Post on 07-Jan-2020
Mladen Pavicic: Graficko racunanjekvantno-mehanickih nelinearnih jednadžbi
Projekt:Kvantno racunanje: paralelnost i vizualizacija(082-0982562-3160); voditelj: M. Pavicic
Program: Distrada i znanstvena vizualizacija(0982562); voditelj: Karolj Skalae-Science day, IRB 17.12.2009
Prof. dr. sc. Mladen Pavicic, Katedra za fiziku, Gradjevinski fakultet u Zagrebu
pavicic@grad.hr ; Web: http://m3k.grad.hr/pavicic
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 1
Pozadina
Gatesova ideja: Riješiti probleme sirovom silom.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 2
Pozadina
Gatesova ideja: Riješiti probleme sirovom silom.
Medjutim: Moore: Hoops! Sorry!:
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 2
Pozadina
Gatesova ideja: Riješiti probleme sirovom silom.
Medjutim: Moore: Hoops! Sorry!:
1993 1996 1999 2002 20050.1
1.0
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Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 2
Pozadina
Gatesova ideja: Riješiti probleme sirovom silom.
Medjutim: Moore: Hoops! Sorry!:
1993 1996 1999 2002 20050.1
1.0
2.0
3.0
3.8
GHz
Intel CPU introductions: 486-50 MHz Jun 1991, DX2-66 Aug 92, P(entium)-100 Mar 94,P-133 Jun 95, P-200 Jun 96, PII-300 May 97, PII-450 Aug 98, PIII-733 Oct 99, PIII-1.0 GHz
Mar 00, P4-1.7 Apr 01, P4-2.0 Aug 01, P4-2.53 May 02, P4-2.8 Aug 02, P4-3.0 Apr 03,P4-3.4 Apr 04, P4-3.6 Jun 04, P4-3.8 Nov 04.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 2
Problem
Pripreme i mjerenja stanja kvantnih sistema ne možemosimulirati klasicnim sistemima.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 3
Problem
Pripreme i mjerenja stanja kvantnih sistema ne možemosimulirati klasicnim sistemima.Npr., jedini stvarni generator nasumicno generiranih brojevasu sami kvantni sistemi. Zatim, jedino kvantni sistemi, kojisu hardware kvantnih kompjutera, mogu direktno simuliratikvantne probleme. Klasicni hardware ih može samorješavati preko binarnog koda jednadžbi.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 3
Problem
Pripreme i mjerenja stanja kvantnih sistema ne možemosimulirati klasicnim sistemima.Npr., jedini stvarni generator nasumicno generiranih brojevasu sami kvantni sistemi. Zatim, jedino kvantni sistemi, kojisu hardware kvantnih kompjutera, mogu direktno simuliratikvantne probleme. Klasicni hardware ih može samorješavati preko binarnog koda jednadžbi.
Izracunavanje orijentacija uredjaja zapripremu i mjerenje kvantnih stanja tražimasivno racunanje pomocu clustera i grida.Rezultati su premnogobrojni → primijen-juje se 2D, 3D i 4D graficka vizualizacija.Priprema stanja koja su iskljucivo kvantna:Kochen-Specker-ova stanja.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 3
PHYSICAL REVIEW A69, 062311 (2004)
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 4
Multiparty entanglement in graph states
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 5
M. Hein, J. Eisert, and H. J. Briegel
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 6
Institut für Physik, Uni Potsdam
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 7
Blackett Lab., Imperial College, London
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 8
Orthogonal Spins
We have to measure spins in 3, 4, 5, . . . dimensions.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 9
Orthogonal Spins
We have to measure spins in 3, 4, 5, . . . dimensions. Ofcourse in a Hilbert space. It corresponds to outgoing portsin our 3-dim lab.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 9
Orthogonal Spins
We have to measure spins in 3, 4, 5, . . . dimensions. Ofcourse in a Hilbert space. It corresponds to outgoing portsin our 3-dim lab. Vectors are orthogonal ⇒ nonlinearequations
aB · aC = aB1aC1 + aB2aC2 + aB3aC3 + aB4aC4 = 0,
aB · aD = aB1aD1 + aB2aD2 + aB3aD3 + aB4aD4 = 0,
aB · aE = aB1aE1 + aB2aE2 + aB3aE3 + aB4aE4 = 0,
aC · aD = aC1aD1 + aC2aD2 + aC3aD3 + aC4aD4 = 0,
aC · aE = aC1aE1 + aC2aE2 + aC3aE3 + aC4aE4 = 0,
aD · aE = aD1aE1 + aD2aE2 + aD3aE3 + aD4aE4 = 0.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 9
Mission Impossible
To solve these equations for all possible combinations for atleast 18 vectors (no solutions below 18) we would need amillion ages of the universe on all today’s processors on theGlobe working in parallel.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 10
Mission Impossible
To solve these equations for all possible combinations for atleast 18 vectors (no solutions below 18) we would need amillion ages of the universe on all today’s processors on theGlobe working in parallel.
D3,1 D3,2
D4,3 D4,4
Figure 1: Generation tree for MMP diagrams with
loops of size 5 for 3-dim vectors.
1
Use MMP hyper-graphs instead ofequations.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 10
Exponential ⇒ Polynomial
We first “translate” nonlinear equations into linearhypergraphs.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 11
Exponential ⇒ Polynomial
We first “translate” nonlinear equations into linearhypergraphs.Next, we impose conditions on generation of hypergraphs.Generation proves to be statistically polynomially complex(SPC).
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 11
Exponential ⇒ Polynomial
We first “translate” nonlinear equations into linearhypergraphs.Next, we impose conditions on generation of hypergraphs.Generation proves to be statistically polynomially complex(SPC).We filter the obtained hypergraphs by additions conditions.The procedure is also SPC.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 11
Exponential ⇒ Polynomial
We first “translate” nonlinear equations into linearhypergraphs.Next, we impose conditions on generation of hypergraphs.Generation proves to be statistically polynomially complex(SPC).We filter the obtained hypergraphs by additions conditions.The procedure is also SPC.In the end we translate a rather “small” number (millions) ofhypergraphs back into equations and solve them by meansof interval analysis method. Its programs are SPC as well.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 11
David A. Richter
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 12
Western Michigan University
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 13
P.K.Aravind
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 14
Worcester Polytechnic Institute
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 15
Norman D. Megill, Boston Information Group
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Jean-Pierre Merlet, INRIA, France
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Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 17
Solutions
\ 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 total
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22 1 9 36 23 12 3 1 85
23 2 19 76 79 58 27 11 3 1 276
24 1 6 39 137 187 188 136 83 40+ 1 18 6 2 1 845
total 1 2 8 24 65 139 228 248 216 147 86 42 18 6 2 1 1233
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 18
Solutions
\ 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 total
18 1 1
19 1 1
20 1 4+ 1 1 7
21 2 11 4 1 18
22 1 9 36 23 12 3 1 85
23 2 19 76 79 58 27 11 3 1 276
24 1 6 39 137 187 188 136 83 40+ 1 18 6 2 1 845
total 1 2 8 24 65 139 228 248 216 147 86 42 18 6 2 1 1233
Boxed in red are the solutionspreviously found by humans.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 18
Solutions
\ 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 total
18 1 1
19 1 1
20 1 4+ 1 1 7
21 2 11 4 1 18
22 1 9 36 23 12 3 1 85
23 2 19 76 79 58 27 11 3 1 276
24 1 6 39 137 187 188 136 83 40+ 1 18 6 2 1 845
total 1 2 8 24 65 139 228 248 216 147 86 42 18 6 2 1 1233
Boxed in red are the solutionspreviously found by humans.
It would take over 150 years togenerate all the solutions on a sin-gle PC
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 18
Brendan D. McKay, Australian National Univ.
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Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 19
Krešimir Fresl, Gra devinski fakultet, Zagreb
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Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 20
A i Danko Bosanac, IRB . . .
je na projektu, ali u njegovim drugim teorijskim dijelovima, ane na konkretnim clusterskim i grid racunanjima.
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 21
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 22
PARALELIZAM & VIZUALIZACIJA . . .
. . . ili kako paralelno vizualizirano grid racunanje gradikvantnu—inherentno paralelnu—simulaciju.
Hvala!
Mladen Pavicic: Graficko racunanje kvantno-mehanickih nelinearnih jednadzbi – p. 23