum 1 (2017) 10, arXiv:1611.08535 [quant-ph]. https://doi.org/10.22331/q-2017-04-26-10 arXiv:1611.08535 [16] A. Vanrietvelde, H. Kristjánsson, and J. Barrett, ``Routed quantum circuits,'' Quantum 5 (Jul, 2021) 503, arXiv:2011.08120 [quant-ph]. https://doi.org/10.22331/q-2021-07-13-503 arXiv:2011.08120 [17] J. Barrett, R. Lorenz, and O. Oreshkov, ``Quantum causal models,'' arXiv:1906.10726 [quant-ph]. arXiv:1906.10726 [18] R. Lorenz and J. Barrett, ``Causal and compositional structure of unitary transformations,'' Quantum 5 (2021) 511, arXiv:2001.07774 [quant-ph]. https://doi.org/10.22331/q-2021-07-28-511 arXiv:2001.07774 [19] A. Vanrietvelde and G. Chiribella, ``Universal control of quantum processes using sector-preserving channels,'' Quantum Information and Computation 21 no. 15-16, (Dec, 2021) 1320–1352, arXiv:2106.12463 [quant-ph]. https://doi.org/10.26421/QIC21.15-16-5 arXiv:2106.12463 [20] M. Wilson and A. Vanrietvelde, ``Composable constraints,'' arXiv:2112.06818 [math.CT]. arXiv:2112.06818 [21] A. Kissinger and S. Uijlen, ``A categorical semantics for causal structure,'' Logical Methods in Computer Science Volume 15, Issue 3 (2019) . https://lmcs.episciences.org/5681. https://doi.org/10.23638/LMCS-15(3:15)2019 https://lmcs.episciences.org/5681 [22] O. Oreshkov, ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics,'' Quantum 3 (2019) 206. http://dx.doi.org/10.22331/q-2019-12-02-206. https://doi.org/10.22331/q-2019-12-02-206 [23] G. Chiribella and H. Kristjánsson, ``Quantum Shannon theory with superpositions of trajectories,'' Proceedings of the Royal Society A 475 (2019) , arXiv:1812.05292 [quant-ph]. https://doi.org/10.1098/rspa.2018.0903 arXiv:1812.05292 [24] Ämin Baumeler and E. Tselentis, ``Equivalence of grandfather and information antinomy under intervention,'' Electronic Proceedings in Theoretical Computer Science 340 (Sep, 2021) 1–12, arXiv:2004.12921 [quant-ph]. https://doi.org/10.4204/eptcs.340.1 arXiv:2004.12921 [25] J. Barrett, R. Lorenz, and O. Oreshkov, ``Cyclic quantum causal models,'' Nature Communications 12 no. 1, (2021) 1–15, arXiv:2002.12157 [quant-ph]. https://doi.org/10.1038/s41467-020-20456-x arXiv:2002.12157 [26] A. Joyal, R. Street, and D. Verity, ``Traced monoidal categories,'' Mathematical Proceedings of the Cambridge Philosophical Society 119 no. 3, (1996) 447–468. https://doi.org/10.1017/S0305004100074338 [27] M. Araújo, P. A. Guérin, and Ä. Baumeler, ``Quantum computation with indefinite causal structures,'' Physical Review A 96 no. 5, (2017) 052315, arXiv:1706.09854 [quant-ph]. https://doi.org/10.1103/PhysRevA.96.052315 arXiv:1706.09854 [28] W. Yokojima, M. T. Quintino, A. Soeda, and M. Murao, ``Consequences of preserving reversibility in quantum superchannels,'' Quantum 5 (2021) 441, arXiv:2003.05682 [quant-ph]. https://doi.org/10.22331/q-2021-04-26-441 arXiv:2003.05682 [29] S. Abramsky and A. Brandenburger, ``The sheaf-theoretic structure of non-locality and contextuality,'' New Journal of Physics 13 no. 11, (2011) 113036, arXiv:1102.0264 [quant-ph]. https://doi.org/10.1088/1367-2630/13/11/113036 arXiv:1102.0264 [30] T. Fritz, ``Possibilistic physics,'' 2009. fqxi.org/community/forum/topic/569. http://fqxi.org/community/forum/topic/569 [31] C. Brukner, ``Bounding quantum correlations with indefinite causal order,'' New J. Phys. 17 no. 8, (2015) 083034, arXiv:1404.0721 [quant-ph]. https://doi.org/10.1088/1367-2630/17/8/083034 arXiv:1404.0721 [32] Z. Liu and G. Chiribella, ``Tsirelson bounds for quantum correlations with indefinite causal order,'' Nature Communications 16 no. 1, (2025) 3314, arXiv:2403.02749 [quant-ph]. https://doi.org/10.1038/s41467-025-58508-9 arXiv:2403.02749 [33] E.-E. Tselentis and Ä. Baumeler, ``Admissible Causal Structures and Correlations,'' PRX Quantum 4 no. 4, (2023) 040307, arXiv:2210.12796 [quant-ph]. https://doi.org/10.1103/PRXQuantum.4.040307 arXiv:2210.12796 [34] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, ``The simplest causal inequalities and their violation,'' New Journal of Physics 18 no. 1, (2015) 013008, arXiv:1508.01704 [quant-ph]. https://doi.org/10.1088/1367-2630/18/1/013008 arXiv:1508.01704 [35] I. Mejdoub and A. Vanrietvelde, ``Unilateral determination of causal order in a cyclic process,'' arXiv:2506.18540 [quant-ph]. arXiv:2506.18540 [36] A. Jamiołkowski, ``Linear transformations which preserve trace and positive semidefiniteness of operators,'' Reports on Mathematical Physics 3 no. 4, (1972) 275–278. https://doi.org/10.1016/0034-4877(72)90011-0 [37] M.-D. Choi, ``Completely positive linear maps on complex matrices,'' Linear algebra and its applications 10 no. 3, (1975) 285–290. https://doi.org/10.1016/0024-3795(75)90075-0 [38] N. Ormrod, A. Vanrietvelde, and J. Barrett, ``Causal structure in the presence of sectorial constraints, with application to the quantum switch,'' Quantum 7 (2023) 1028, arXiv:2204.10273 [quant-ph]. https://doi.org/10.22331/q-2023-06-01-1028 arXiv:2204.10273Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-02 13:46:55: Could not fetch cited-by data for 10.22331/q-2025-12-02-1923 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-02 13:46:55: 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.