Quantum Clustering Algorithms Esma A¨ımeur
[email protected] Gilles Brassard
[email protected] S´ebastien Gambs
[email protected] Universit´ede Montr´eal, D´epartement d’informatique et de recherche op´erationnelle C.P. 6128, Succursale Centre-Ville, Montr´eal (Qu´ebec), H3C 3J7 Canada Abstract Multidisciplinary by nature, Quantum Information By the term “quantization”, we refer to the Processing (QIP) is at the crossroads of computer process of using quantum mechanics in order science, mathematics, physics and engineering. It con- to improve a classical algorithm, usually by cerns the implications of quantum mechanics for infor- making it go faster. In this paper, we initiate mation processing purposes (Nielsen & Chuang, 2000). the idea of quantizing clustering algorithms Quantum information is very different from its classi- by using variations on a celebrated quantum cal counterpart: it cannot be measured reliably and it algorithm due to Grover. After having intro- is disturbed by observation, but it can exist in a super- duced this novel approach to unsupervised position of classical states. Classical and quantum learning, we illustrate it with a quantized information can be used together to realize wonders version of three standard algorithms: divisive that are out of reach of classical information processing clustering, k-medians and an algorithm for alone, such as being able to factorize efficiently large the construction of a neighbourhood graph. numbers, with dramatic cryptographic consequences We obtain a significant speedup compared to (Shor, 1997), search in a unstructured database with a the classical approach. quadratic speedup compared to the best possible clas- sical algorithms (Grover, 1997) and allow two people to communicate in perfect secrecy under the nose of an 1.