Quant2a
- 1. Cryptography
- 4. • I,H matrices 2X2 • Tensor(I,H) (I11H I12H • I21H I22H) • I11 I22 I21 I22 are all numbers! • The outcome is a matrix of 4x4
- 8. • The best-known classical algorithm for factoring N is ”General number field sieve” • Shor’s Algorithm is polynomial in log𝑁 and uses log 𝑁 space Factorization
- 9. • Let N a huge integer s.t. • p, q are primes • N= p*q • We wish to find p, q Reformulation of factorization
- 10. • Let N as before. • Pick randomly a number a< N • If GCD(a, N )>1 so we achieved the required (we can do this step using the Euclidean algorithm) • GCD(a, N ) = 1 Finding an Even Period
- 11. • Consider the following function : F(x) = 𝑎𝑥 mod(N) for x integer • Assume r is the smallest period of F: F(x+r)=F(x) ∀ m<r F(x+m)≠F(x) Even Period
- 12. • N=15 • 7^2=49 4(15) • 7^3= 13(15) • 7^4 =1 (15) • 7^8 =1 (15)
- 13. • a^m = a^(m+r) => a^r=1 a^r-1 =0 mod(N) • If r is even : a^2-b^2 –(a-b)(a+b) • 0=a^r-1 = (a^(r/2) -1)(a^(r/2)+1) • If a^(r/2) =-1Mod(N) =>a^(r/2)=N-1 mod(N)
- 14. • If r is odd we cannot do anything (need a new a) • If r is even but 𝑎 𝑟 2 ≡ -1mod(N) we cant factorize N need a new a We have r is an even period 𝑎𝑟 ≡ 1 (N) When r is bad?
- 15. Even Period 𝑎 r 2 ∓ 1 ≡ 0 mod(N) We factorized N !!!
- 17. Examples The simple N=15 , a=7 The period: 7,49, 343, 2401 Convert to mod (15): 7,4, 13, 1 ➢ r =4 ➢GCD(72 +1,15) = 5 ➢GCD(72 -1,15) = 3 Wow it works!
- 18. Examples Tedious one N=35 , a=12 Period mod 35: 12,4,13,16,17,29,33,11,27,9,3,1 ➢ r =12, we need 126 mod(35)= 29 ➢ GCD(126 +1,35) = 5 GCD(126 -1,35) = 7
- 19. Well…. If some one provides you the period r ,yes. But no oracles in algorithms. We need to find it ourselves Is it that easy?
- 20. The Quantum Part • We have a function F s.t. • F : {0,1}𝑛 -> {0,1}𝑛 • Find an even number r s.t. • F(x)= F(x+r) ∀ x • ∀ m< r F(x+m)≠F(x)
- 24. The Black box
- 26. • We need to ask the oracle twice • Can't solve it faster!! Black box – Classical Case
- 27. Quantum Case • We wish to find a unitary gate that allows us to find the function faster. • This operator maps one Qbit to one Qbit Does it work?
- 28. F constant • U maps 0 & 1to the same values so it is not unitary
- 31. Hadamard • (1 1 • (1-1)
- 32. Examples
- 36. • The left side is the scheme • The right is one of the four functions we may have Remark
- 38. • If f is constant, get |+> and after Hadamard |0> • If f isn’t constant, get |-> and after Hadamard |1> What do we have?
- 39. Quantum Fourier Transform (QFT)
- 40. QFT Properties • QFT is unitary : ||x||=1 => ||QFT|x>||=1 ||x||=1,||y||=1 ,<x,y>=0 => <QFT|x>,QFT|y> > =0 • The order of QFT is a power of 2(2𝑙 ) • Commonuse for FFT is finding periods (convert a signal from time axis to frequency)
- 41. QFT – Matrix representation • QFT is a matrix that its (a, c) entry is: • We can show that it is a unitary matrix • A common notation is 𝐴𝑞
- 42. Or Tensor (with n qubits )
- 43. Phase Operator
- 45. Shor’s Algorithm
- 47. Measuring the second register Now, QFT Shor algorithm
- 48. • We map a state |a> 0<a<q-1 as follow: |a> → Quantum Fourier Transform (QFT)
- 50. Probability Analysis for a state |y>
- 51. Searching for good y’s
- 53. • https://www.quantiki.org/wiki/shors-factoring-algorithm • https://www.kau.edu.sa/Files/830001/Files/57627_Algorithms_Part17.pdf • https://people.eecs.berkeley.edu/~luca/quantum/lecture08.pdf • https://courses.edx.org/c4x/BerkeleyX/CS191x/asset/chap5.pdf • https://www.cl.cam.ac.uk/teaching/1617/QuantComp/slides7.pdf • https://arcb.csc.ncsu.edu/~mueller/qc/qc19/readings/Quantum%20Fourier %20Transforms%20191119.pdf • https://arxiv.org/pdf/1408.6252.pdf • https://www.youtube.com/watch?v=YoWBUhbrRn0 bib