Instructor: Shengyu Zhang

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CSC3160 Design and Analysis of Algorithms
Week 13 Quantum Algorithms

• Instructor Shengyu Zhang

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Content

• Quantum mechanics
• Mathematical formulation
• Efficient quantum algorithm for Factoring
• Reduction of Factoring to Period Finding
• Efficient quantum algorithm for Period Finding by
  quantum Fourier transform (QFT)

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Quantum mechanics

• Quantum mechanics study the behaviors of systems
  at atomic scales (and smaller).
• Many classical rules do not hold any more.
• Applications to
• chemistry,
• technology (such as laser, transistor, the
  electron microscope, and magnetic resonance
  imaging, etc)

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Mathematical model for quantum mechanics

• Despite all the mysteries, quantum mechanics has
  a clean simple and beautiful math framework.
• 4 postulates

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Postulate 1

• State space Every isolated physical system
  corresponds to a unit vector in a complex vector
  space.
• We use a weird ket notation ? to denote such
  a state.
• For example, for an N-dimensional space, 0?, ,
  N-1? is a complete basis states.
• For 1 quantum bit, or qubit a0?ß1?, where
  a2 ß2 1
• That is, it can sits anywhere between 0 and 1 (on
  the unit circle)!

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Postulate 2

• Evolution/Operation The evolution of a closed
  quantum system is described by a unitary
  transformation.
• That is, if the system is a1? at time t1, and
  a2? at time t2, then there is a unitary
  transformation U s.t. a2? Ua1?.
• Think of a unitary transformation as a simple
  rotation. Definition coming up later.
• For 1 qubit, its simply a rotation on the plane.

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Postulate 3

• We can only observe a quantum system by measuring
  it.
• The outcome of the measure is random in general.
• For 1 qubit state a0?ß1?, the probability of
  getting 0 is a2 and probability of getting 1 is
  ß2.
• The system is changed by the measurement.
• For 1 qubit state a0?ß1?, the system becomes
  0? if 0 is the outcome, and 1? if 1 is the
  outcome.

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Postulate 4

• The state of two systems a1? and a2?, put
  together, is a1??a2?.
• ? tensor product.
• (a,b) ? (c,d,e) (ac, ad, ae, bc, bd, be)
• In general, dimension is the product of the two
  dimensions
• Notation 0??n 0???0?, n times

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Simple facts

• Controlled operation Given implementation of any
  operation U, we can implement the controlled-U,
  an quantum analog of if clause.
• For example, if we have a single-qubit unitary
  operator U, then we can implement the operator
  s.t. a0?a0? ß1?a1? ? a0?a0? ß1? Ua1?

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Factoring

• Recall that in the very first class of this
  course, we mentioned that though multiplication
  is easy, its inverse, factoring seems very hard
• only for classical computers!
• Best known (About) 2n1/3
• Quantum? (About) n2 or n3

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Map of solution
efficient classical reduction
Order Finding
Factoring
Phase Estimation
Quantum Fourier Transform (QFT)
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Order Finding

• Given positive integers x and N with no common
  factors, the order of x is the least positive
  integer r s.t. xr 1 mod N.
• Well talk about the reduction of Factoring to
  Order Finding given time.
• Next Efficient quantum algorithms for Order
  Finding, using two tools
• Quantum Fourier Transform (QFT)
• Phase Estimation

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Fourier Transform (over ZN)

• Recall that the Fourier Transform (over ZN) maps
  the vector (x0, , xN-1) to (y0, , yN-1), where
  yk N-1/2?j0,,N-1 ?Njkxj
• ?N ei(2p/N)
• Quantum Fourier Transform for N-dimensional
  state x? (where x?0, , N-1)
• QFTx? N-1/2?j0,,N-1 ?Njxj?
• That is, N-1/2?j0,,N-1 e2pijx/Nj?
• In particular, 0??n ?2-n/2?j0,,N-1j?

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• For a matrix A and a vector x, if Ax ?xthen
  ? is an eigenvalue and x is an eigenvector.
• A matrix is unitary if all its eigenvalues have
  unit length. That is, of the form ei?.
• ?/2p is called phase.

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Phase Estimation

• Phase Estimation For a unitary matrix U and a
  given eigenvector u?, suppose we can apply all
  U, U2, U4, , U2n efficiently, then we can find
  the eigenvalue corresponding to u?.
• If ei2pf is the eigenvalue, then Phase Estimation
  gives 0? u? ? f? u?

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Phase Estimation by QFT

• For simplicity, suppose the phase f has an n-bit
  representation, i.e. f 0.f1 fn
• Algorithm
• 0??n u?QFT? 2-n/2?j0,,N-1j? u?C-Uj?
  2-n/2?j0,,N-1j? Uju? 2-n/2?j0,,N-1j?
  e2pijfu? (2-n/2?j0,,N-1 e2pijf j?) u?
  IQFT? f1 fn? u? !
• Last step Try QFT on f1 fn?
• It gives 2-n/2?j0,,N-1 e2pijf j?

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• QFT on f1 fn? 2-n/2?j0,,N-1 e2pijf j?
• Recall that f 0.f1 fn
• QFT on f1 fn? QFT on fN? 2-n/2?j0,,N-1
  e2pijfN/N j? 2-n/2?j0,,N-1 e2pijf j?

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Solving Order Finding by Phase Estimation

• Recall that the Order Finding (x,N) problem is to
  find the min r s.t. xr 1 mod N.
• Define unitary transform UN?N Uy? xy mod
  N?
• U is unitary xy mod N y0, , N-1 are
  distinct if x and N are relatively prime.
• Consider the state (for any s?0, , N-1) us?
  r-1/2?k0, , r-1 e-2pisk/r xk mod N?
• Try to compute Uus? What do you find?

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• Uy? xy mod N?,
• us? r-1/2?k0, , r-1 e-2pisk/r xk mod N?
• Uus? U (r-1/2?k0, , r-1 e-2pisk/r xk mod
  N?) r-1/2?k0, , r-1 e-2pisk/r Uxk mod N?
  r-1/2?k0, , r-1 e-2pisk/r xxk mod N?
  r-1/2?k0, , r-1 e-2pisk/r xk1 mod N?
  r-1/2?k1, , r e-2pis(k-1)/r xk mod N? //
  k k1 r-1/2?k1, , r e-2pisk/r2pis/r
  xk mod N? e2pis/r r-1/2?k1, , r
  e-2pisk/r xk mod N? e2pis/r r-1/2?k0, ,
  r-1 e-2pisk/r xk mod N? e2pis/rus?
• So us? is an eigenvector of U w.r.t. eigenvalue
  e2pis/r
• In other words, the phase is s/r.

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• So us? is an eigenvector of U w.r.t. eigenvalue
  e2pis/r
• In other words, the phase is s/r.
• Thus if we can use Phase Estimation to find the
  phase s/r, then we can find the order r.
• Some details hidden here since Phase Estimation
  only gives an approximation to s/r.
• But it turns out that an approximation to s/r is
  enough for us to retrieve r

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Preparations for Phase Estimation

• To use Phase Estimation, we need
• Prepare a state in us?
• Apply U2j for j 1, 2, , n.
• Actually Not exactly n, but similar.
• The second issue though U2j looks scary, we can
  implement it efficiently

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• Recall that Uy? xy mod N?
• Thus U2jy? x2jy mod N? (x2j mod N) (y
  mod N)?
• ab mod N (a mod N)(b mod N).
• So its enough to compute x2j mod N, which an be
  done easily
• Compute x1 x mod N
• Compute x2 x12 mod N ( x2 mod N)
• Compute x3 x22 mod N ( x4 mod N)
• Compute x4 x32 mod N ( x8 mod N)

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The first issue Prepare a state in us?

• Recall us? r-1/2?k0, , r-1 e-2pisk/r xk
  mod N?
• This looks more serious After all, preparing
  us? needs knowledge of r, which is exactly what
  we are looking for.
• Key Observation Though we dont know each us?,
  we know their sum r-1/2?s0, , r-1 us?
  1?

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• r-1/2?s0, , r-1 us? r-1/2?s0, , r-1
  r-1/2?k0, , r-1 e-2pisk/r xk mod N? r-1
  ?k0, , r-1 (?s0, , r-1 e-2pisk/r) xk mod
  N? 0 unless k 0 (in
  which case the sum r) x0 mod N? 1?

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• Thus, if we prepare 0? 1? in the Phase
  Estimation, then we will have the results of
  phases in superposition
• 0? 1? r-1/2?s0, , r-1 0? us? PE?
  r-1/2?s0, , r-1 s/r? us? Measure ? s/r? for
  a random s.
• It turns out that from s/r, one can get r easily.
  
• using continued fraction algorithm

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About the final

• Open book / lecture notes. No Internet.
• Covering the whole semester, but mainly the
  second half such as DP, Network Flow, FFT/QFT,
  Approximation Algorithm,
• The last problem you need to think about which
  method to use (Divide-and-Conquer? DP? Greedy?
  Reduction? )