Lecture note 8: Quantum Algorithms

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Lecture note 8 Quantum Algorithms

• Jian-Wei Pan

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Outline

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

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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.

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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.

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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.

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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

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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!!!

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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

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A little number theory

• Consider the equations
• then we have

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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.
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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
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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.

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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.

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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.

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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)

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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.
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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.

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Shors algorithm

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

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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)

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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.

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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

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Shors algorithm

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

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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

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

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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

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Grovers algorithm

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

0
1
2
3
4
5
0
1
2
3
4
5
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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
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Grovers algorithm

• Initial state
• After n iteration, we have
• Considering

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Grovers algorithm

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

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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

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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).