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ELE 523E COMPUTATIONAL NANOELECTRONICS

Mustafa Altun Electronics & Communication Engineering Istanbul Technical University Web: http://www.ecc.itu.edu.tr/. ELE 523E COMPUTATIONAL NANOELECTRONICS. W3: Quantum and Molecular Computing, 30/9/2013. FALL 2013. Outline. Quantum computing Gates and circuits Algorithms

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ELE 523E COMPUTATIONAL NANOELECTRONICS

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  1. MustafaAltun Electronics & Communication Engineering Istanbul Technical University • Web: http://www.ecc.itu.edu.tr/ ELE 523E COMPUTATIONALNANOELECTRONICS W3: Quantum and Molecular Computing, 30/9/2013 FALL 2013

  2. Outline • Quantum computing • Gates and circuits • Algorithms • Molecular (including DNA) computing • Fundementals • Implementing basic operations • DNA strand displacament

  3. Quantum Gates - NOT Quantumgatesarereversible

  4. Quantum Gates - Hadamard

  5. Quantum Gates - CNOT A A′ B B′

  6. Quantum Gates - CNOT A A′ B B′ Truth table

  7. Quantum gates – CCNOT A A′ B B′ C C′

  8. Quantum gates – CCNOT A A′ B B′ C C′ Toffoli gate

  9. Quantum Circuits • There is no way of efficiently preparing the input state. • There is no way of reading out the output precisely. • This is in the nature of measurement in quantum mechanics. Measurement symbol

  10. Quantum Circuits Severe restrictions on measuring and copying signals

  11. Quantum Circuits Example: Find the output quantum states and probabilities. ? ? Howaboutthetruthtable?

  12. Quantum Circuits Example: Find the output quantum states and probabilities. ? ? Howaboutthetruthtable?

  13. Factorizing RSA Numbers • 15 = 3 × 5 • 77 = 7 × 11 • 529 = 23 × 23 • 4633 = 41 × 113 • RSA-100=15226050279225333605356183781326374297180681 14961380688657908494580122963258952897654000350692006139= 37975227936943673922808872755445627854565536638199× 40094690950920881030683735292761468389214899724061 • The prize for RSA-1024 is $100.000. • RSA-2048 takes approximately 10 billion years with the bestknown algorithm. RSAnumbersaresemi-primenumbers. For each RSA number n, there exist primenumbers p and q such that n = p × q.

  14. ClassicalFactorizing The Classical Algorithm Input: A semi-prime numbern. Output: Prime numberspandqsuchthatn = p × q. Steps: Arbitrarirlyselect a numberxsuchthat1<x<n. Find ximodnfori = 1,2,3,…untilfindingtheperiodR. Calculate greatest common divisor (GCD) of (xR/2 -1, n) and (xR/2 +1, n). GCDsgivethenumbersporq. WhatifR is odd? Whatifthealgorithmresults in 1andn ?

  15. ClassicalFactorizing Example:Ifn=15whatarepandq? • Arbitrarirlyselect a numberxsuchthat1<x<n. • x=7 • Find ximodnfori = 1,2,3,…untilfindingtheperiodR. • 71mod 15 = 7 • 72mod 15 = 4 • 73mod 15 = 13 • 74mod 15 = 1 • 75mod 15 = 7 • FindGCDs. • GCD(74/2-1, 15) = 3,GCD(74/2+1, 15) = 5 R= 4

  16. ClassicalFactorizing Example:Ifn=15whatarepandq? • Arbitrarirlyselect a numberxsuchthat1<x<n. • x=9 • Find ximodnfori = 1,2,3,…untilfindingtheperiodR. • 91mod 15 = 9 • 92mod 15 = 6 • 93mod 15 = 9 • FindGCDs. • GCD(92/2-1, 15) = 1, GCD(92/2+1, 15) = 5 R= 2

  17. ClassicalFactorizing Example:Ifn=15whatarepandq? • Arbitrarirlyselect a numberxsuchthat1<x<n. • x=14 • Find ximodnfori = 1,2,3,…untilfindingtheperiodR. • 141mod 15 = 14 • 142mod 15 = 1 • 143mod 15 = 14 • FindGCDs. • GCD(142/2-1, 15) = 1, GCD(142/2+1, 15) = 15 R= 2

  18. ClassicalFactorizing Example:Ifn=15whatarepandq? • Arbitrarirlyselect a numberxsuchthat1<x<n. • x=10 • Find ximodnfori = 1,2,3,…untilfindingtheperiodR. • 101mod 15 = 10 • 102mod 15 = 10 R= 1 Is there a way of properly selecting x?

  19. ClassicalFactorizing • Step-2 is problematic for large numbers. • Shor’salgorithmcompletes step-2 efficiently. • Step-3 can be done polinomially in time with using Euclidean algorithm. The ClassicalAlgorithm Arbitrarirlyselect a numberxsuchthat1<x<n. Find ximodnfori = 1,2,3,…untilfindingtheperiodR. Calculate greatest common divisor (GCD) of (xR/2 -1, n) and (xR/2 +1, n).

  20. Euclidean Algorithm GCD(7854, 4746) = ? Euclid, 300 BC Euclid’sElements

  21. Shor’sAlgorithm • Time complexity: O((log N)3). • N is number of digits of thegiven semi-prime number. • Based on quantum Fourier transformand modular exponentiation. • Success rate is 50%. • Currently no practicalandusefulimplementation.

  22. + x 2z y + MolecularComputing types count cell/test tube x 9 8 y 6 5 z 7 9 Discrete quantity of molecules

  23. + x 2z y + Molecular Computing types count x 9 3 Whatarethe final quantities of molecules? y 6 0 Write an equationwithinitialand final quantities. z 19 7

  24. + + + Molecular Computing slow medium fast Reactionratesusedfor step-by-step procedure. UNI or BI directional

  25. + + + Molecular Computing test tube Initial condition slow medium fast 10 blue and 5 black What will happen? Explain step-by-step.

  26. + + + Molecular Computing test tube Initial condition slow medium fast 6 blue, 6 black, and 6 orange What will happen? Explain step-by-step.

  27. Addition Howtoselectreactionrates (sloworfast)? biochemical code

  28. Multiplication pseudo-code biochemical code

  29. Exponentiation pseudo-code biochemical code

  30. DNA Strand Displacement x • Outputs can be used as inputs. • Different types of strand displacements with different reaction rates. • Waste molecules! x y y

  31. DNA StrandDisplacement x y z + x y z

  32. DNA Strand Displacement x y x y z z z + AND OR

  33. Suggested Readings • Soloveichik, D., Seelig, G., & Winfree, E. (2010). DNA as a universal substrate for chemical kinetics. Proceedings of the National Academy of Sciences, 107(12), 5393-5398. • Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information. Cambridge university press.

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