{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:19:33Z","timestamp":1760242773629,"version":"build-2065373602"},"reference-count":65,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2016,6,14]],"date-time":"2016-06-14T00:00:00Z","timestamp":1465862400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states\u2014defined as states smoothly peaked in some wave-vector eigenstate of the walk\u2014is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations.<\/jats:p>","DOI":"10.3390\/e18060228","type":"journal-article","created":{"date-parts":[[2016,6,14]],"date-time":"2016-06-14T11:12:12Z","timestamp":1465902732000},"page":"228","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Discrete Time Dirac Quantum Walk in 3+1 Dimensions"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0602-5519","authenticated-orcid":false,"given":"Giacomo","family":"D\u2019Ariano","sequence":"first","affiliation":[{"name":"QUIT group, Dipartimento di Fisica, via Bassi 6, Pavia 27100, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicola","family":"Mosco","sequence":"additional","affiliation":[{"name":"INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia 27100, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paolo","family":"Perinotti","sequence":"additional","affiliation":[{"name":"INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia 27100, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessandro","family":"Tosini","sequence":"additional","affiliation":[{"name":"INFN Gruppo IV, Sezione di Pavia, via Bassi 6, Pavia 27100, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,6,14]]},"reference":[{"key":"ref_1","unstructured":"Van Kampen, N.G. (1992). Stochastic Processes in Physics and Chemistry, Elsevier."},{"key":"ref_2","unstructured":"Weiss, G. (2005). Aspects and Applications of the Random Walk (Random Materials & Processes S.), North-Holland."},{"key":"ref_3","unstructured":"Cox, D.R. (1962). Renewal Theory, Methuen."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1103\/PhysRevA.48.1687","article-title":"Quantum random walks","volume":"48","author":"Aharonov","year":"1993","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Aharonov, D., Ambainis, A., Kempe, J., and Vazirani, U. (2001, January 6\u20138). Quantum walks on graphs. Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing\u2014STOC \u201901, Heraklion, Greece.","DOI":"10.1145\/380752.380758"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., and Watrous, J. (2001, January 6\u20138). One-dimensional quantum walks. Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing\u2014STOC \u201901, Heraklion, Greece.","DOI":"10.1145\/380752.380757"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., and Spielman, D.A. (2003, January 9\u201311). Exponential algorithmic speedup by a quantum walk. Proceedings of the Thirty-Fifth ACM Symposium on Theory of Computing\u2014STOC \u201903, San Diego, CA, USA.","DOI":"10.1145\/780551.780552"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1137\/S0097539705447311","article-title":"Quantum walk algorithm for element distinctness","volume":"37","author":"Ambainis","year":"2007","journal-title":"SIAM J. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1137\/050643684","article-title":"Quantum algorithms for the triangle problem","volume":"37","author":"Magniez","year":"2007","journal-title":"SIAM J. Comput."},{"key":"ref_10","unstructured":"Farhi, E., Goldstone, J., and Gutmann, S. (2007). A quantum algorithm for the Hamiltonian NAND tree. arXiv:quant-ph\/0702144."},{"key":"ref_11","unstructured":"Santha, M. (2008, January 25\u201329). Quantum walk based search algorithms. Proceedings of the 5th International Conference Theory and Applications of Models of Computation: (TAMC 2008), Xi\u2019an, China."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Portugal, R. (2013). Quantum Walks and Search Algorithms, Springer.","DOI":"10.1007\/978-1-4614-6336-8"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"435304","DOI":"10.1088\/1751-8113\/48\/43\/435304","article-title":"Grover search with lackadaisical quantum walks","volume":"48","author":"Wong","year":"2015","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_14","unstructured":"Nayak, A., and Vishwanath, A. (2000). Quantum Walk on the Line. arXiv:quant-ph\/0010117."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1080\/00107151031000110776","article-title":"Quantum random walks: An introductory overview","volume":"44","author":"Kempe","year":"2003","journal-title":"Contemp. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0167-2789(93)90096-J","article-title":"Lattice Boltzmann equation for quantum mechanics","volume":"69","author":"Succi","year":"1993","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6920","DOI":"10.1103\/PhysRevD.49.6920","article-title":"Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata","volume":"49","year":"1994","journal-title":"Phys. Rev. D"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/BF02199356","article-title":"From quantum cellular automata to quantum lattice gases","volume":"85","author":"Meyer","year":"1996","journal-title":"J. Stat. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"054302","DOI":"10.1103\/PhysRevA.73.054302","article-title":"Relativistic quantum walks","volume":"73","author":"Strauch","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"046702","DOI":"10.1103\/PhysRevE.63.046702","article-title":"Quantum lattice-gas model for computational fluid dynamics","volume":"63","author":"Yepez","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/j.physleta.2011.12.021","article-title":"The quantum field as a quantum computer","volume":"376","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.aop.2014.12.016","article-title":"Quantum field as a quantum cellular automaton: The Dirac free evolution in one dimension","volume":"354","author":"Bisio","year":"2015","journal-title":"Ann. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"032301","DOI":"10.1103\/PhysRevA.88.032301","article-title":"Dirac quantum cellular automaton in one dimension: Zitterbewegung and scattering from potential","volume":"88","author":"Bisio","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"465302","DOI":"10.1088\/1751-8113\/47\/46\/465302","article-title":"The Dirac equation as a quantum walk: Higher dimensions, observational convergence","volume":"47","author":"Arrighi","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"60004","DOI":"10.1209\/0295-5075\/104\/60004","article-title":"Decoupled quantum walks, models of the Klein-Gordon and wave equations","volume":"104","author":"Arrighi","year":"2013","journal-title":"Europhys. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"062109","DOI":"10.1103\/PhysRevA.89.062109","article-title":"Discrete spacetime and relativistic quantum particles","volume":"89","author":"Farrelly","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"012302","DOI":"10.1103\/PhysRevA.89.012302","article-title":"Causal fermions in discrete space-time","volume":"89","author":"Farrelly","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012316","DOI":"10.1103\/PhysRevA.72.012316","article-title":"Quantum walks and orbital states of a Weyl particle","volume":"72","author":"Katori","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"022322","DOI":"10.1103\/PhysRevA.75.022322","article-title":"Free-Dirac-particle evolution as a quantum random walk","volume":"75","author":"Bracken","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"062106","DOI":"10.1103\/PhysRevA.90.062106","article-title":"Derivation of the Dirac equation from principles of information processing","volume":"90","author":"Perinotti","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.aop.2016.02.009","article-title":"Quantum cellular automaton theory of light","volume":"368","author":"Bisio","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"50003","DOI":"10.1209\/0295-5075\/109\/50003","article-title":"Doubly special relativity from quantum cellular automata","volume":"109","author":"Bisio","year":"2015","journal-title":"Europhys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1007\/s10701-015-9927-0","article-title":"Weyl, Dirac and Maxwell Quantum Cellular Automata","volume":"45","author":"Bisio","year":"2015","journal-title":"Found. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1007\/s10701-015-9934-1","article-title":"Free Quantum Field Theory from Quantum Cellular Automata","volume":"45","author":"Bisio","year":"2015","journal-title":"Found. Phys."},{"key":"ref_35","unstructured":"Schr\u00f6dinger, E. (1930). \u00dcber die kr\u00e4ftefreie Bewegung in der relativistischen Quantenmechanik, Akademie der wissenschaften in kommission bei W. de Gruyter u. Company. (In German)."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"8599","DOI":"10.1103\/PhysRevB.44.8599","article-title":"Effects of the nonrelativistic Zitterbewegung on the electron-phonon interaction in two-band systems","volume":"44","author":"Cannata","year":"1991","journal-title":"Phys. Rev. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"7454","DOI":"10.1103\/PhysRevB.42.7454","article-title":"Nonrelativistic zitterbewegung in two-band systems","volume":"42","author":"Ferrari","year":"1990","journal-title":"Phys. Rev. B"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0038-1098(90)90192-E","article-title":"Dirac-like behaviour of a non-relativistic tight binding Hamiltonian in one dimension","volume":"74","author":"Cannata","year":"1990","journal-title":"Solid State Commun."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/0031-8914(70)90004-2","article-title":"Zitterbewegung of quasiparticles in superconductors","volume":"50","author":"Cremer","year":"1970","journal-title":"Physica"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"187203","DOI":"10.1103\/PhysRevLett.95.187203","article-title":"Spin Transverse Force on Spin Current in an Electric Field","volume":"95","author":"Shen","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"076603","DOI":"10.1103\/PhysRevLett.99.076603","article-title":"Spin-Orbit Interaction in Symmetric Wells with Two Subbands","volume":"99","author":"Bernardes","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"253005","DOI":"10.1103\/PhysRevLett.98.253005","article-title":"Dirac equation and quantum relativistic effects in a single trapped ion","volume":"98","author":"Lamata","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1038\/nature08688","article-title":"Quantum simulation of the Dirac equation","volume":"463","author":"Gerritsma","year":"2010","journal-title":"Nature"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"172305","DOI":"10.1103\/PhysRevB.74.172305","article-title":"Unified description of Zitterbewegung for spintronic, graphene, and superconducting systems","volume":"74","author":"Cserti","year":"2006","journal-title":"Phys. Rev. B"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"195439","DOI":"10.1103\/PhysRevB.76.195439","article-title":"Transient Zitterbewegung of charge carriers in mono-and bilayer graphene, and carbon nanotubes","volume":"76","author":"Rusin","year":"2007","journal-title":"Phys. Rev. B"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"206801","DOI":"10.1103\/PhysRevLett.94.206801","article-title":"Zitterbewegung of electronic wave packets in III-V zinc-blende semiconductor quantum wells","volume":"94","author":"Schliemann","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"085217","DOI":"10.1103\/PhysRevB.72.085217","article-title":"Zitterbewegung and its effects on electrons in semiconductors","volume":"72","author":"Zawadzki","year":"2005","journal-title":"Phys. Rev. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"3533","DOI":"10.1016\/j.physleta.2010.06.028","article-title":"Nature of electron Zitterbewegung in crystalline solids","volume":"374","author":"Zawadzki","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1038\/nmat1849","article-title":"The rise of graphene","volume":"6","author":"Geim","year":"2007","journal-title":"Nat. Mater."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"143201","DOI":"10.1088\/0953-8984\/23\/14\/143201","article-title":"Zitterbewegung (trembling motion) of electrons in semiconductors: A review","volume":"23","author":"Zawadzki","year":"2011","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"153002","DOI":"10.1103\/PhysRevLett.100.153002","article-title":"Observing zitterbewegung with ultracold atoms","volume":"100","author":"Vaishnav","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"113903","DOI":"10.1103\/PhysRevLett.100.113903","article-title":"Observing Zitterbewegung for Photons near the Dirac Point of a Two-Dimensional Photonic Crystal","volume":"100","author":"Zhang","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1023\/A:1023413713008","article-title":"Quantum Random Walks in One Dimension","volume":"1","author":"Konno","year":"2002","journal-title":"Quantum Inf. Process."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"3165","DOI":"10.1016\/j.physleta.2014.09.020","article-title":"Path-integral solution of the one-dimensional Dirac quantum cellular automaton","volume":"378","author":"Mosco","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"40012","DOI":"10.1209\/0295-5075\/109\/40012","article-title":"Discrete Feynman propagator for the Weyl quantum walk in 2 + 1 dimensions","volume":"109","author":"Mosco","year":"2015","journal-title":"Europhys. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1090\/S0025-5718-1965-0178586-1","article-title":"An algorithm for the machine calculation of complex Fourier series","volume":"19","author":"Cooley","year":"1965","journal-title":"Math. Comput."},{"key":"ref_57","unstructured":"Alim, U., and M\u00f6ller, T. (2009, January 18\u201322). A Fast Fourier Transform with Rectangular Output on the BCC and FCC Lattices. Proceedings of the International Conference on Sampling Theory and Applications (SampTA), Marseille, France."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1119\/1.1933296","article-title":"On the zitterbewegung of the Dirac electron","volume":"20","author":"Huang","year":"1952","journal-title":"Am. J. Phys."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1119\/1.11697","article-title":"The Zitterbewegung of a free localized Dirac particle","volume":"47","author":"Lock","year":"1979","journal-title":"Am. J. Phys"},{"key":"ref_60","unstructured":"Bisio, A., D\u2019Ariano, G.M., and Perinotti, P. (2015). Lorentz symmetry for 3d Quantum Cellular Automata. arXiv:1503.01017."},{"key":"ref_61","unstructured":"Thaller, B. (2004). Visualizing the kinematics of relativistic wave packets. arXiv:quant-ph\/0409079."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1103\/PhysRevA.59.604","article-title":"Numerical approach to solve the time-dependent Dirac equation","volume":"59","author":"Braun","year":"1999","journal-title":"Phys. Rev. A"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1016\/j.cpc.2008.01.042","article-title":"FFT-split-operator code for solving the Dirac equation in 2+1 dimensions","volume":"178","author":"Mocken","year":"2008","journal-title":"Comput. Phys. Commun."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1016\/j.cpc.2012.02.012","article-title":"Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling","volume":"183","author":"Lorin","year":"2012","journal-title":"Comput. Phys. Commun."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2454","DOI":"10.1016\/j.cpc.2011.07.003","article-title":"Accelerating the Fourier split operator method via graphics processing units","volume":"182","author":"Bauke","year":"2011","journal-title":"Comput. Phys. Commun."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/6\/228\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:25:26Z","timestamp":1760210726000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/6\/228"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,14]]},"references-count":65,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2016,6]]}},"alternative-id":["e18060228"],"URL":"https:\/\/doi.org\/10.3390\/e18060228","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,6,14]]}}}