My lecture for Edusec on Quantum comp

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

7. Shor’s Algorithm

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)

22. Deutsch’s Problem
Quantum Oracle

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

30. The Hadamard Case

31. Hadamard
• (1 1
• (1-1)

32. Examples

35. Deutsch’s Problem

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
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