an introduction to quantum algorithm

7/26/2019 An Introduction to Quantum Algorithm

1/41

Quantum Algorithm

and its applications

Ritajit Majumdar

MTech 2nd YearRoll Number : 97/CSM/140001

Registration Number : 0029169 of 2008-2009

3rd semester seminar talk

Ritajit Majumdar (CU) Quantum Algorithm 3rd semester seminar talk 1 / 23
http://find/


2/41

1 Introduction

2 Lets Go Quantum

3 Consequence of Shor Algorithm

4 So what Now????

http://find/


3/41

Introduction

1 Introduction

2 Lets Go Quantum


4 So what Now????

http://find/


4/41

Introduction Factorisation is hard

How it all began


I d i F i i i h d
http://find/


5/41

Introduction Factorisation is hard

How it all began


I t d ti F t i ti b d d t i d fi di
http://find/


6/41

Introduction Factorisation can be reduced to period finding

Factorisation to Period Finding

Let N = 15, x = 2

r 0 1 2 3 4 5 6 7 8 ...xr (mod N) 1 2 4 8 1 2 4 8 1 ...

The period of repetition is one less than a factor of N

The problem ofPrime Factorization can be reduced to the problem ofPeriod Finding


http://find/


7/41



Let N = 15, x = 2

r 0 1 2 3 4 5 6 7 8 ...xr (mod N) 1 2 4 8 1 2 4 8 1 ...




http://find/


8/41



Let N = 15, x = 2

r 0 1 2 3 4 5 6 7 8 ...xr (mod N) 1 2 4 8 1 2 4 8 1 ...




http://find/


9/41


Is this reduction any good?


Lets Go Quantum
http://find/


10/41

Let s Go Quantum

1 Introduction

2 Lets Go Quantum


4 So what Now????


Lets Go Quantum Quantum Fourier Transform
http://find/


11/41

Q Q

Quantum Fourier Transform

FN= 1N

1 1 1 1 . . . 11 w w2 w3 . . . wN1

1 w2 w4 w6 . . . w2(N

1)

... ...

... ...

. . . ...

1 wN1 w2(N1) w3(N1) . . . w(N1)2

Hence FNij is wij


http://find/


12/41

Q Q

Quantum Parallelism

Let N = 4 w=exp(i2 ) =cos(2 ) +i.sin(2 ) =i

Consider state|1 = 0 1 0 0T

F4 |1 = 12

1 1 1 11 i 1 i1 1 1 11

i

1 i

0100

= 12

1i

1

i

= 12 |0 + i2 |1 12 |2 i2 |3


http://find/


13/41

Quantum Parallelism

Let N = 4

w=exp(i2 ) =cos(

2 ) +i.sin(

2 ) =i

Consider state|1 = 0 1 0 0T

F4 |1 = 12

1 1 1 1

1 i 1 i1 1 1 11 i 1 i

0

100

= 12

1

i1i

= 12|0

+ i2

|1

12

|2

i2

|3

Inference

Quantum Fourier Transform produces an equal superposition of all thebasis states


Lets Go Quantum The Algorithm
http://find/


14/41

Shor Algorithm

Choose qsuch that N2

q


15/41

Shor Algorithm


q


16/41

Shor Algorithm


q


17/41

Shor Algorithm


q


18/41

Shor Algorithm


q


19/41

Shor Algorithm


q


20/41

Shor Algorithm

The first register is in state - 1qr

qr1j=0|j.r+l

Apply QFT again 1r

r1k=0w

kl |k.qr

Measure the statecollapses to k.qr

where 0 k r 1

If gcd(k, qr) = 1computing gcd(k.q

r, q) gives q

r

q is already knownfind r

If gcd(k, qr)= 1, repeat the algorithm


http://find/


21/41

Shor Algorithm


qr

1

j=0|j.r+l

Apply QFT again 1r

r1k=0w

kl |k.qr


where 0 k r 1


r, q) gives q

r




http://find/http://goback/


22/41

Shor Algorithm


qr

1

j=0|j.r+l

Apply QFT again 1r

r1k=0w

kl |k.qr


where 0 k r 1


r, q) gives q

r




http://goforward/http://find/http://goback/


23/41

Shor Algorithm


qr

1

j=0|j.r+l

Apply QFT again 1r

r1k=0w

kl |k.qr


where 0 k r 1


r, q) gives q

r






24/41

Shor Algorithm


qr

1

j=0|j.r+l

Apply QFT again 1r

r1k=0w

kl |k.qr


where 0 k r 1


r, q) gives q

r






25/41

Shor Algorithm


qr

1

j=0|j.r+l

Apply QFT again 1r

r1k=0w

kl |k.qr


where 0 k r 1


r, q) gives q

r




Consequence of Shor Algorithm


26/41

1 Introduction

2 Lets Go Quantum


4 So what Now????


Consequence of Shor Algorithm Public Key Cryptography is insecure
http://find/


27/41

RSA is insecure

Run time of Shor algorithm : O((log N)2(log log N)(log log log N))[1]


http://find/


28/41

Other public key ciphers...

Shors Algorithm has been experimentally realized using NMR[2]

It has been realised using Ion Trap with only 5 qubits[5]

Counter Argument: SO WHAT? We can use cryptosystems likeElliptic Curve, Elgamal etc.

These algorithms rely on Discrete Logarithm Problem

Well, no... Shor himself has given a quantum algorithm that can solveDiscrete Logarithm problem in polynomial time[1]



O
http://find/


29/41









O h bli k i h
http://find/


30/41









O h bli k i h
http://find/


31/41




Counter Argument: SO WHAT? We can use cryptosystems like

Elliptic Curve, Elgamal etc.



Inference

Public key cryptography is INSECURE against quantum attacks


Consequence of Shor Algorithm Private Key Cryptography is insecure too

P i t k i h
http://find/


32/41

Private key ciphers...


Consequence of Shor Algorithm Private Key Cryptography is insecure too

P i t k i h
http://find/


33/41

Private key ciphers...

A recent paper published on February 18, 2016 claims that Symmetric KeyCiphers can be broken in (n) time using Simons Algorithm[4]


So what Now????
http://find/


34/41

1 Introduction

2 Lets Go Quantum


4 So what Now????


So what Now???? Quantum Cryptography

BB84 Protocol


35/41

BB84 Protocol

Use two conjugate basis

|+

=

{,

}and

|=

{,

}to establish a

secret key between two parties at a distance[3]


So what Now???? Quantum Cryptography

Other variants of Quantum Key Distributions
http://find/


36/41

Other variants of Quantum Key Distributions

E91 protocol [Ekert, PRL 1991]

Semi Quantum QKD [Boyer, Kenigsberg and Mor, PRL 2007]

Device Independent (DI) QKD

Idea by Mayers and Yao [FOCS, 1998]

Measurement Device Independent (MDI) QKD [Lo, Curty and Qi,PRL, 2012]

Side Channel Free (SCF) QKD [Braunstein and Pirandola, PRL, 2012]

Fully Device Independent (FDI) QKD [Vazirani and Vidick, PRL, 2014]


So what Now???? Rebirth of Classical Cryptography

Post Quantum Cryptography
http://find/


37/41

Post Quantum Cryptography

Lattice-based cryptography (e.g., NTRU)

Multivariate cryptography (e.g., Rainbow)

Hash-based cryptography (e.g., Lamport, Merkle)

Code-based cryptography (e.g., McEliece, Niederreiter)

Supersingular ECC



Acknowledgement


38/41

Acknowledgement

I would like to acknowledge the following whom I referred to for preparingthis talk -

1 Pedagogical Talk on Death and Re-birth of Classical Cryptographyin Quantum Era by Dr. Gautam Paul, ISI Kolkata, at ISCQI 2016.

2 Online course on Quantum Mechanics and Quantum Computationby Prof. Umesh Vazirani of University of California, Berkeley, from

www.coursera.org.



Reference
http://find/


39/41

Reference

[1] Peter W Shor. Algorithms for quantum computation: Discrete logarithms and

factoring. In Foundations of Computer Science, 1994 Proceedings., 35th AnnualSymposium on, pages 124 134. IEEE, 1994

[2] Vandersypen et al. Experimental realization of Shors quantum factoringalgorithm using nuclear magnetic resonance. Nature 414, 883-887 (20 December2001)

[3] Charles H Bennett. Quantum cryptography: Public key distribution and cointossing. In International Conference on Computer System and Signal Processing,IEEE, 1984, pages 175 179, 1984

[4] Kaplan et al. Breaking symmetric cryptosystems using quantum periodfinding. arXiv:1602.05973, 2016

[5] Monz et al. Realization of a scalable Shor algorithm. Science 04 Mar 2016:

Vol. 351, Issue 6277, pp. 1068-1070



Image Courtesy


40/41

Image Courtesy

Boromir: generator-meme.com/meme/one-does-not-simply/

Shor: www-math.mit.edu/shor/

Batman: bgr.com/2015/08/07/batman-movies-dc-ben-affleck/

Boromir facepalm: generator-meme.com/meme/own/

Grumpy cat: www.dailymail.co.uk


http://find/


41/41


an introduction to quantum algorithm

Documents