{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T00:13:38Z","timestamp":1784160818770,"version":"3.55.0"},"reference-count":53,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T00:00:00Z","timestamp":1748390400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MUR project Progetti di Ricerca d\u2019Interesse Nazionale (PRIN) QCAPP","award":["2022LCEA9Y"],"award-info":[{"award-number":["2022LCEA9Y"]}]},{"name":"MUR project Progetti di Ricerca d\u2019Interesse Nazionale (PRIN) DISTRUCT","award":["P2022T2JZ9"],"award-info":[{"award-number":["P2022T2JZ9"]}]},{"name":"National Research Centre for HPC, Big Data and Quantum Computing, PNRR MUR Project","award":["CN0000013-ICSC"],"award-info":[{"award-number":["CN0000013-ICSC"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We study a coarse-graining procedure for quantum cellular automata on hypercubic lattices that consists in grouping neighboring cells into tiles and selecting a subspace within each tile. This is done in such a way that multiple evolution steps applied to this subspace can be viewed as a single evolution step of a new quantum cellular automaton, whose cells are the subspaces themselves. We derive a necessary and sufficient condition for renormalizability and use it to investigate the renormalization flow of cellular automata on a line, where the cells are qubits and the tiles are composed of two neighboring cells. The problem is exhaustively solved, and the fixed points of the renormalization flow are highlighted.<\/jats:p>","DOI":"10.22331\/q-2025-05-28-1756","type":"journal-article","created":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T15:50:48Z","timestamp":1748447448000},"page":"1756","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Renormalisation of Quantum Cellular Automata"],"prefix":"10.22331","volume":"9","author":[{"given":"Lorenzo Siro","family":"Trezzini","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica dell'Universit\u00e0 di Pavia, via Bassi 6, 27100 Pavia"},{"name":"Istituto Nazionale di Fisica Nucleare, Gruppo IV, via Bassi 6, 27100 Pavia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alessandro","family":"Bisio","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica dell'Universit\u00e0 di Pavia, via Bassi 6, 27100 Pavia"},{"name":"Istituto Nazionale di Fisica Nucleare, Gruppo IV, via Bassi 6, 27100 Pavia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paolo","family":"Perinotti","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica dell'Universit\u00e0 di Pavia, via Bassi 6, 27100 Pavia"},{"name":"Istituto Nazionale di Fisica Nucleare, Gruppo IV, via Bassi 6, 27100 Pavia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2025,5,28]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Richard P. Feynman. ``Simulating physics with computers''. International Journal of Theoretical Physics 21, 467\u2013488 (1982).","DOI":"10.1007\/BF02650179"},{"key":"1","unstructured":"John Von Neumann and Arthur W. Burks. ``Theory of self-reproducing automata''. University of Illinois Press. USA, (1966)."},{"key":"2","doi-asserted-by":"crossref","unstructured":"K.P. Hadeler and J. M\u00fcller. ``Cellular automata: Analysis and applications''. Springer Monographs in Mathematics. Springer International Publishing. (2017). url: https:\/\/books.google.it\/books?id=L4olDwAAQBAJ.","DOI":"10.1007\/978-3-319-53043-7"},{"key":"3","doi-asserted-by":"crossref","unstructured":"Gerhard Gr\u00f6ssing and Anton Zeilinger. ``Quantum cellular automata''. Complex Syst. 2, 197\u2013208, (1988).","DOI":"10.1016\/0378-4363(88)90196-9"},{"key":"4","doi-asserted-by":"publisher","unstructured":"J. Watrous. ``On one-dimensional quantum cellular automata''. In Proceedings of IEEE 36th Annual Foundations of Computer Science. Pages 528\u2013537. (1995).","DOI":"10.1109\/SFCS.1995.492583"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Christoph D\u00fcrr, Huong L\u00ea Thanh, and Miklos Santha. ``A decision procedure for well-formed linear quantum cellular automata''. In Claude Puech and R\u00fcdiger Reischuk, editors, STACS 96. Pages 281\u2013292. Berlin, Heidelberg (1996).","DOI":"10.48550\/arXiv.cs\/9906024"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Christoph D\u00fcrr and Miklos Santha. ``A decision procedure for unitary linear quantum cellular automata''. SIAM Journal on Computing 31, 1076\u20131089 (2002).","DOI":"10.1137\/S0097539797327702"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Jozef Gruska. ``Quantum computing''. Pages 2297\u20132310. John Wiley & Sons, Ltd. (2009).","DOI":"10.1002\/9780470050118.ecse720"},{"key":"8","doi-asserted-by":"publisher","unstructured":"B. Schumacher and R. F. Werner. ``Reversible quantum cellular automata''. quant-ph\/0405174 (2004).","DOI":"10.48550\/arXiv.quant-ph\/0405174"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Pablo Arrighi, Vincent Nesme, and Reinhard Werner. ``One-dimensional quantum cellular automata over finite, unbounded configurations''. In Carlos Mart\u00edn-Vide, Friedrich Otto, and Henning Fernau, editors, Language and Automata Theory and Applications. Pages 64\u201375. Berlin, Heidelberg (2008).","DOI":"10.1007\/978-3-540-88282-4_8"},{"key":"10","doi-asserted-by":"publisher","unstructured":"D. J. Shepherd, T. Franz, and R. F. Werner. ``Universally programmable quantum cellular automaton''. Phys. Rev. Lett. 97, 020502 (2006).","DOI":"10.1103\/PhysRevLett.97.020502"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Pablo Arrighi, Renan Fargetton, and Zizhu Wang. ``Intrinsically universal one-dimensional quantum cellular automata in two flavours''. Fundamenta Informaticae 91, 197\u2013230 (2009).","DOI":"10.3233\/FI-2009-0041"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Giacomo Mauro D'Ariano and Paolo Perinotti. ``Derivation of the dirac equation from principles of information processing''. Phys. Rev. A 90, 062106 (2014).","DOI":"10.1103\/PhysRevA.90.062106"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Alessandro Bisio, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Quantum cellular automaton theory of light''. Annals of Physics 368, 177\u2013190.","DOI":"10.1016\/j.aop.2016.02.009"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Arindam Mallick and C. M. Chandrashekar. ``Dirac cellular automaton from split-step quantum walk''. Scientific Reports 6, 25779 (2016).","DOI":"10.1038\/srep25779"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Pablo Arrighi, C\u00e9dric B\u00e9ny, and Terry Farrelly. ``A quantum cellular automaton for one-dimensional qed''. Quantum Information Processing 19, 88 (2020).","DOI":"10.1007\/s11128-019-2555-4"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Alessandro Bisio, Giacomo Mauro D'Ariano, Paolo Perinotti, and Alessandro Tosini. ``Thirring quantum cellular automaton''. Physical Review A 97, 032132.","DOI":"10.1103\/PhysRevA.97.032132"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Alessandro Bisio, Nicola Mosco, and Paolo Perinotti. ``Scattering and perturbation theory for discrete-time dynamics''. Physical Review Letters 126, 250503.","DOI":"10.1103\/PhysRevLett.126.250503"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Nathana\u00ebl Eon, Giuseppe Di Molfetta, Giuseppe Magnifico, and Pablo Arrighi. ``A relativistic discrete spacetime formulation of 3+1 QED''. Quantum 7, 1179 (2023).","DOI":"10.22331\/q-2023-11-08-1179"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Lorenzo Piroli and J. Ignacio Cirac. ``Quantum cellular automata, tensor networks, and area laws''. Phys. Rev. Lett. 125, 190402 (2020).","DOI":"10.1103\/PhysRevLett.125.190402"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Alexander Jahn, Zolt\u00e1n Zimbor\u00e1s, and Jens Eisert. ``Tensor network models of AdS\/qCFT''. Quantum 6, 643 (2022).","DOI":"10.22331\/q-2022-02-03-643"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Logan E. Hillberry, Lorenzo Piroli, Eric Vernier, Nicole Yunger Halpern, Toma\u017e Prosen, and Lincoln D. Carr. ``Integrability of goldilocks quantum cellular automata'' (2024). https:\/\/doi.org\/10.48550\/arXiv.2404.02994.","DOI":"10.48550\/arXiv.2404.02994"},{"key":"22","doi-asserted-by":"publisher","unstructured":"David T Stephen, Hendrik Poulsen Nautrup, Juani Bermejo-Vega, Jens Eisert, and Robert Raussendorf. ``Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter''. Quantum 3, 142.","DOI":"10.22331\/q-2019-05-20-142"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Hoi Chun Po, Lukasz Fidkowski, Takahiro Morimoto, Andrew C Potter, and Ashvin Vishwanath. ``Chiral floquet phases of many-body localized bosons''. Physical Review X 6, 041070.","DOI":"10.1103\/PhysRevX.6.041070"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Lukasz Fidkowski, Hoi Chun Po, Andrew C Potter, and Ashvin Vishwanath. ``Interacting invariants for floquet phases of fermions in two dimensions''. Physical Review B 99, 085115.","DOI":"10.1103\/PhysRevB.99.085115"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Alexei Kitaev and John Preskill. ``Topological entanglement entropy''. Physical review letters 96, 110404.","DOI":"10.1103\/PhysRevLett.96.110404"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Paolo Perinotti. ``Cellular automata in operational probabilistic theories''. Quantum 4, 294 (2020).","DOI":"10.22331\/q-2020-07-09-294"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Jean Zinn-Justin. ``Phase Transitions and Renormalization Group''. Oxford University Press. (2007).","DOI":"10.1093\/acprof:oso\/9780199227198.001.0001"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Leo P. Kadanoff. ``Scaling laws for ising models near ${T}_{c}$''. Physics Physique Fizika 2, 263\u2013272 (1966).","DOI":"10.1103\/PhysicsPhysiqueFizika.2.263"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Kenneth G. Wilson. ``Renormalization group and critical phenomena. i. renormalization group and the kadanoff scaling picture''. Phys. Rev. B 4, 3174\u20133183 (1971).","DOI":"10.1103\/PhysRevB.4.3174"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Kenneth G. Wilson. ``Renormalization group and critical phenomena. ii. phase-space cell analysis of critical behavior''. Phys. Rev. B 4, 3184\u20133205 (1971).","DOI":"10.1103\/PhysRevB.4.3184"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Navot Israeli and Nigel Goldenfeld. ``Computational irreducibility and the predictability of complex physical systems''. Phys. Rev. Lett. 92, 074105 (2004).","DOI":"10.1103\/PhysRevLett.92.074105"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Navot Israeli and Nigel Goldenfeld. ``Coarse-graining of cellular automata, emergence, and the predictability of complex systems''. Phys. Rev. E 73, 026203 (2006).","DOI":"10.1103\/PhysRevE.73.026203"},{"key":"33","doi-asserted-by":"publisher","unstructured":"O. Duranthon and Giuseppe Di Molfetta. ``Coarse-grained quantum cellular automata''. Physical Review A 103 (2021).","DOI":"10.1103\/physreva.103.032224"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Stefan Boettcher, Stefan Falkner, and Renato Portugal. ``Renormalization and scaling in quantum walks''. Phys. Rev. A 90, 032324 (2014).","DOI":"10.1103\/PhysRevA.90.032324"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Pierre-Gabriel Rozon, Michael J. Gullans, and Kartiek Agarwal. ``Constructing quantum many-body scar hamiltonians from floquet automata''. Phys. Rev. B 106, 184304 (2022).","DOI":"10.1103\/PhysRevB.106.184304"},{"key":"36","doi-asserted-by":"publisher","unstructured":"John Preskill. ``Quantum Computing in the NISQ era and beyond''. Quantum 2, 79 (2018).","DOI":"10.22331\/q-2018-08-06-79"},{"key":"37","doi-asserted-by":"publisher","unstructured":"T. L. M. Guedes, D. Winter, and M. M\u00fcller. ``Quantum cellular automata for quantum error correction and density classification''. Phys. Rev. Lett. 133, 150601 (2024).","DOI":"10.1103\/PhysRevLett.133.150601"},{"key":"38","unstructured":"Wim van Dam. ``Quantum cellular automata''. Master's Thesis, Computer Science Institute, University of Nijmegen, The Netherlands, (1996)."},{"key":"39","doi-asserted-by":"publisher","unstructured":"Mauro Faccin, Michael T Schaub, and Jean-Charles Delvenne. ``Entrograms and coarse graining of dynamics on complex networks''. Journal of Complex Networks 6, 661\u2013678 (2017).","DOI":"10.1093\/comnet\/cnx055"},{"key":"40","doi-asserted-by":"publisher","unstructured":"D. Gross, V. Nesme, H. Vogts, and R. F. Werner. ``Index theory of one dimensional quantum walks and cellular automata''. Communications in Mathematical Physics 310, 419\u2013454 (2012).","DOI":"10.1007\/s00220-012-1423-1"},{"key":"41","doi-asserted-by":"crossref","unstructured":"Pablo Arrighi, S. Martiel, and V. Nesme. ``Cellular automata over generalized Cayley graphs''. Mathematical Structures in Computer Science 18, 340\u2013383 (2018). url: https:\/\/hal.science\/hal-01785458.","DOI":"10.1017\/S0960129517000044"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Giacomo Mauro D'Ariano and Paolo Perinotti. ``Quantum cellular automata and free quantum field theory''. Frontiers of Physics 12, 120301 (2016).","DOI":"10.1007\/s11467-016-0616-z"},{"key":"43","doi-asserted-by":"crossref","unstructured":"O. Bratteli and D.W. Robinson. ``Operator algebras and quantum statistical mechanics 1: C*- and w*-algebras. symmetry groups. decomposition of states''. Operator Algebras and Quantum Statistical Mechanics. Springer. (1987). url: https:\/\/books.google.it\/books?id=YuR4VQOQQUIC.","DOI":"10.1007\/978-3-662-02520-8"},{"key":"44","unstructured":"R. Haag. ``Local quantum physics: Fields, particles, algebras''. R.Balian, W.Beiglbock, H.Grosse. Springer-Verlag. (1992). url: https:\/\/books.google.it\/books?id=Op_vAAAAMAAJ."},{"key":"45","doi-asserted-by":"publisher","unstructured":"Paolo Zanardi. ``Stabilizing quantum information''. Phys. Rev. A 63, 012301 (2000).","DOI":"10.1103\/PhysRevA.63.012301"},{"key":"46","doi-asserted-by":"crossref","unstructured":"Jean-Luc Brylinski and Ranee Brylinski. ``Universal quantum gates''. In Mathematics of quantum computation. Pages 117\u2013134. Chapman and Hall\/CRC (2002).","DOI":"10.1201\/9781420035377-5"},{"key":"47","doi-asserted-by":"publisher","unstructured":"Alexander Stottmeister, Vincenzo Morinelli, Gerardo Morsella, and Yoh Tanimoto. ``Operator-algebraic renormalization and wavelets''. Phys. Rev. Lett. 127, 230601 (2021).","DOI":"10.1103\/PhysRevLett.127.230601"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Tobias J. Osborne and Alexander Stottmeister. ``Conformal field theory from lattice fermions''. Communications in Mathematical Physics 398, 219\u2013289 (2023).","DOI":"10.1007\/s00220-022-04521-8"},{"key":"49","doi-asserted-by":"publisher","unstructured":"Lauritz van Luijk, Alexander Stottmeister, and Reinhard F. Werner. ``Convergence of dynamics on inductive systems of banach spaces''. Annales Henri Poincar\u00e9 25, 4931\u20134986 (2024).","DOI":"10.1007\/s00023-024-01413-6"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Alexander Stottmeister and Tobias J. Osborne. ``On the renormalization group fixed point of the two-dimensional ising model at criticality''. Scientific Reports 13, 14859 (2023).","DOI":"10.1038\/s41598-023-42005-4"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Pablo Arrighi, Stefano Facchini, and M. Forets. ``Discrete lorentz covariance for quantum walks and quantum cellular automata''. New Journal of Physics 16 (2014). url: https:\/\/api.semanticscholar.org\/CorpusID:3944811.","DOI":"10.1088\/1367-2630\/16\/9\/093007"},{"key":"52","unstructured":"Antonio F. Rotundo, Paolo Perinotti, and Alessandro Bisio. ``Effective dynamics from minimising dissipation'' (2024). arXiv:2412.10216."}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-05-28-1756\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T15:50:50Z","timestamp":1748447450000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2025-05-28-1756\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,28]]},"references-count":53,"URL":"https:\/\/doi.org\/10.22331\/q-2025-05-28-1756","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,28]]},"article-number":"1756"}}