{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:11:16Z","timestamp":1760148676689,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,29]],"date-time":"2023-05-29T00:00:00Z","timestamp":1685318400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT, Funda\u00e7\u00e3o para a Ci"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We introduce a quantum Lernmatrix based on the Monte Carlo Lernmatrix in which n units are stored in the quantum superposition of log2(n) units representing On2log(n)2 binary sparse coded patterns. During the retrieval phase, quantum counting of ones based on Euler\u2019s formula is used for the pattern recovery as proposed by Trugenberger. We demonstrate the quantum Lernmatrix by experiments using qiskit. We indicate why the assumption proposed by Trugenberger, the lower the parameter temperature t; the better the identification of the correct answers; is not correct. Instead, we introduce a tree-like structure that increases the measured value of correct answers. We show that the cost of loading L sparse patterns into quantum states of a quantum Lernmatrix are much lower than storing individually the patterns in superposition. During the active phase, the quantum Lernmatrices are queried and the results are estimated efficiently. The required time is much lower compared with the conventional approach or the of Grover\u2019s algorithm.<\/jats:p>","DOI":"10.3390\/e25060871","type":"journal-article","created":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T02:33:27Z","timestamp":1685414007000},"page":"871","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quantum Lernmatrix"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2179-4378","authenticated-orcid":false,"given":"Andreas","family":"Wichert","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, INESC-ID & Instituto Superior T\u00e9cnico, University of Lisbon, 2740-122 Porto Salvo, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,29]]},"reference":[{"key":"ref_1","unstructured":"Ventura, D., and Martinez, T. (1988, January 4\u20139). Quantum associative memory with exponential capacity. Proceedings of the 1998 IEEE International Joint Conference on Neural Networks Proceedings. IEEE World Congress on Computational Intelligence, Anchorage, AK, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/S0020-0255(99)00101-2","article-title":"Quantum associative memory","volume":"124","author":"Ventura","year":"2000","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s12559-010-9047-2","article-title":"Face Recognition with Quantum Associative Networks Using Overcomplete Gabor Wavelet","volume":"2","author":"Tay","year":"2010","journal-title":"Cogn. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"067901","DOI":"10.1103\/PhysRevLett.87.067901","article-title":"Probabilistic Quantum Memories","volume":"87","author":"Trugenberger","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1023\/A:1024022632303","article-title":"Quantum Pattern Recognition","volume":"1","author":"Trugenberger","year":"2003","journal-title":"Quantum Inf. Process."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Schuld, M., and Petruccione, F. (2018). Supervised Learning with Quantum Computers, Springer.","DOI":"10.1007\/978-3-319-96424-9"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Grover, L.K. (1996, January 22\u201324). A fast quantum mechanical algorithm for database search. Proceedings of the STOC\u201996: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, USA.","DOI":"10.1145\/237814.237866"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1103\/PhysRevLett.79.325","article-title":"Quantum Mechanics helps in searching for a needle in a haystack","volume":"79","author":"Grover","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Grover, L.K. (1998, January 24\u201326). A framework for fast quantum mechanical algorithms. Proceedings of the STOC\u201998: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, Dallas, TX, USA.","DOI":"10.1145\/276698.276712"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4329","DOI":"10.1103\/PhysRevLett.80.4329","article-title":"Quantum Computers Can Search Rapidly by Using Almost Any Transformation","volume":"80","author":"Grover","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/s10994-012-5316-5","article-title":"Quantum speed-up for unsupervised learning","volume":"90","author":"Brassard","year":"2013","journal-title":"Mach. Learn."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wittek, P. (2014). Quantum Machine Learning, What Quantum Computing Means to Data Mining, Academic Press. Elsevier Insights.","DOI":"10.1016\/B978-0-12-800953-6.00004-9"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1038\/nphys3272","article-title":"Quantum Machine Learning Algorithms: Read the Fine Print","volume":"11","author":"Aaronson","year":"2015","journal-title":"Nat. Phys."},{"key":"ref_14","unstructured":"Diamantini, M.C., and Trugenberger, C.A. (2022). Mirror modular cloning and fast quantum associative retrieval. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"209801","DOI":"10.1103\/PhysRevLett.91.209801","article-title":"Comment on \u201cProbabilistic Quantum Memories\u201d","volume":"91","author":"Brun","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"150502","DOI":"10.1103\/PhysRevLett.103.150502","article-title":"Quantum algorithm for solving linear systems of equations","volume":"103","author":"Harrow","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"040504","DOI":"10.1103\/PhysRevLett.122.040504","article-title":"Quantum Machine Learning in Feature Hilbert Spaces","volume":"122","author":"Schuld","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","unstructured":"Palm, G. (1982). Neural Assemblies, an Alternative Approach to Artificial Intelligence, Springer."},{"key":"ref_19","unstructured":"Hecht-Nielsen, R. (1989). Neurocomputing, Addison-Wesley."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1007\/BF00293853","article-title":"Die Lernmatrix","volume":"1","author":"Steinbuch","year":"1961","journal-title":"Kybernetik"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Steinbuch, K. (1971). Automat und Mensch, Springer. [4th ed.].","DOI":"10.1007\/978-3-642-65070-3"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1038\/222960a0","article-title":"Nonholgraphic associative memory","volume":"222","author":"Willshaw","year":"1969","journal-title":"Nature"},{"key":"ref_23","unstructured":"Contributors, Q. (2023). Qiskit: An Open-source Framework for Quantum Computing. Zenodo."},{"key":"ref_24","unstructured":"Palm, G. (1990). Gehirn und Kognition, Spektrum der Wissenschaft."},{"key":"ref_25","unstructured":"Churchland, P.S., and Sejnowski, T.J. (1994). The Computational Brain, The MIT Press."},{"key":"ref_26","unstructured":"Fuster, J. (1995). Memory in the Cerebral Cortex, The MIT Press."},{"key":"ref_27","unstructured":"Squire, L.R., and Kandel, E.R. (1999). Memory: From Mind to Moleculus, Scientific American Library."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kohonen, T. (1989). Self-Organization and Associative Memory, Springer. [3rd ed.].","DOI":"10.1007\/978-3-642-88163-3"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hertz, J., Krogh, A., and Palmer, R.G. (1991). Introduction to the Theory of Neural Computation, Addison-Wesley.","DOI":"10.1063\/1.2810360"},{"key":"ref_30","unstructured":"Anderson, J.R. (1995). Cognitive Psychology and Its Implications, W. H. Freeman and Company. [4th ed.]."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1109\/T-C.1972.223477","article-title":"Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements","volume":"100","author":"Amari","year":"1972","journal-title":"IEEE Trans. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Anderson, J.A. (1995). An Introduction to Neural Networks, The MIT Press.","DOI":"10.7551\/mitpress\/3905.001.0001"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Ballard, D.H. (1997). An Introduction to Natural Computation, The MIT Press.","DOI":"10.7551\/mitpress\/3917.001.0001"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"Hopfield","year":"1982","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"McClelland, J., and Rumelhart, D. (1986). Parallel Distributed Processing, The MIT Press.","DOI":"10.7551\/mitpress\/5236.001.0001"},{"key":"ref_36","unstructured":"OFTA (1991). Les Re\u00b4seaux de Neurones, Masson."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Bol, G., Nakhaeizadeh, G., and Vollmer, K. (1996). Finanzmarktanalyse und-Prognose mit Innovativen und Quantitativen Verfahren, Physica-Verlag.","DOI":"10.1007\/978-3-642-61489-7"},{"key":"ref_38","unstructured":"Sommer, F.T. (1993). Theorie Neuronaler Assoziativspeicher. [Ph.D. Thesis, Heinrich-Heine-Universit\u00e4t D\u00fcsseldorf]."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1037\/h0026823","article-title":"Context-Sensitive Coding, Associative Memory, and Serial Order in (Speech) Behavior","volume":"76","author":"Wickelgren","year":"1969","journal-title":"Psychol. Rev."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"6207","DOI":"10.1007\/s00521-021-06759-0","article-title":"\u201cWhat-Where\u201d sparse distributed invariant representations of visual patterns","volume":"34","author":"Wichert","year":"2022","journal-title":"Neural Comput. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Sa-Couto, L., and Wichert, A. (2023). Competitive learning to generate sparse representations for associative memory. arXiv.","DOI":"10.2139\/ssrn.4342082"},{"key":"ref_42","unstructured":"Marcinowski, M. (1987). Codierungsprobleme beim Assoziativen Speichern. [Master\u2019s Thesis, Fakult\u00e4t f\u00fcr Physik der Eberhard-Karls-Universit\u00e4t T\u00fcbingen]."},{"key":"ref_43","unstructured":"Freeman, J.A. (1994). Simulating Neural Networks with Mathematica, Addison-Wesley."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/j.neunet.2010.09.012","article-title":"Tree-like hierarchical associative memory structures","volume":"24","author":"Sacramento","year":"2011","journal-title":"Neural Netw."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.neunet.2011.07.005","article-title":"Regarding the temporal requirements of a hierarchical Willshaw network","volume":"25","author":"Sacramento","year":"2012","journal-title":"Neural Netw."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/6\/871\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:44:45Z","timestamp":1760125485000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/6\/871"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,29]]},"references-count":45,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["e25060871"],"URL":"https:\/\/doi.org\/10.3390\/e25060871","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,5,29]]}}}