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## Lecture note 8: Quantum Algorithms

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### Lecture note 8: Quantum Algorithms Jian-Wei Pan Outline Quantum Parallelism Shor s quantum factoring algorithm Grover s quantum search algorithm Quantum ... – PowerPoint PPT presentation

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Title: Lecture note 8: Quantum Algorithms

1
Lecture note 8 Quantum Algorithms
• Jian-Wei Pan

2
Outline
• Quantum Parallelism
• Shors quantum factoring algorithm
• Grovers quantum search algorithm

3
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 Shors algorithm for
factoring, or Grovers algorithm for searching.

4
RSA encryption and factoring
• RSA is named after Riverst, Shamir and Adleman,
• who came up with the scheme
• m1m2 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.

5
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,
• 1143816257578888676692357799761466120102182967212
42362562561842935706935245733897830597123
63958705058989075147599290026879543541
• 34905295108476509491478496199038981334177
64638493387843990820577
• x 3276913299326670954996198819083446141317
7642967992942539798288533
• 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.

6
Shors algorithm
• Algorithms for quantum
• computation discrete
• logarithms and factoring
• Foundations of Computer
• Science, 1994 Proceedings.,
• 35th Annual Symposium on
• Publication Date 20-22 Nov 1994. On pages
124-134
• Shor, P.W.
• ATT Bell Labs., Murray Hill, NJ

7
Shors algorithm
• Shors code-breaking Quantum Algorithm
• -How fast can you factor a number?
• E.g. factor a 300-digit number
• Classical THz computer
• - steps
• - 150,000 years
• Quantum THz computer
• - steps
• - 1 second
• Code-breaking can be done in minutes, not in
millennia
• Public key encryption, based on factoring, will
be vulnerable!!!

8
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

9
A little number theory
• Consider the equations
• then we have

10
A little number theory
We acquire a trivial solution and the desired
solution
Note that gcd can be calculated efficiently. If
we can find r, and r is even Then provided we
dont get trivial solutions. If r is an odd
number, change x, try again.
11
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. N15, x7.
period r 4
• Factoring reduces to period finding.

r 0 1 2 3 4
1 7 4 13 1
12
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, N15
• Choose a random integer x satisfying gcd(x,N)1,
• x7 in our case.

13
Shors 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.

14
Shors algorithm
• a input register, 0- output register
• Apply 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.

15
Shors 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)

16
Shors 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
Now we measure the output registers, this will
collapse the superposition state to one of the
outputs 1gt,7gt4gt,13gt, for example 1gt.
17
Shors algorithm
• Measure the output register will collapse the
input register into an equal superposition state.
• which is a periodic function of period r4.
• We now apply a quantum Fourier transform on the
collapsed input register to increase the
probability of some states.

18
Shors algorithm
• Here f(k) can be easily calculated
• For simplicity, we have assumed M/r is an integer

19
Shors algorithm
• The QFT essentially peaks the probability
amplitudes at integer multiples of M/r. When we
measure the input registers, we randomly get
cjM/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 1r is coprime to r is
greater than 1/logr. Thus we repeat the
computation O(logr)ltO(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)

20
Shors algorithm
• In our case, c0, 64, 128,192, M256 then
c/M0, ¼, ½, ¾.
• We can obtain the correct period r4 from ¼ and ¾
• and incorrect period r2 from ½ . The results
can be
• easily checked from
• Now that we have the period r4, the factors of
N15 can be determined. This computation will be
done on a classical computer.

21
Shors 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

22
Shors algorithm
• N1553, the simplest meaningful instance of
Shors algorithm
• Input register 3 qubits
• output register 4 qubits
• (Nature 414, 883, 2001)

23
Grovers 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 takes
• steps
• How quickly can you find a needle in a haystack

24
Grovers search algorithm
L.K. Grover, Phys. Rev. Lett., 79,325, (1997)
25
Grovers search algorithm
• Problem given a Quantum oracle,
• try to find one specific state , satisfying
• R is a NN diagonal matrix, satisfying Rii-1,
if ix 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.

26
Grovers 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

27
Grovers algorithm
• In fact, R is a phase
• rotate operator
• e.g.

0
1
2
3
4
5
0
1
2
3
4
5
28
Grovers algorithm
• Diffusion operator
• The successive operation of Rotate phase and
Diffusion
• operator will increase the probability amplitude
of the
• desired state.

0
1
2
3
4
29
Grovers algorithm
• Initial state
• After n iteration, we have
• Considering

30
Grovers algorithm
• Finally, we get
• The probability to collapse into the x
• We choose iteration steps
• the probability of failure

31
Grovers 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
• Good Grovers is Optimal
• Bad No logarithmic time algorithm
• Limits of Black-Box quantum
computing

32
Grovers 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).