Shors Algorithm - Advanced Theory of Computation - Lecture Slides, Slides of Theory of Computation

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Shor’s Algorithm

Overview 

Shor's algorithm

is a quantum algorithm for

factoring a number

N

in O((log

N

) time and

O(log

N

) space, named after Peter Shor.



The algorithm is significant because it impliesthat RSA, a popular public-key cryptographymethod, might be easily broken, given asufficiently large quantum computer

Overview 

Like many quantum computer algorithms, Shor'salgorithm is probabilistic



It gives the correct answer with high probability,and the probability of failure can be decreased byrepeating the algorithm.

Overview 

Shor's algorithm was discovered in 1994 by PeterShor, but the classical part was known before.



it is credited to G. L. Miller. Seven years later, in2001.



it was demonstrated by a group at IBM, whichfactored 15 into 3 and 5, using a quantumcomputer with 7 qubits.

Shor’s Algorithm - Periodicity ^

Choose N = 15 and x = 7 and we get the following:

mod 15 = 1 7

mod 15 = 7 7

mod 15 = 4 7

mod 15 = 13 7

0 1 2 3 4 mod 15 = 1

^

An important result from Number Theory:

F(a) = x mod N

is a periodic function

a

Shor’s Algorithm - In Depth Analysis

To Factor an odd integer N (Let’s choose 15) :

Determine if the number

n

is a prime, a even number, or an

integer power of a prime number. If it is we will not use Shor'salgorithm. There are efficient classical methods for determining ifa integer

n

belongs to one of the above groups, and providing

factors for it if it is. This step would be performed on a classicalcomputer.

Choose an integer

q

such that N

q

< 2N

2

let’s pick 256

Choose a random integer

x

such that GCD(

x , N) = 1

let’s pick 7

2

2

Shor’s Algorithm - Preparing Data^ 5.

Load the input register with an equally weightedsuperposition of all integers from 0 to

q

0 to 255

Load the output register with all zeros.

The total state of the system at this point will be:

1 √^256

∑

|a, 000> 255 a=

InputRegister

OutputRegister

Note: the comma heredenotes that theregisters are entangled

Shor’s Algorithm - Modular Arithmetic

Apply the transformation

x

mod N to each number in

the input register, storing the result of each computationin the output register.

a

Input Register

7

Mod 15

Output Register

|0>

7

Mod 15

1

|1>

7

Mod 15

7

|2>

7

Mod 15

4

|3>

7

Mod 15

13

|4>

7

Mod 15

1

|5>

7

Mod 15

7

|6>

7

Mod 15

4

|7>

7

Mod 15

13

a 0 1 2 3 4 5 6 7

Note that we are using decimalnumbers here only for simplicity.

..

Shor’s Algorithm - Entanglement 9.

Since the two registers are entangled, measuring the outputregister will have the effect of partially collapsing the inputregister into an

equal superposition

of each state between 0

and

q

-1 that yielded

c

(the value of the collapsed output

register.)

Now things really get interesting!

Since the output register collapsed to |1>, the input registerwill partially collapse to:

The probabilities in this case are

since our register is

now in an equal superposition of 64 values (0, 4, 8,... 252)

1 √^64

1 √^64

1 √^64

1 √^64 1 √^64

Shor’s Algorithm - QFT We now apply the Quantum Fourier transform on thepartially collapsed input register. The fourier transform hasthe effect of taking a state |a> and transforming it into astate given by:

1 √ q

∑

**|c> ***

e

q-1 c=

2 

i ac /

q

Shor’s Algorithm - QFT

The QFT will essentially peak the probability amplitudes atinteger multiples of

q

/r , where

r is the desired period in our

case r is 4.

So we no longer have an equal superposition of states, theprobability amplitudes of the above states are now higherthan the other states in our register.Measure the state of register one, call this value

m

, this

integer

m

has a very high probability of being a multiple of

q /

r With our knowledge of q, and m, there are methods ofcalculating the period (one method is the continuous fractionexpansion of the ratio between q and m.)

Shor’s Algorithm - The Factors :)^ 10. Now that we have the period, the factors of N can be

determined by taking the greatest common divisor of Nwith respect to

x ^ (P/2) +

1 and

x ^ (P/2)

.^

The idea

here is that this computation will be done on a classicalcomputer.

We compute:Gcd(

^5

Gcd(

^3

We have successfully factored 15!

4/2 4/

References 

Peter W. Shor “Polynomial-Time Algorithms forPrime Factorization and Discrete Logarithms on aQuantum Computer “, SIAM Journal onComputing (1997)



www.eecis.udel.edu/~saunders/courses/879-03s/



http://en.wikipedia.org/wiki/Shor's_algorithm

Q & A

Thank You