Lecture note 8: Quantum Algorithms

Jian-Wei Pan

Physikalisches Institut der Universität Heidelberg Philosophenweg 12, 69120 Heidelberg, Germany

Outline

• Quantum Parallelism• Shor’s quantum factoring algorithm• Grover’s quantum search algorithm

Quantum Algorithm

• Quantum Parallelism- Fundamental feature of many quantum algorithms- it allows a quantum computer to evaluate a function f(x) for many different values of x simultaneously.- This is what makes famous quantum algorithms, such as Shor’s algorithm for factoring, or Grover’s algorithm for searching.

| |i ii i

U a i aU i=∑ ∑

RSA encryption and factoring

• RSA is named after Riverst, Shamir and Adleman, who came up with the scheme

m1×m2 = N, (with m1 and m2 primes)• Based on the ease with which N can be calculated

from m1 and m2.• And the difficulty of calculating m1 and m2 from N.• N is made public available and is used to encrypt data.• m1 and m2 are the secret keys which enable one to

decrypt the data.• To crack a code, a code breaker needs to factor N.

RSA encryption and factoring

• Problem: given a number, what are its prime factors ?e.g. a 129-digit odd number which is the product of

two large primes,

114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123 63958705058989075147599290026879543541=3490529510847650949147849619903898133417764638493387843990820577x 32769132993266709549961988190834461413177642967992942539798288533

• Best factoring algorithm requires sources that grow exponentially in the size of the number

- , with n the length of N• Difficulty of factoring is the basis of security for the

RSA encryption scheme used.

))log(exp( 3/23/1 nnO

Shor’s algorithm

Algorithms for quantum computation: discrete logarithms and factoring

- Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium onPublication Date: 20-22 Nov 1994. On pages: 124-134

Shor, P.W. AT&T Bell Labs., Murray Hill, NJ;

Shor’s algorithm• Shor’s code-breaking

Quantum Algorithm-How fast can you factor a number?

- Quantum computer advantage

• Code-breaking can be done in minutes, not in millennia

• Public key encryption, based on factoring, will be vulnerable!!!

• E.g. factor a 300-digit number

• Classical THz computer- steps- 150,000 years

• Quantum THz computer- steps- 1 second

2410

1010

How to factor an odd number – a little number theory

• Modular Arithmetic

simply means

where k is an integer.

• Consider- where x and N are co-primes, i.e. greatest common

divisor gcd(a,N)=1. No factors in common.

It will be demonstrated in the following that finding r is equivalent to factoring N

mod moda b N b a N= ⇔ =

kNba +=

1modrx N=

A little number theory

1 2 1modrm m N x N× = ⇔ =

• Consider the equations

then we have

2

2

1 2

1mod1 0 mod

( 1)( 1) 0mod( 1)( 1) 0mod

y Ny Ny y Ny y m m

=

− =+ − =+ − =

1

2

1 0 mod 1 0 mod;or

1 0 mod 1 0 mod1y m y Ny m y+ = + =⎧ ⎧

⎨ ⎨− = − =⎩⎩

A little number theorygcd( 1, ) ,gcd( 1, ) 1.

y N Ny N+ =⎧

⎨ − =⎩

1

2

gcd( 1, ) ,gcd( 1, ) .

y N my N m+ =⎧

⎨ − =⎩

Note that gcd can be calculated efficiently.If we can find r, and r is evenThen

provided we don’t get trivial solutions.If r is an odd number, change x, try again.

2 / 2 2( ) 1modry x N= =

/ 21

/ 22

gcd( 1, )

gcd( 1, )

r

r

m x N

m x N

= +

= −

We acquire a trivial solution

and the desired solution

A little number theory

• Finding r is equivalent to factoring N- It takes operations to find r using classical computer. (n the digits of N)

• An important result from number theory,

is a periodic function. E.g. N=15, x=7. period r= 4• Factoring reduces to period finding.

)2( nO

( ) modrf r x N=

r 0 1 2 3 41 7 4 13 1modrx N

Shor algorithm

• Using quantum computer to find the period r.• The algorithm is dependent on

– Modular Arithmetic– Quantum Parallelism– Quantum Fourier Transform

• Illustration• To factor an odd integer, N=15• Choose a random integer x satisfying gcd(x,N)=1,x=7 in our case.

Shor’s algorithm

• Create two quantum registers, - input registers: contain enough qubits to represent r , ( 8 qubits up to 255)- output registers: contain enough qubits to represent (we need 4 qubits )

• Load the input registers an equally weighted superposition state of all integers (0-255) . The output registers are zero.

( ) modrf r x N N= <

Shor’s algorithm

a – input register, 0- output registerApply a controlled unitary transformation to the input register , storing the results in the output registers. • From quantum Parallelism, this unitary

transformation can be implemented on all the states simultaneously.

255

0

1| | ,0256 a

aψ=

= ∑

| , 0 | , modaU a a x N=

255 255

0 0

1 1| | ,0 | , mod256 256

a

a a

U U a a x Nψ= =

= =∑ ∑

Shor’s algorithm• The unitary transformation U consists of a series

of elementary quantum gates, single-, two-qubit...• The sequence of these quantum gates that are

applied to the quantum input depends on the classical variables x and N complicatedly.

• We need a classical computer processes the classical variables and produces an output that is a program for the quantum computer, i.e. the number and sequence of elementary quantum operations. This can be performed efficiently on a classical computer.(see details, PRA, 54, 1034, (1996);

Shor’s algorithm

Assume we applied U on the quantum registers.

in 0 1 2 3 4 5 6 7 8 9 10 11 12 …out 1 7 4 13 1 7 4 13 1 7 4 13 1 …

1| [(| 0 | 4 | 8 |12 ... | 252 ) |1256

(|1 | 5 | 9 |13 ... | 253 ) | 7

(| 2 | 6 |10 |14 ... | 254 ) | 4

(| 3 | 7 |11 |15 ... | 255 ) |13 ]

U ψ = + + + +

+ + + + +

+ + + + +

+ + + + +

Now we measure the output registers, this will collapse the superposition state to one of the outputs |1>,|7>|4>,|13>, for example |1>.

Shor’s algorithm

• Measure the output register will collapse the input register into an equal superposition state.which is a periodic function of period r=4.

• We now apply a quantum Fourier transform on the collapsed input register to increase the probability of some states.

/ 1

0

1| (| 0 | 4 | 8 |12 ... | 252 ) |64

M r

j

jrφ−

=

= + + + + ∝ ∑

1

0

21| | , (M=256)akMM

k

iT a e k

Mπ−

=

> = >∑

Shor’s algorithm

/ 1 / 1 1

0 0 0

1 / 1 1

0 0 0

2| | |

2( ) | = ( ) |

jr kM r M r MM

j j k

jr kM M r MM

k j k

iT T jr e k

ie k f k k

πφ

π

− − −

= = =

− − −

= = =

∝ = >

= > >

∑ ∑ ∑

∑ ∑ ∑

Here f(k) can be easily calculated

For simplicity, we have assumed M/r is an integer

2/ 1

2 /

0

1 0, / integer2 / ( ) 1/ , / =integer

ikM r

ikr M

j

e kr Mi jkr Mf k e eM r kr M

π

ππ−

=

⎧ −= ≠⎪= = −⎨

⎪⎩∑

Shor’s algorithm

• The QFT essentially peaks the probability amplitudes at integer multiples of M/r. When we measure the input registers, we randomly get c=jM/r, with .

• If gcd(j,r)=1, we can determine r by canceling to an irreducible fraction.

• From number theory, the probability that a number chosen randomly from 1…r is coprime to r is greater than 1/logr. Thus we repeat the computation O(logr)<O(logN) times , we will find the period r with probability close to 1.

• This gives an efficient determination of r.(see more details in Rev. Mod. Phys., 68, 733 (1996)

1 1

0 0| ( ) | | / | 0 | 64 |128 |192

M r

k jT f k k jM rφ

− −

= =

∝ > = = + + +∑ ∑

0 1j r≤ ≤ −c jM r

=

Shor’s algorithm

• In our case, c=0, 64, 128,192, M=256; then c/M=0, ¼, ½, ¾.

• We can obtain the correct period r=4 from ¼ and ¾and incorrect period r=2 from ½ . The results can be easily checked from

• Now that we have the period r=4, the factors of N=15 can be determined. This computation will be done on a classical computer.

mod 1rx N =

3)15,17gcd(),1gcd(

5)15,17gcd(),1gcd(2/42/

2

2/42/1

=−=−=

=+=+=

Nxm

Nxmr

r

Shor’s algorithm

– Generate random x∈{1, …, N-1};– Check if gcd(x, N)=1;– r = period(x);

(The period can be evaluated in polynomial time on a quantum computer.)

- Prime factors are calculated by classical computer

),1gcd(

),1gcd(2/

2

2/1

Nxm

Nxmr

r

−=

+=

Shor’s algorithm

• N=15=5×3, the simplest meaningful instance of Shor’s algorithm

• Input register 3 qubitsoutput register 4 qubits(Nature 414, 883, 2001)

Grover’s algorithm

• Classical search- sequentially try all N possibilities

- average search takes N/2 steps

• Quantum search- simultaneously try all possibilities

- refining process reveals answer

- average search takessteps1/2N

• How quickly can you find a needle in a haystack

Grover’s search algorithm

L.K. Grover, Phys. Rev. Lett., 79,325, (1997)

Grover’s search algorithm

• Problem : given a Quantum oracle, try to find one specific state , satisfying

R is a N×N diagonal matrix, satisfying Rii=-1, if i=x; Rii =1, other diagonal elements. To find x is equivalent to find which diagonal element of R is -1, i.e. x .

• Classically, we have to go through every diagonal element. We expect to find the -1 term after N/2 queries to all the diagonal elements.

1, 2... ... ,...i x Nx

| , | {

| , otherwisei i x

R ii− =

=

Grover’s algorithm

• Take a m-qubit register, assume• Prepare the registers in an equal superposition

state of all the states.

• Iterations of Rotate Phase and Diffusion operator• Measure the register to get the specific state

2m N=

N1

0

1| |N

ii

Nψ

−

=

= ∑

x

Grover’s algorithm

• In fact, R is a phaserotate operator

e.g.

| , | {

| , otherwisei i x

R ii− =

=

1x = 00 11 22 33 44 55

00 11 22 33 44 55

ψ|

ψ|R

Grover’s algorithm

• Diffusion operator

The successive operation of Rotate phase and Diffusion operator will increase the probability amplitude of the desired state.

2 2 21 ...

2 2 21 ...

2 2 2... 1

N N N

D N N N

N N N

⎧ ⎫− +⎪ ⎪⎪ ⎪⎪ ⎪− +⎪ ⎪= ⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪− +⎪ ⎪⎩ ⎭

M M O M

00 11 22 33 44

|DR ψ

Grover’s algorithm

• Initial state

• After n iteration, we have

• Considering

| ( ) | ,nn DRψ ψ=

2 2ˆ ˆ 1

2 2ˆ ˆ 2 1

DRN N

DRN N

α α β

β α β

⎧ ⎛ ⎞= − −⎜ ⎟⎪⎪ ⎝ ⎠⎨

⎛ ⎞ ⎛ ⎞⎪ = − + −⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎩

1 1

0 0( )

1 1 1|N N

i ii x

i x iN N N

ψ α β− −

= =≠

= = + ≡ +∑ ∑

| | | ,n n na bψ α β= + 1 0

1 0

2 11 , 2, with

1, 1

n n

n n

a a aN

b b bε

ε ε

−

−

⎛ ⎞ =− −⎛ ⎞ ⎛ ⎞ ⎧⎜ ⎟= ⎨⎜ ⎟ ⎜ ⎟⎜ ⎟ =⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎩− −⎝ ⎠

1N

Grover’s algorithm

• Finally, we get

• The probability to collapse into the x

• We choose iteration steps the probability of failure

( )

1

0

2cos2sin

N

nii x

nn Nx iN N

ψ−

=≠

⎛ ⎞⎜ ⎟⎛ ⎞ ⎝ ⎠≈ +⎜ ⎟

⎝ ⎠∑

22sin nP nN

⎛ ⎞= ⎜ ⎟⎝ ⎠

[ ]4

n Nπ=

( ) 2 1 11 cos 02

Np n ONN

π →∞⎛ ⎞ ⎛ ⎞− ≤ − = ⎯⎯⎯→⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

Grover’s algorithm

• Can we do better than a quadratic speed up for Quantum Searches.

• No! Grover algorithm is optimal. Any quantum algorithm, with respect to an Oracle, can not do better that Quadratic time.

• Good and Bad– Good: Grover’s is Optimal– Bad: No logarithmic time algorithm

:Limits of “Black-Box” quantum computing

Grover’s algorithm

• Experiment realization- Nuclear magnetic resonance

I. L .Chuang et. al. PRL, 80, 3408 (1998).- Linear optics

P.G. Kwait et. al. J. Mod. Opt. 47, 257 (2000).- individual atom

J. Ahn et. al. Science, 287, 463 (2000).- trapped ion

M. Feng, PRA, 63, 052308 (2001).