{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:46:19Z","timestamp":1760060779010,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T00:00:00Z","timestamp":1757980800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princeton University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL dits, as higher-dimensional analogs of QL bits. We employ systems of k-regular graphs to construct QL-dits for arbitrary dimensions, where the emergent eigenspectrum of their adjacency matrices defines the QL-state space. The tensor product structure of multi-QL dit systems is realized through the Cartesian product of graphs. Furthermore, we examine the potential computational advantages of employing d-nary QL systems over two-level QL bit systems, particularly in terms of classical resource efficiency. Overall, this study generalizes the paradigm of using synchronized network dynamics for QL information processing to include higher-dimensional QL resources.<\/jats:p>","DOI":"10.3390\/e27090963","type":"journal-article","created":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T11:19:31Z","timestamp":1758021571000},"page":"963","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multi-Dimensional Quantum-like Resources from Complex Synchronized Networks"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-8094-2400","authenticated-orcid":false,"given":"Debadrita","family":"Saha","sequence":"first","affiliation":[{"name":"Department of Chemistry, Princeton University, Princeton, NJ 08544, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3336-7960","authenticated-orcid":false,"given":"Gregory D.","family":"Scholes","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Princeton University, Princeton, NJ 08544, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1126\/science.1173440","article-title":"Emulation of a Quantum Spin with a Superconducting Phase Qudit","volume":"325","author":"Neeley","year":"2009","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"024062","DOI":"10.1103\/PhysRevApplied.17.024062","article-title":"Experimental High-Dimensional Greenberger-Horne-Zeilinger Entanglement with Superconducting Transmon Qutrits","volume":"17","author":"Krenn","year":"2022","journal-title":"Phys. Rev. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"180504","DOI":"10.1103\/PhysRevLett.125.180504","article-title":"Implementation of a Walsh-Hadamard Gate in a Superconducting Qutrit","volume":"125","author":"Yurtalan","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1038\/s41567-022-01658-0","article-title":"A universal qudit quantum processor with trapped ions","volume":"18","author":"Ringbauer","year":"2022","journal-title":"Nat. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1186\/s43593-024-00066-6","article-title":"Integrated preparation and manipulation of high-dimensional flying structured photons","volume":"4","author":"Zhao","year":"2024","journal-title":"eLight"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1038\/nature22986","article-title":"On-chip generation of high-dimensional entangled quantum states and their coherent control","volume":"546","author":"Kues","year":"2017","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"062619","DOI":"10.1103\/PhysRevA.110.062619","article-title":"Optical Gottesman-Kitaev-Preskill qubit generation via approximate squeezed coherent state superposition breeding","volume":"110","author":"Pizzimenti","year":"2024","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1038\/s41467-022-28767-x","article-title":"A programmable qudit-based quantum processor","volume":"13","author":"Chi","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3452","DOI":"10.1016\/j.physleta.2014.10.003","article-title":"Determining the parity of a permutation using an experimental NMR qutrit","volume":"378","author":"Dogra","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Gedik, Z., Silva, I.A., \u00c7akmak, B., Karpat, G., Vidoto, E.L.G., Soares-Pinto, D.O., deAzevedo, E.R., and Fanchini, F.F. (2015). Computational speed-up with a single qudit. Sci. Rep., 5.","DOI":"10.1038\/srep14671"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1380","DOI":"10.1038\/s41467-024-45368-y","article-title":"Navigating the 16-dimensional Hilbert space of a high-spin donor qudit with electric and magnetic fields","volume":"15","author":"Botzem","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1038\/35071024","article-title":"Quantum computing in molecular magnets","volume":"410","author":"Leuenberger","year":"2001","journal-title":"Nature"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Gokhale, P., Baker, J.M., Duckering, C., Brown, N.C., Brown, K.R., and Chong, F.T. (2019, January 22\u201326). Asymptotic improvements to quantum circuits via qutrits. Proceedings of the 46th International Symposium on Computer Architecture, Phoenix, AZ, USA.","DOI":"10.1145\/3307650.3322253"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1038\/nphys1150","article-title":"Simplifying quantum logic using higher-dimensional Hilbert spaces","volume":"5","author":"Lanyon","year":"2009","journal-title":"Nat. Phys."},{"key":"ref_15","unstructured":"Galda, A., Cubeddu, M., Kanazawa, N., Narang, P., and Earnest-Noble, N. (2021). Implementing a ternary decomposition of the toffoli gate on fixed-frequency transmon qutrits. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1038\/s41534-024-00829-6","article-title":"Noisy qudit vs multiple qubits: Conditions on gate efficiency for enhancing fidelity","volume":"10","author":"Hartmann","year":"2024","journal-title":"Npj Quantum Inf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"012311","DOI":"10.1103\/PhysRevA.67.012311","article-title":"Security of quantum key distribution with entangled qutrits","volume":"67","author":"Durt","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wang, Y., Hu, Z., Sanders, B.C., and Kais, S. (2020). Qudits and High-Dimensional Quantum Computing. Front. Phys., 8.","DOI":"10.3389\/fphy.2020.589504"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"20150245","DOI":"10.1098\/rsta.2015.0245","article-title":"Quantum Bayesianism as the basis of general theory of decision-making","volume":"374","author":"Khrennikov","year":"2016","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_20","first-page":"20240209","article-title":"Quantum-like states on complex synchronized networks","volume":"480","author":"Scholes","year":"2024","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"062203","DOI":"10.1103\/PhysRevA.111.062203","article-title":"Quantum information with quantumlike bits","volume":"111","author":"Amati","year":"2025","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"060202","DOI":"10.1103\/PhysRevLett.134.060202","article-title":"Quantum-like Product States Constructed from Classical Networks","volume":"134","author":"Scholes","year":"2025","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Amati, G., and Scholes, G.D. (2025). Encoding quantum-like information in classical synchronizing dynamics. arXiv.","DOI":"10.1103\/nvdp-sr11"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Scholes, G.D. (2023). Large coherent states formed from disordered k-regular random graphs. Entropy, 25.","DOI":"10.3390\/e25111519"},{"key":"ref_25","unstructured":"Scholes, G.D. (2025). Graphs that predict exciton delocalization. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1016\/j.tics.2010.05.004","article-title":"Probabilistic models of cognition: Exploring representations and inductive biases","volume":"14","author":"Griffiths","year":"2010","journal-title":"Trends Cogn. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1279","DOI":"10.1126\/science.1192788","article-title":"How to grow a mind: Statistics, structure, and abstraction","volume":"331","author":"Tenenbaum","year":"2011","journal-title":"Science"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1086\/338631","article-title":"Measuring New Types of Question-Order Effects: Additive and Subtractive","volume":"66","author":"Moore","year":"2002","journal-title":"Public Opin. Q."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1037\/0033-295X.90.4.293","article-title":"Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment","volume":"90","author":"Tversky","year":"1983","journal-title":"Psychol. Rev."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1037\/dec0000110","article-title":"A quantum model of strategic decision-making explains the disjunction effect in the Prisoner\u2019s Dilemma game","volume":"7","year":"2020","journal-title":"Decision"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2010). Ubiquitous Quantum Structure, Springer.","DOI":"10.1007\/978-3-642-05101-2"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1023\/A:1018885632116","article-title":"Classical and quantum mechanics on information spaces with applications to cognitive, psychological, social, and anomalous phenomena","volume":"29","author":"Khrennivov","year":"1999","journal-title":"Found. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/j.jmp.2009.01.007","article-title":"Quantum mechanics and violations of the sure-thing principle: The use of probability interference and other concepts","volume":"53","author":"Khrennikov","year":"2009","journal-title":"J. Math. Psychol."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/BF00208726","article-title":"Applications of quantum statistics in psychological studies of decision processes","volume":"1","author":"Aerts","year":"1995","journal-title":"Found. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1023\/A:1014809312397","article-title":"Weak quantum theory: Complementarity and entanglement in physics and beyond","volume":"32","author":"Atmanspacher","year":"2002","journal-title":"Found. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.jmp.2009.06.002","article-title":"Introduction to the special issue on quantum cognition","volume":"53","author":"Bruza","year":"2009","journal-title":"J. Math. Psychol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1146\/annurev-psych-033020-123501","article-title":"Quantum cognition","volume":"73","author":"Pothos","year":"2022","journal-title":"Annu. Rev. Psychol."},{"key":"ref_38","unstructured":"Osgood, C., Suci, G., and Tannenbaum, P. (1957). The Measurement of Meaning, University of Illinois Press. Number 47 in Illini Book, IB 47."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.3758\/s13423-017-1371-z","article-title":"A quantum theory account of order effects and conjunction fallacies in political judgments","volume":"25","author":"Yearsley","year":"2018","journal-title":"Psychon. Bull. Rev."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2123","DOI":"10.1038\/s41467-024-46402-9","article-title":"Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges","volume":"15","author":"Fauseweh","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"12685","DOI":"10.1021\/acs.chemrev.9b00829","article-title":"Quantum algorithms for quantum chemistry and quantum materials science","volume":"120","author":"Bauer","year":"2020","journal-title":"Chem. Rev."},{"key":"ref_42","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.aop.2014.06.013","article-title":"A practical introduction to tensor networks: Matrix product states and projected entangled pair states","volume":"349","year":"2014","journal-title":"Ann. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1126\/science.aag2302","article-title":"Solving the quantum many-body problem with artificial neural networks","volume":"355","author":"Carleo","year":"2017","journal-title":"Science"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","article-title":"Quantum computing in the NISQ era and beyond","volume":"2","author":"Preskill","year":"2018","journal-title":"Quantum"},{"key":"ref_46","unstructured":"Scholes, G.D. (2025). Dynamics in an emergent quantum-like state space generated by a nonlinear classical network. arXiv."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"2959","DOI":"10.1063\/1.1476391","article-title":"Noncommutative tori and universal sets of nonbinary quantum gates","volume":"43","author":"Vlasov","year":"2002","journal-title":"J. Math. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"052318","DOI":"10.1103\/PhysRevA.71.052318","article-title":"Criteria for exact qudit universality","volume":"71","author":"Brennen","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"052309","DOI":"10.1103\/PhysRevA.62.052309","article-title":"Multivalued logic gates for quantum computation","volume":"62","author":"Muthukrishnan","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"235303","DOI":"10.1088\/1751-8113\/41\/23\/235303","article-title":"Bloch vectors for qudits","volume":"41","author":"Bertlmann","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"010301","DOI":"10.1103\/PhysRevA.94.010301","article-title":"Heisenberg-Weyl Observables: Bloch vectors in phase space","volume":"94","author":"Asadian","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/09500340.2011.632097","article-title":"Parametrizations of density matrices","volume":"59","author":"Messina","year":"2012","journal-title":"J. Mod. Opt."},{"key":"ref_53","unstructured":"Pudda, F., Chizzini, M., and Crippa, L. (2024). Generalised Quantum Gates for Qudits and their Application in Quantum Fourier Transform. arXiv."},{"key":"ref_54","unstructured":"Kuramoto, Y. (1975, January 23\u201329). Self-entrainment of a population of coupled non-linear oscillators. Proceedings of the International Symposium on Mathematical Problems in Theoretical Physics, Kyoto, Japan."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2015.10.008","article-title":"The Kuramoto model in complex networks","volume":"610","author":"Rodrigues","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1038\/nphys3801","article-title":"Topological mechanics","volume":"12","author":"Huber","year":"2016","journal-title":"Nat. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1038\/nphoton.2014.248","article-title":"Topological photonics","volume":"8","author":"Lu","year":"2014","journal-title":"Nat. Photonics"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"023211","DOI":"10.1103\/PhysRevResearch.4.023211","article-title":"Topological synchronization of coupled nonlinear oscillators","volume":"4","author":"Sone","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1038\/s42005-025-01999-1","article-title":"Observation of non-Hermitian topological synchronization","volume":"8","author":"Di","year":"2025","journal-title":"Commun. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1698","DOI":"10.1103\/PhysRevLett.42.1698","article-title":"Solitons in Polyacetylene","volume":"42","author":"Su","year":"1979","journal-title":"Phys. Rev. Lett."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRevLett.49.405","article-title":"Quantized Hall Conductance in a Two-Dimensional Periodic Potential","volume":"49","author":"Thouless","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/S0003-4916(02)00018-0","article-title":"Fault-tolerant quantum computation by anyons","volume":"303","author":"Kitaev","year":"2003","journal-title":"Ann. Phys."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1103\/RevModPhys.80.1083","article-title":"Non-Abelian anyons and topological quantum computation","volume":"80","author":"Nayak","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_64","unstructured":"Allam, A., Filanovsky, I.M., Oliveira, L.B., and Fernandes, J.R. (2006, January 21\u201324). Synchronization of mutually coupled LC-oscillators. Proceedings of the 2006 IEEE International Symposium on Circuits and Systems, Kos, Greece."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1591","DOI":"10.1109\/TCSI.2015.2418851","article-title":"Oscillator Array Models for Associative Memory and Pattern Recognition","volume":"62","author":"Maffezzoni","year":"2015","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1109\/JPROC.2018.2878854","article-title":"Computing With Networks of Oscillatory Dynamical Systems","volume":"107","author":"Raychowdhury","year":"2019","journal-title":"Proc. IEEE"},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Flovik, V., Macia, F., and Wahlstr\u00f6m, E. (2016). Describing synchronization and topological excitations in arrays of magnetic spin torque oscillators through the Kuramoto model. Sci. Rep., 6.","DOI":"10.1038\/srep32528"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/9\/963\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:46:22Z","timestamp":1760035582000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/9\/963"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,16]]},"references-count":67,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["e27090963"],"URL":"https:\/\/doi.org\/10.3390\/e27090963","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,9,16]]}}}