Paul Benioff

**Paul Anthony Benioff** (New York, 1930eko maiatzaren 1a - New York, 2022ko martxoaren 29a) estatubatuar fisikaria izan zen, konputazio kuantikoaren arloan aitzindaria izan zena. Benioff, batez ere, 1970 eta 80ko hamarkadetan informazio kuantikoaren teorian egindako ikerketengatik izan zen ezaguna, ordenagailu kuantikoen aukera teorikoa frogatu zuen, ordenagailu baten lehen eredu mekaniko kuantikoa deskribatuz. Lan horretan, Benioffek erakutsi zuen ordenagailu batek mekanika kuantikoaren legeen arabera funtziona zezakeela, Turingen makinarako  deskribapen bat egin zuen Schrödingerren ekuazioa erabiliz. Benioffek informazio kuantikoaren teorian egindako lanaren multzoak konputagailu kuantikoak, robot kuantikoak eta logikaren, matematikaren eta fisikaren oinarrien arteko erlazioa biltzen zituen.

## Bizitza goiztiarra eta hezkuntza 
Benioff 1930eko maiatzaren 1ean jaio zen, Pasadenan, Kalifornian. Aita, Hugo Benioff, sismologiako irakaslea izan zen Kaliforniako Teknologia Institutuan, eta ama, Alice Pauline Silverman, ingeleseko masterra ikasi zuen Kaliforniako Unibertsitatean, Berkeleyn.  

Benioff ere Berkeleyra joan zen, eta 1951n botanikako graduko titulua lortu zuen. Tracerlab-entzat kimika nuklearrean lanean bi urtez aritu ondoren, Berkeleyra itzuli zen. 1959an, kimika nuklearreko doktoretza lortu zuen.

## Lan zientifiko hautatuak 
* ”Cosmic-ray production rate and mean removal time of beryllium-7 from the atmosphere," *Physical Review*, Vol. 104, 1956, pp. 1122–1130.
* ”Information theory in quantum statistical mechanics," *Physics Letters*, Vol. 14, 1965, pp. 196–197.
* ”Some aspects of the relationship between mathematical logic and physics. I," *Journal of Mathematical Physics*, Vol. 11, 1970, pp. 2553–2569.
* ”Some aspects of the relationship between mathematical logic and physics. II," *Journal of Mathematical Physics*, Vol. 12, 1971, pp. 360–376.
* ”Operator valued measures in quantum mechanics: finite and infinite processes," *Journal of Mathematical Physics*, Vol. 13, 1972, pp. 231–242.
* ”Decision procedures in quantum mechanics," *Journal of Mathematical Physics*, Vol. 13, 1972, pp. 908–915.
* ”Procedures in quantum mechanics without Von Neumann's projection axiom," *Journal of Mathematical Physics*, Vol. 13, 1972, pp. 1347–1355.
* ”Some consequences of the strengthened interpretative rules of quantum mechanics," *Journal of Mathematical Physics*, Vol. 15, 1974, pp. 552–559.
* ”Models of Zermelo Frankel set theory as carriers for the mathematics of physics. I", *Journal of Mathematical Physics*, Vol. 17, 1976, pp. 618–628.
* ”Models of Zermelo Frankel set theory as carriers for the mathematics of physics. II," *Journal of Mathematical Physics*, Vol. 17, 1976, pp. 629–640.
* ”Finite and infinite measurement sequences in quantum mechanics and randomness: The Everett interpretation," *Journal of Mathematical Physics*, Vol. 18, 1977, pp. 2289–2295.
* "The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines", *Journal of Statistical Physics*, Vol. 22, 1980, pp. 563–591.
* "Quantum mechanical hamiltonian models of turing machines", *Journal of Statistical Physics*, Vol. 29, 1982, pp. 515–546.
* "Quantum Mechanical Models of Turing Machines That Dissipate No Energy", *Phys. Rev. Lett.*, Vol. 48, 1982, pp. 1581–1585.
* "Quantum mechanical Hamiltonian models of discrete processes that erase their own histories: Application to Turing machines, Int. J". *Theor. Phys.*, Vol. 21, 1982, pp. 177–201.
* "Comment on 'Dissipation in Computation'," *Physical Review Letters*, Vol. 53, 1984, pp. 1203.
* "Quantum Mechanical Hamiltonian Models of Computers", *Annals New York Academy of Sciences*, Vol. 480, 1986, pp. 475–486.
* "Quantum ballistic evolution in quantum mechanics: Application to quantum computers", *Phys. Rev. A*, Vol. 54, 1996, pp. 1106–1123, Arxiv.
* "Tight binding Hamiltonians and Quantum Turing Machines", *Phys. Rev. Lett.*, Vol. 78, 1997, pp. 590–593.
* "Transmission and spectral aspects of tight binding hamiltonians for the counting quantum turing machine," *Physical Review B*, Vol. 55, 1997, pp. 9482–9493.
* "Models of Quantum Turing Machines", *Fortschritte der Physik*, Vol. 46, 1998, pp. 423–441, Arxiv.
* "Quantum robots and environments", *Phys. Rev. A*, Vol. 58, 1998, pp. 893–904, Arxiv.
* "Quantum Robots and Quantum Computers", in: A. J. G. Hey (Hrsg.), *Feynman and Computation*, Perseus Books 1999, pp. 155–176, Arxiv.
* "A simple example of definitions of truth, validity, consistency, and completeness in quantum mechanics," *Physical Review A*, Vol. 59, 1999, pp. 4223–4252.
* "The Representation of Natural Numbers in Quantum Mechanics", *Phys. Rev. A*, Vol. 63, 2001, 032305, Arxiv.
* "Efficient Implementation and the Product State Representation of Numbers", *Phys. Rev. A*, Vol. 64, 2001, pp. 052310, Arxiv.
* "Language is physical," *Quantum Information Proceedings*, Vol. 1, 2002, pp. 495–509.
* "Use of mathematical logical concepts in quantum mechanics: an example," *Journal of Physics A: Mathematical and General*, Vol. 35, 2002, pp. 5843–5857.
* "Towards a Coherent Theory of Physics and Mathematics", *Found. Phys.*, Vol. 32, 2002, pp. 989–1029, Arxiv.
* "The Representation of Numbers in Quantum Mechanics", *Algorithmica*, Vol. 34, 2002, pp. 529–559, Arxiv.
* "Towards a Coherent Theory of Physics and Mathematics: The Theory-Experiment Connection", *Foundations of Physics*, Vol. 35, 2005, pp. 1825–1856, Arxiv.
* "Representation of complex rational numbers in quantum mechanics", *Phys. Rev. A*, Vol. 72, 2005, pp. 032314, Arxiv.
* “Fields of quantum reference frames based on different representations of rational numbers as states of qubit strings.” Submitted to proceedings, 3rd Feynman Festival, University of Maryland, 2006, *Journal of Physics*: Conference Series 70 (2007) 012003.
* "A representation of real and complex numbers in quantum theory," *International Journal of Pure and Applied Mathematics*, Vol. 39, 2007, pp. 297–339.
* "Reference frame fields based on quantum theory representations of real and complex numbers," *Advances in Quantum Computation*, Vol. 482, 2009, pp. 125–163.
* “Effects on quantum physics of the local availability of mathematics and space time dependent scaling factors for number systems.” Chapter 2, in *Advances in Quantum Theory*, I. I. Cotaescu (Ed.), Intech open access publisher, 2012.
* “Gauge theory extension to include number scaling by boson field: Effects on some aspects of physics and geometry.” Chapter in *Recent Developments in Bosons Research*, Ignace Tremblay (Ed.), Nova Press, 2013.
* "Fiber bundle description of number scaling in gauge theory and geometry," *Quantum Studies: Mathematics and Foundations*, Vol. 2, 2015, pp. 289–313.
* "Effects of a scalar scaling field on quantum mechanics," *Quantum Information Processing*, Vol. 15(7), 2016, pp. 3005–3034.
* "The no information at a distance principle and local mathematics: some effects on physics and geometry," *Theoretical Information Studies*, submitted.

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## Kanpo estekak 
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