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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?
  • - Quantum computer advantage
  • 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
    better that Quadratic time.
  • Good and Bad
  • 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).
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